| Line 2,526: |
Line 2,526: |
| | * Raskin, Marcus G., and Bernstein, Herbert J. (1987, eds.), ''New Ways of Knowing : The Sciences, Society, and Reconstructive Knowledge'', Rowman and Littlefield, Totowa, NJ. | | * Raskin, Marcus G., and Bernstein, Herbert J. (1987, eds.), ''New Ways of Knowing : The Sciences, Society, and Reconstructive Knowledge'', Rowman and Littlefield, Totowa, NJ. |
| | | | |
| − | ==Notes Found in a Cactus Patch==
| |
| | | | |
| − | : '''''Note.''' This is a collection of fragments from previous discussions that I plan to use in documenting the cactus graph syntax for propositional logic.''
| |
| − |
| |
| − | ===Cactus Language===
| |
| − |
| |
| − | Table 13 illustrates the ''existential interpretation'' of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
| |
| − |
| |
| − | Even though I do most of my thinking in the existential interpretation, I will continue to speak of these forms as ''logical graphs'', because I think it is an important fact about them that the formal validity of the axioms and theorems is not dependent on the choice between the entitative and the existential interpretations.
| |
| − |
| |
| − | The first extension is the ''reflective extension of logical graphs'' (RefLog). It is obtained by generalizing the negation operator "<math>\texttt{(~)}</math>" in a certain way, calling "<math>\texttt{(~)}</math>" the ''controlled'', ''moderated'', or ''reflective'' negation operator of order 1, then adding another such operator for each finite <math>k = 2, 3, \ldots .</math>
| |
| − |
| |
| − | In sum, these operators are symbolized by bracketed argument lists as follows: "<math>\texttt{(~)}</math>", "<math>\texttt{(~,~)}</math>", "<math>\texttt{(~,~,~)}</math>", …, where the number of slots is the order of the reflective negation operator in question.
| |
| − |
| |
| − | The cactus graph and the cactus expression shown here are both described as a ''spike''.
| |
| − |
| |
| − | {| align="center" cellpadding="6" width="90%"
| |
| − | | align="center" |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | o |
| |
| − | | | |
| |
| − | | @ |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | | ( ) |
| |
| − | o---------------------------------------o
| |
| − | </pre>
| |
| − | |}
| |
| − |
| |
| − | The rule of reduction for a lobe is:
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| − |
| |
| − | {| align="center" cellpadding="6" width="90%"
| |
| − | | align="center" |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | x_1 x_2 ... x_k |
| |
| − | | o-----o--- ... ---o |
| |
| − | | \ / |
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| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
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| − | | \ / |
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| − | | @ = @ |
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| − | | |
| |
| − | o---------------------------------------o
| |
| − | </pre>
| |
| − | |}
| |
| − |
| |
| − | if and only if exactly one of the <math>x_j\!</math> is a spike.
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| − |
| |
| − | In Ref Log, an expression of the form <math>\texttt{((}~ e_1 ~\texttt{),(}~ e_2 ~\texttt{),(}~ \ldots ~\texttt{),(}~ e_k ~\texttt{))}</math>
| |
| − | expresses the fact that ''exactly one of the <math>e_j\!</math> is true''. Expressions of this form are called ''universal partition'' expressions, and
| |
| − | they parse into a type of graph called a ''painted and rooted cactus'' (PARC):
| |
| − |
| |
| − | {| align="center" cellpadding="6" width="90%"
| |
| − | | align="center" |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | e_1 e_2 ... e_k |
| |
| − | | o o o |
| |
| − | | | | | |
| |
| − | | o-----o--- ... ---o |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | @ |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | </pre>
| |
| − | |}
| |
| − |
| |
| − | {| align="center" cellpadding="6" width="90%"
| |
| − | | align="center" |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | ( x1, x2, ..., xk ) = [blank] |
| |
| − | | |
| |
| − | | iff |
| |
| − | | |
| |
| − | | Just one of the arguments |
| |
| − | | x1, x2, ..., xk = () |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | </pre>
| |
| − | |}
| |
| − |
| |
| − | The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
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| − |
| |
| − | {| align="center" cellpadding="6" width="90%"
| |
| − | | Existential Interpretation:
| |
| − | | Just one of the k argument is false.
| |
| − | |-
| |
| − | | Entitative Interpretation:
| |
| − | | Not just one of the k arguments is true.
| |
| − | |}
| |
| − |
| |
| − | {| align="center" cellpadding="6" width="90%"
| |
| − | | align="center" |
| |
| − | <pre>
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | Graph | String | Translation |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | @ | " " | true. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | o | | |
| |
| − | | | | | |
| |
| − | | @ | ( ) | untrue. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | r | | |
| |
| − | | @ | r | r. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | r | | |
| |
| − | | o | | |
| |
| − | | | | | |
| |
| − | | @ | (r) | not r. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | r s t | | |
| |
| − | | @ | r s t | r and s and t. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | r s t | | |
| |
| − | | o o o | | |
| |
| − | | \|/ | | |
| |
| − | | o | | |
| |
| − | | | | | |
| |
| − | | @ | ((r)(s)(t)) | r or s or t. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | r implies s. |
| |
| − | | r s | | |
| |
| − | | o---o | | if r then s. |
| |
| − | | | | | |
| |
| − | | @ | (r (s)) | no r sans s. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | r s | | |
| |
| − | | o---o | | r exclusive-or s. |
| |
| − | | \ / | | |
| |
| − | | @ | (r , s) | r not equal to s. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | r s | | |
| |
| − | | o---o | | |
| |
| − | | \ / | | |
| |
| − | | o | | r if & only if s. |
| |
| − | | | | | |
| |
| − | | @ | ((r , s)) | r equates with s. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | r s t | | |
| |
| − | | o--o--o | | |
| |
| − | | \ / | | |
| |
| − | | \ / | | just one false |
| |
| − | | @ | (r , s , t) | out of r, s, t. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | r s t | | |
| |
| − | | o o o | | |
| |
| − | | | | | | | |
| |
| − | | o--o--o | | |
| |
| − | | \ / | | |
| |
| − | | \ / | | just one true |
| |
| − | | @ | ((r),(s),(t)) | among r, s, t. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | genus t over |
| |
| − | | r s | | species r, s. |
| |
| − | | o o | | |
| |
| − | | t | | | | partition t |
| |
| − | | o--o--o | | among r & s. |
| |
| − | | \ / | | |
| |
| − | | \ / | | whole pie t: |
| |
| − | | @ | ( t ,(r),(s)) | slices r, s. |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | </pre>
| |
| − | |}
| |
| − |
| |
| − | {| align="center" cellpadding="6" width="90%"
| |
| − | | align="center" |
| |
| − | <pre>
| |
| − | Table 13. The Existential Interpretation
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | Cactus Graph | Cactus Expression | Existential |
| |
| − | | | | Interpretation |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | @ | " " | true. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | o | | |
| |
| − | | | | | |
| |
| − | | @ | ( ) | untrue. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a | | |
| |
| − | | @ | a | a. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a | | |
| |
| − | | o | | |
| |
| − | | | | | |
| |
| − | | @ | (a) | not a. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b c | | |
| |
| − | | @ | a b c | a and b and c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b c | | |
| |
| − | | o o o | | |
| |
| − | | \|/ | | |
| |
| − | | o | | |
| |
| − | | | | | |
| |
| − | | @ | ((a)(b)(c)) | a or b or c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | | | a implies b. |
| |
| − | | a b | | |
| |
| − | | o---o | | if a then b. |
| |
| − | | | | | |
| |
| − | | @ | (a (b)) | no a sans b. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b | | |
| |
| − | | o---o | | a exclusive-or b. |
| |
| − | | \ / | | |
| |
| − | | @ | (a , b) | a not equal to b. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b | | |
| |
| − | | o---o | | |
| |
| − | | \ / | | |
| |
| − | | o | | a if & only if b. |
| |
| − | | | | | |
| |
| − | | @ | ((a , b)) | a equates with b. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b c | | |
| |
| − | | o--o--o | | |
| |
| − | | \ / | | |
| |
| − | | \ / | | just one false |
| |
| − | | @ | (a , b , c) | out of a, b, c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b c | | |
| |
| − | | o o o | | |
| |
| − | | | | | | | |
| |
| − | | o--o--o | | |
| |
| − | | \ / | | |
| |
| − | | \ / | | just one true |
| |
| − | | @ | ((a),(b),(c)) | among a, b, c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | | | genus a over |
| |
| − | | b c | | species b, c. |
| |
| − | | o o | | |
| |
| − | | a | | | | partition a |
| |
| − | | o--o--o | | among b & c. |
| |
| − | | \ / | | |
| |
| − | | \ / | | whole pie a: |
| |
| − | | @ | ( a ,(b),(c)) | slices b, c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | </pre>
| |
| − | |}
| |
| − |
| |
| − | {| align="center" cellpadding="6" width="90%"
| |
| − | | align="center" |
| |
| − | <pre>
| |
| − | Table 14. The Entitative Interpretation
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | Cactus Graph | Cactus Expression | Entitative |
| |
| − | | | | Interpretation |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | @ | " " | untrue. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | o | | |
| |
| − | | | | | |
| |
| − | | @ | ( ) | true. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a | | |
| |
| − | | @ | a | a. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a | | |
| |
| − | | o | | |
| |
| − | | | | | |
| |
| − | | @ | (a) | not a. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b c | | |
| |
| − | | @ | a b c | a or b or c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b c | | |
| |
| − | | o o o | | |
| |
| − | | \|/ | | |
| |
| − | | o | | |
| |
| − | | | | | |
| |
| − | | @ | ((a)(b)(c)) | a and b and c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | | | a implies b. |
| |
| − | | | | |
| |
| − | | o a | | if a then b. |
| |
| − | | | | | |
| |
| − | | @ b | (a) b | not a, or b. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b | | |
| |
| − | | o---o | | a if & only if b. |
| |
| − | | \ / | | |
| |
| − | | @ | (a , b) | a equates with b. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b | | |
| |
| − | | o---o | | |
| |
| − | | \ / | | |
| |
| − | | o | | a exclusive-or b. |
| |
| − | | | | | |
| |
| − | | @ | ((a , b)) | a not equal to b. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b c | | |
| |
| − | | o--o--o | | |
| |
| − | | \ / | | |
| |
| − | | \ / | | not just one true |
| |
| − | | @ | (a , b , c) | out of a, b, c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a b c | | |
| |
| − | | o--o--o | | |
| |
| − | | \ / | | |
| |
| − | | \ / | | |
| |
| − | | o | | |
| |
| − | | | | | just one true |
| |
| − | | @ | ((a , b , c)) | among a, b, c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | a | | |
| |
| − | | o | | genus a over |
| |
| − | | | b c | | species b, c. |
| |
| − | | o--o--o | | |
| |
| − | | \ / | | partition a |
| |
| − | | \ / | | among b & c. |
| |
| − | | o | | |
| |
| − | | | | | whole pie a: |
| |
| − | | @ | ( a ,(b),(c)) | slices b, c. |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | </pre>
| |
| − | |}
| |
| − |
| |
| − | {| align="center" cellpadding="6" width="90%"
| |
| − | | align="center" |
| |
| − | <pre>
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | Graph | String | Entitative | Existential |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | @ | " " | untrue. | true. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | o | | | |
| |
| − | | | | | | |
| |
| − | | @ | ( ) | true. | untrue. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | r | | | |
| |
| − | | @ | r | r. | r. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | r | | | |
| |
| − | | o | | | |
| |
| − | | | | | | |
| |
| − | | @ | (r) | not r. | not r. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | r s t | | | |
| |
| − | | @ | r s t | r or s or t. | r and s and t. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | r s t | | | |
| |
| − | | o o o | | | |
| |
| − | | \|/ | | | |
| |
| − | | o | | | |
| |
| − | | | | | | |
| |
| − | | @ | ((r)(s)(t)) | r and s and t. | r or s or t. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | r implies s. |
| |
| − | | | | | |
| |
| − | | o r | | | if r then s. |
| |
| − | | | | | | |
| |
| − | | @ s | (r) s | not r, or s | no r sans s. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | r implies s. |
| |
| − | | r s | | | |
| |
| − | | o---o | | | if r then s. |
| |
| − | | | | | | |
| |
| − | | @ | (r (s)) | | no r sans s. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | r s | | | |
| |
| − | | o---o | | |r exclusive-or s.|
| |
| − | | \ / | | | |
| |
| − | | @ | (r , s) | |r not equal to s.|
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | r s | | | |
| |
| − | | o---o | | | |
| |
| − | | \ / | | | |
| |
| − | | o | | |r if & only if s.|
| |
| − | | | | | | |
| |
| − | | @ | ((r , s)) | |r equates with s.|
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | r s t | | | |
| |
| − | | o--o--o | | | |
| |
| − | | \ / | | | |
| |
| − | | \ / | | | just one false |
| |
| − | | @ | (r , s , t) | | out of r, s, t. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | |
| |
| − | | r s t | | | |
| |
| − | | o o o | | | |
| |
| − | | | | | | | | |
| |
| − | | o--o--o | | | |
| |
| − | | \ / | | | |
| |
| − | | \ / | | | just one true |
| |
| − | | @ | ((r),(s),(t)) | | among r, s, t. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | | | | | genus t over |
| |
| − | | r s | | | species r, s. |
| |
| − | | o o | | | |
| |
| − | | t | | | | | partition t |
| |
| − | | o--o--o | | | among r & s. |
| |
| − | | \ / | | | |
| |
| − | | \ / | | | whole pie t: |
| |
| − | | @ | ( t ,(r),(s)) | | slices r, s. |
| |
| − | o-----------------o-----------------o-----------------o-----------------o
| |
| − | </pre>
| |
| − | |}
| |
| − |
| |
| − | ===Zeroth Order Logic===
| |
| − |
| |
| − | <pre>
| |
| − | Here is a scaled-down version of one of my very first applications,
| |
| − | having to do with the demographic variables in a survey data base.
| |
| − |
| |
| − | This Example illustrates the use of 2-variate logical forms
| |
| − | for expressing and reasoning about the logical constraints
| |
| − | that are involved in the following types of situations:
| |
| − |
| |
| − | 1. Distinction: A =/= B
| |
| − | Also known as: logical inequality, exclusive disjunction
| |
| − | Represented as: ( A , B )
| |
| − | Graphed as:
| |
| − | |
| |
| − | | A B
| |
| − | | o---o
| |
| − | | \ /
| |
| − | | @
| |
| − |
| |
| − | 2. Equality: A = B
| |
| − | Also known as: logical equivalence, if and only if, A <=> B
| |
| − | Represented as: (( A , B ))
| |
| − | Graphed as:
| |
| − | |
| |
| − | | A B
| |
| − | | o---o
| |
| − | | \ /
| |
| − | | o
| |
| − | | |
| |
| − | | @
| |
| − |
| |
| − | 3. Implication: A => B
| |
| − | Also known as: entailment, if-then
| |
| − | Represented as: ( A ( B ))
| |
| − | Graphed as:
| |
| − | |
| |
| − | | A B
| |
| − | | o---o
| |
| − | | |
| |
| − | | @
| |
| − |
| |
| − | Example of a proposition expressing a "zeroth order theory" (ZOT):
| |
| − |
| |
| − | Consider the following text, written in what I am calling "Ref Log",
| |
| − | also known as the "Cactus Language" synpropositional logic:
| |
| − |
| |
| − | | ( male , female )
| |
| − | | (( boy , male child ))
| |
| − | | (( girl , female child ))
| |
| − | | ( child ( human ))
| |
| − |
| |
| − | Graphed as:
| |
| − |
| |
| − | | boy male girl female
| |
| − | | o---o child o---o child
| |
| − | | male female \ / \ / child human
| |
| − | | o---o o o o---o
| |
| − | | \ / | | |
| |
| − | | @ @ @ @|
| |
| − |
| |
| − | Nota Bene. Due to graphic constraints -- no, the other
| |
| − | kind of graphic constraints -- of the immediate medium,
| |
| − | I am forced to string out the logical conjuncts of the
| |
| − | actual cactus graph for this situation, one that might
| |
| − | sufficiently be reasoned out from the exhibit supra by
| |
| − | fusing together the four roots of the severed cactus.
| |
| − |
| |
| − | Either of these expressions, text or graph, is equivalent to
| |
| − | what would otherwise be written in a more ordinary syntax as:
| |
| − |
| |
| − | | male =/= female
| |
| − | | boy <=> male child
| |
| − | | girl <=> female child
| |
| − | | child => human
| |
| − |
| |
| − | This is a actually a single proposition, a conjunction of four lines:
| |
| − | one distinction, two equations, and one implication. Together these
| |
| − | amount to a set of definitions conjointly constraining the logical
| |
| − | compatibility of the six feature names that appear. They may be
| |
| − | thought of as sculpting out a space of models that is some subset
| |
| − | of the 2^6 = 64 possible interpretations, and thereby shaping some
| |
| − | universe of discourse.
| |
| − |
| |
| − | Once this backdrop is defined, it is possible to "query" this universe,
| |
| − | simply by conjoining additional propositions in further constraint of
| |
| − | the underlying set of models. This has many uses, as we shall see.
| |
| − |
| |
| − | We are considering an Example of a propositional expression
| |
| − | that is formed on the following "alphabet" or "lexicon" of
| |
| − | six "logical features" or "boolean variables":
| |
| − |
| |
| − | $A$ = {"boy", "child", "female", "girl", "human", "male"}.
| |
| − |
| |
| − | The expression is this:
| |
| − |
| |
| − | | ( male , female )
| |
| − | | (( boy , male child ))
| |
| − | | (( girl , female child ))
| |
| − | | ( child ( human ))
| |
| − |
| |
| − | Putting it very roughly -- and putting off a better description
| |
| − | of it till later -- we may think of this expression as notation
| |
| − | for a boolean function f : %B%^6 -> %B%. This is what we might
| |
| − | call the "abstract type" of the function, but we will also find
| |
| − | it convenient on many occasions to represent the points of this
| |
| − | particular copy of the space %B%^6 in terms of the positive and
| |
| − | negative versions of the features from $A$ that serve to encase
| |
| − | them as logical "cells", as they are called in the venn diagram
| |
| − | picture of the corresponding universe of discourse X = [$A$].
| |
| − |
| |
| − | Just for concreteness, this form of representation begins and ends:
| |
| − |
| |
| − | <0,0,0,0,0,0> = (boy)(child)(female)(girl)(human)(male),
| |
| − | <0,0,0,0,0,1> = (boy)(child)(female)(girl)(human) male ,
| |
| − | <0,0,0,0,1,0> = (boy)(child)(female)(girl) human (male),
| |
| − | <0,0,0,0,1,1> = (boy)(child)(female)(girl) human male ,
| |
| − | ...
| |
| − | <1,1,1,1,0,0> = boy child female girl (human)(male),
| |
| − | <1,1,1,1,0,1> = boy child female girl (human) male ,
| |
| − | <1,1,1,1,1,0> = boy child female girl human (male),
| |
| − | <1,1,1,1,1,1> = boy child female girl human male .
| |
| − |
| |
| − | I continue with the previous Example, that I bring forward and sum up here:
| |
| − |
| |
| − | | boy male girl female
| |
| − | | o---o child o---o child
| |
| − | | male female \ / \ / child human
| |
| − | | o---o o o o--o
| |
| − | | \ / | | |
| |
| − | | @ @ @ @
| |
| − | |
| |
| − | | (male , female)((boy , male child))((girl , female child))(child (human))
| |
| − |
| |
| − | For my master's piece in Quantitative Psychology (Michigan State, 1989),
| |
| − | I wrote a program, "Theme One" (TO) by name, that among its other duties
| |
| − | operates to process the expressions of the cactus language in many of the
| |
| − | most pressing ways that we need in order to be able to use it effectively
| |
| − | as a propositional calculus. The operational component of TO where one
| |
| − | does the work of this logical modeling is called "Study", and the core
| |
| − | of the logical calculator deep in the heart of this Study section is
| |
| − | a suite of computational functions that evolve a particular species
| |
| − | of "normal form", analogous to a "disjunctive normal form" (DNF),
| |
| − | from whatever expression they are prebendered as their input.
| |
| − |
| |
| − | This "canonical", "normal", or "stable" form of logical expression --
| |
| − | I'll refine the distinctions among these subforms all in good time --
| |
| − | permits succinct depiction as an "arboreal boolean expansion" (ABE).
| |
| − |
| |
| − | Once again, the graphic limitations of this space prevail against
| |
| − | any disposition that I might have to lay out a really substantial
| |
| − | case before you, of the brand that might have a chance to impress
| |
| − | you with the aptitude of this ilk of ABE in rooting out the truth
| |
| − | of many a complexly obscurely subtly adamant whetstone of our wit.
| |
| − |
| |
| − | So let me just illustrate the way of it with one conjunct of our Example.
| |
| − | What follows will be a sequence of expressions, each one after the first
| |
| − | being logically equal to the one that precedes it:
| |
| − |
| |
| − | Step 1
| |
| − |
| |
| − | | g fc
| |
| − | | o---o
| |
| − | | \ /
| |
| − | | o
| |
| − | | |
| |
| − | | @
| |
| − |
| |
| − | Step 2
| |
| − |
| |
| − | | o
| |
| − | | fc | fc
| |
| − | | o---o o---o
| |
| − | | \ / \ /
| |
| − | | o o
| |
| − | | | |
| |
| − | | g o-------------o--o g
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | @
| |
| − |
| |
| − | Step 3
| |
| − |
| |
| − | | f c
| |
| − | | o
| |
| − | | | f c
| |
| − | | o o
| |
| − | | | |
| |
| − | | g o-------------o--o g
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | @
| |
| − |
| |
| − | Step 4
| |
| − |
| |
| − | | o
| |
| − | | |
| |
| − | | c o o c o
| |
| − | | | | |
| |
| − | | o o c o o c
| |
| − | | | | | |
| |
| − | | f o---o--o f f o---o--o f
| |
| − | | \ / \ /
| |
| − | | g o-------------o--o g
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | @
| |
| − |
| |
| − | Step 5
| |
| − |
| |
| − | | o c o
| |
| − | | c | |
| |
| − | | f o---o--o f f o---o--o f
| |
| − | | \ / \ /
| |
| − | | g o-------------o--o g
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | @
| |
| − |
| |
| − | Step 6
| |
| − |
| |
| − | | o
| |
| − | | |
| |
| − | | o o o
| |
| − | | | | |
| |
| − | | c o---o--o c o c o---o--o c
| |
| − | | \ / | \ /
| |
| − | | f o-------------o--o f f o-------------o--o f
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | g o---------------------------o--o g
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | @
| |
| − |
| |
| − | Step 7
| |
| − |
| |
| − | | o o
| |
| − | | | |
| |
| − | | c o---o--o c o c o---o--o c
| |
| − | | \ / | \ /
| |
| − | | f o-------------o--o f f o-------------o--o f
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | \ / \ /
| |
| − | | g o---------------------------o--o g
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | @
| |
| − |
| |
| − | This last expression is the ABE of the input expression.
| |
| − | It can be transcribed into ordinary logical language as:
| |
| − |
| |
| − | | either girl and
| |
| − | | either female and
| |
| − | | either child and true
| |
| − | | or not child and false
| |
| − | | or not female and false
| |
| − | | or not girl and
| |
| − | | either female and
| |
| − | | either child and false
| |
| − | | or not child and true
| |
| − | | or not female and true
| |
| − |
| |
| − | The expression "((girl , female child))" is sufficiently evaluated
| |
| − | by considering its logical values on the coordinate tuples of %B%^3,
| |
| − | or its indications on the cells of the associated venn diagram that
| |
| − | depicts the universe of discourse, namely, on these eight arguments:
| |
| − |
| |
| − | <1, 1, 1> = girl female child ,
| |
| − | <1, 1, 0> = girl female (child),
| |
| − | <1, 0, 1> = girl (female) child ,
| |
| − | <1, 0, 0> = girl (female)(child),
| |
| − | <0, 1, 1> = (girl) female child ,
| |
| − | <0, 1, 0> = (girl) female (child),
| |
| − | <0, 0, 1> = (girl)(female) child ,
| |
| − | <0, 0, 0> = (girl)(female)(child).
| |
| − |
| |
| − | The ABE output expression tells us the logical values of
| |
| − | the input expression on each of these arguments, doing so
| |
| − | by attaching the values to the leaves of a tree, and acting
| |
| − | as an "efficient" or "lazy" evaluator in the sense that the
| |
| − | process that generates the tree follows each path only up to
| |
| − | the point in the tree where it can determine the values on the
| |
| − | entire subtree beyond that point. Thus, the ABE tree tells us:
| |
| − |
| |
| − | girl female child -> 1
| |
| − | girl female (child) -> 0
| |
| − | girl (female) -> 0
| |
| − | (girl) female child -> 0
| |
| − | (girl) female (child) -> 1
| |
| − | (girl)(female) -> 1
| |
| − |
| |
| − | Picking out the interpretations that yield the truth of the expression,
| |
| − | and expanding the corresponding partial argument tuples, we arrive at
| |
| − | the following interpretations that satisfy the input expression:
| |
| − |
| |
| − | girl female child -> 1
| |
| − | (girl) female (child) -> 1
| |
| − | (girl)(female) child -> 1
| |
| − | (girl)(female)(child) -> 1
| |
| − |
| |
| − | In sum, if it's a female and a child, then it's a girl,
| |
| − | and if it's either not a female or not a child or both,
| |
| − | then it's not a girl.
| |
| − |
| |
| − | Brief Automata
| |
| − |
| |
| − | By way of providing a simple illustration of Cook's Theorem,
| |
| − | that "Propositional Satisfiability is NP-Complete", here is
| |
| − | an exposition of one way to translate Turing Machine set-ups
| |
| − | into propositional expressions, employing the Ref Log Syntax
| |
| − | for Prop Calc that I described in a couple of earlier notes:
| |
| − |
| |
| − | Notation:
| |
| − |
| |
| − | Stilt(k) = Space and Time Limited Turing Machine,
| |
| − | with k units of space and k units of time.
| |
| − |
| |
| − | Stunt(k) = Space and Time Limited Turing Machine,
| |
| − | for computing the parity of a bit string,
| |
| − | with Number of Tape cells of input equal to k.
| |
| − |
| |
| − | I will follow the pattern of the discussion in the book of
| |
| − | Herbert Wilf, 'Algorithms & Complexity' (1986), pages 188-201,
| |
| − | but translate into Ref Log, which is more efficient with respect
| |
| − | to the number of propositional clauses that are required.
| |
| − |
| |
| − | Parity Machine
| |
| − |
| |
| − | | 1/1/+1
| |
| − | | ------->
| |
| − | | /\ / \ /\
| |
| − | | 0/0/+1 ^ 0 1 ^ 0/0/+1
| |
| − | | \/|\ /|\/
| |
| − | | | <------- |
| |
| − | | #/#/-1 | 1/1/+1 | #/#/-1
| |
| − | | | |
| |
| − | | v v
| |
| − | | # *
| |
| − |
| |
| − | o-------o--------o-------------o---------o------------o
| |
| − | | State | Symbol | Next Symbol | Ratchet | Next State |
| |
| − | | Q | S | S' | dR | Q' |
| |
| − | o-------o--------o-------------o---------o------------o
| |
| − | | 0 | 0 | 0 | +1 | 0 |
| |
| − | | 0 | 1 | 1 | +1 | 1 |
| |
| − | | 0 | # | # | -1 | # |
| |
| − | | 1 | 0 | 0 | +1 | 1 |
| |
| − | | 1 | 1 | 1 | +1 | 0 |
| |
| − | | 1 | # | # | -1 | * |
| |
| − | o-------o--------o-------------o---------o------------o
| |
| − |
| |
| − | The TM has a "finite automaton" (FA) as its component.
| |
| − | Let us refer to this particular FA by the name of "M".
| |
| − |
| |
| − | The "tape-head" (that is, the "read-unit") will be called "H".
| |
| − | The "registers" are also called "tape-cells" or "tape-squares".
| |
| − |
| |
| − | In order to consider how the finitely "stilted" rendition of this TM
| |
| − | can be translated into the form of a purely propositional description,
| |
| − | one now fixes k and limits the discussion to talking about a Stilt(k),
| |
| − | which is really not a true TM anymore but a finite automaton in disguise.
| |
| − |
| |
| − | In this example, for the sake of a minimal illustration, we choose k = 2,
| |
| − | and discuss Stunt(2). Since the zeroth tape cell and the last tape cell
| |
| − | are occupied with bof and eof marks "#", this amounts to only one digit
| |
| − | of significant computation.
| |
| − |
| |
| − | To translate Stunt(2) into propositional form we use
| |
| − | the following collection of propositional variables:
| |
| − |
| |
| − | For the "Present State Function" QF : P -> Q,
| |
| − |
| |
| − | {p0_q#, p0_q*, p0_q0, p0_q1,
| |
| − | p1_q#, p1_q*, p1_q0, p1_q1,
| |
| − | p2_q#, p2_q*, p2_q0, p2_q1,
| |
| − | p3_q#, p3_q*, p3_q0, p3_q1}
| |
| − |
| |
| − | The propositional expression of the form "pi_qj" says:
| |
| − |
| |
| − | | At the point-in-time p_i,
| |
| − | | the finite machine M is in the state q_j.
| |
| − |
| |
| − | For the "Present Register Function" RF : P -> R,
| |
| − |
| |
| − | {p0_r0, p0_r1, p0_r2, p0_r3,
| |
| − | p1_r0, p1_r1, p1_r2, p1_r3,
| |
| − | p2_r0, p2_r1, p2_r2, p2_r3,
| |
| − | p3_r0, p3_r1, p3_r2, p3_r3}
| |
| − |
| |
| − | The propositional expression of the form "pi_rj" says:
| |
| − |
| |
| − | | At the point-in-time p_i,
| |
| − | | the tape-head H is on the tape-cell r_j.
| |
| − |
| |
| − | For the "Present Symbol Function" SF : P -> (R -> S),
| |
| − |
| |
| − | {p0_r0_s#, p0_r0_s*, p0_r0_s0, p0_r0_s1,
| |
| − | p0_r1_s#, p0_r1_s*, p0_r1_s0, p0_r1_s1,
| |
| − | p0_r2_s#, p0_r2_s*, p0_r2_s0, p0_r2_s1,
| |
| − | p0_r3_s#, p0_r3_s*, p0_r3_s0, p0_r3_s1,
| |
| − | p1_r0_s#, p1_r0_s*, p1_r0_s0, p1_r0_s1,
| |
| − | p1_r1_s#, p1_r1_s*, p1_r1_s0, p1_r1_s1,
| |
| − | p1_r2_s#, p1_r2_s*, p1_r2_s0, p1_r2_s1,
| |
| − | p1_r3_s#, p1_r3_s*, p1_r3_s0, p1_r3_s1,
| |
| − | p2_r0_s#, p2_r0_s*, p2_r0_s0, p2_r0_s1,
| |
| − | p2_r1_s#, p2_r1_s*, p2_r1_s0, p2_r1_s1,
| |
| − | p2_r2_s#, p2_r2_s*, p2_r2_s0, p2_r2_s1,
| |
| − | p2_r3_s#, p2_r3_s*, p2_r3_s0, p2_r3_s1,
| |
| − | p3_r0_s#, p3_r0_s*, p3_r0_s0, p3_r0_s1,
| |
| − | p3_r1_s#, p3_r1_s*, p3_r1_s0, p3_r1_s1,
| |
| − | p3_r2_s#, p3_r2_s*, p3_r2_s0, p3_r2_s1,
| |
| − | p3_r3_s#, p3_r3_s*, p3_r3_s0, p3_r3_s1}
| |
| − |
| |
| − | The propositional expression of the form "pi_rj_sk" says:
| |
| − |
| |
| − | | At the point-in-time p_i,
| |
| − | | the tape-cell r_j bears the mark s_k.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~INPUTS~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Here are the Initial Conditions
| |
| − | for the two possible inputs to the
| |
| − | Ref Log redaction of this Parity TM:
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~INPUT~0~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Initial Conditions:
| |
| − |
| |
| − | p0_q0
| |
| − |
| |
| − | p0_r1
| |
| − |
| |
| − | p0_r0_s#
| |
| − | p0_r1_s0
| |
| − | p0_r2_s#
| |
| − |
| |
| − | The Initial Conditions are given by a logical conjunction
| |
| − | that is composed of 5 basic expressions, altogether stating:
| |
| − |
| |
| − | | At the point-in-time p_0, M is in the state q_0, and
| |
| − | | At the point-in-time p_0, H is on the cell r_1, and
| |
| − | | At the point-in-time p_0, cell r_0 bears the mark "#", and
| |
| − | | At the point-in-time p_0, cell r_1 bears the mark "0", and
| |
| − | | At the point-in-time p_0, cell r_2 bears the mark "#".
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~INPUT~1~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Initial Conditions:
| |
| − |
| |
| − | p0_q0
| |
| − |
| |
| − | p0_r1
| |
| − |
| |
| − | p0_r0_s#
| |
| − | p0_r1_s1
| |
| − | p0_r2_s#
| |
| − |
| |
| − | The Initial Conditions are given by a logical conjunction
| |
| − | that is composed of 5 basic expressions, altogether stating:
| |
| − |
| |
| − | | At the point-in-time p_0, M is in the state q_0, and
| |
| − | | At the point-in-time p_0, H is on the cell r_1, and
| |
| − | | At the point-in-time p_0, cell r_0 bears the mark "#", and
| |
| − | | At the point-in-time p_0, cell r_1 bears the mark "1", and
| |
| − | | At the point-in-time p_0, cell r_2 bears the mark "#".
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~PROGRAM~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | And here, yet again, just to store it nearby,
| |
| − | is the logical rendition of the TM's program:
| |
| − |
| |
| − | Mediate Conditions:
| |
| − |
| |
| − | ( p0_q# ( p1_q# ))
| |
| − | ( p0_q* ( p1_q* ))
| |
| − |
| |
| − | ( p1_q# ( p2_q# ))
| |
| − | ( p1_q* ( p2_q* ))
| |
| − |
| |
| − | Terminal Conditions:
| |
| − |
| |
| − | (( p2_q# )( p2_q* ))
| |
| − |
| |
| − | State Partition:
| |
| − |
| |
| − | (( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* ))
| |
| − | (( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* ))
| |
| − | (( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* ))
| |
| − |
| |
| − | Register Partition:
| |
| − |
| |
| − | (( p0_r0 ),( p0_r1 ),( p0_r2 ))
| |
| − | (( p1_r0 ),( p1_r1 ),( p1_r2 ))
| |
| − | (( p2_r0 ),( p2_r1 ),( p2_r2 ))
| |
| − |
| |
| − | Symbol Partition:
| |
| − |
| |
| − | (( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# ))
| |
| − | (( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# ))
| |
| − | (( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# ))
| |
| − |
| |
| − | (( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# ))
| |
| − | (( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# ))
| |
| − | (( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# ))
| |
| − |
| |
| − | (( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# ))
| |
| − | (( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# ))
| |
| − | (( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# ))
| |
| − |
| |
| − | Interaction Conditions:
| |
| − |
| |
| − | (( p0_r0 ) p0_r0_s0 ( p1_r0_s0 ))
| |
| − | (( p0_r0 ) p0_r0_s1 ( p1_r0_s1 ))
| |
| − | (( p0_r0 ) p0_r0_s# ( p1_r0_s# ))
| |
| − |
| |
| − | (( p0_r1 ) p0_r1_s0 ( p1_r1_s0 ))
| |
| − | (( p0_r1 ) p0_r1_s1 ( p1_r1_s1 ))
| |
| − | (( p0_r1 ) p0_r1_s# ( p1_r1_s# ))
| |
| − |
| |
| − | (( p0_r2 ) p0_r2_s0 ( p1_r2_s0 ))
| |
| − | (( p0_r2 ) p0_r2_s1 ( p1_r2_s1 ))
| |
| − | (( p0_r2 ) p0_r2_s# ( p1_r2_s# ))
| |
| − |
| |
| − | (( p1_r0 ) p1_r0_s0 ( p2_r0_s0 ))
| |
| − | (( p1_r0 ) p1_r0_s1 ( p2_r0_s1 ))
| |
| − | (( p1_r0 ) p1_r0_s# ( p2_r0_s# ))
| |
| − |
| |
| − | (( p1_r1 ) p1_r1_s0 ( p2_r1_s0 ))
| |
| − | (( p1_r1 ) p1_r1_s1 ( p2_r1_s1 ))
| |
| − | (( p1_r1 ) p1_r1_s# ( p2_r1_s# ))
| |
| − |
| |
| − | (( p1_r2 ) p1_r2_s0 ( p2_r2_s0 ))
| |
| − | (( p1_r2 ) p1_r2_s1 ( p2_r2_s1 ))
| |
| − | (( p1_r2 ) p1_r2_s# ( p2_r2_s# ))
| |
| − |
| |
| − | Transition Relations:
| |
| − |
| |
| − | ( p0_q0 p0_r1 p0_r1_s0 ( p1_q0 p1_r2 p1_r1_s0 ))
| |
| − | ( p0_q0 p0_r1 p0_r1_s1 ( p1_q1 p1_r2 p1_r1_s1 ))
| |
| − | ( p0_q0 p0_r1 p0_r1_s# ( p1_q# p1_r0 p1_r1_s# ))
| |
| − | ( p0_q0 p0_r2 p0_r2_s# ( p1_q# p1_r1 p1_r2_s# ))
| |
| − |
| |
| − | ( p0_q1 p0_r1 p0_r1_s0 ( p1_q1 p1_r2 p1_r1_s0 ))
| |
| − | ( p0_q1 p0_r1 p0_r1_s1 ( p1_q0 p1_r2 p1_r1_s1 ))
| |
| − | ( p0_q1 p0_r1 p0_r1_s# ( p1_q* p1_r0 p1_r1_s# ))
| |
| − | ( p0_q1 p0_r2 p0_r2_s# ( p1_q* p1_r1 p1_r2_s# ))
| |
| − |
| |
| − | ( p1_q0 p1_r1 p1_r1_s0 ( p2_q0 p2_r2 p2_r1_s0 ))
| |
| − | ( p1_q0 p1_r1 p1_r1_s1 ( p2_q1 p2_r2 p2_r1_s1 ))
| |
| − | ( p1_q0 p1_r1 p1_r1_s# ( p2_q# p2_r0 p2_r1_s# ))
| |
| − | ( p1_q0 p1_r2 p1_r2_s# ( p2_q# p2_r1 p2_r2_s# ))
| |
| − |
| |
| − | ( p1_q1 p1_r1 p1_r1_s0 ( p2_q1 p2_r2 p2_r1_s0 ))
| |
| − | ( p1_q1 p1_r1 p1_r1_s1 ( p2_q0 p2_r2 p2_r1_s1 ))
| |
| − | ( p1_q1 p1_r1 p1_r1_s# ( p2_q* p2_r0 p2_r1_s# ))
| |
| − | ( p1_q1 p1_r2 p1_r2_s# ( p2_q* p2_r1 p2_r2_s# ))
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~INTERPRETATION~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Interpretation of the Propositional Program:
| |
| − |
| |
| − | Mediate Conditions:
| |
| − |
| |
| − | ( p0_q# ( p1_q# ))
| |
| − | ( p0_q* ( p1_q* ))
| |
| − |
| |
| − | ( p1_q# ( p2_q# ))
| |
| − | ( p1_q* ( p2_q* ))
| |
| − |
| |
| − | In Ref Log, an expression of the form "( X ( Y ))"
| |
| − | expresses an implication or an if-then proposition:
| |
| − | "Not X without Y", "If X then Y", "X => Y", etc.
| |
| − |
| |
| − | A text string expression of the form "( X ( Y ))"
| |
| − | parses to a graphical data-structure of the form:
| |
| − |
| |
| − | X Y
| |
| − | o---o
| |
| − | |
| |
| − | @
| |
| − |
| |
| − | All together, these Mediate Conditions state:
| |
| − |
| |
| − | | If at p_0 M is in state q_#, then at p_1 M is in state q_#, and
| |
| − | | If at p_0 M is in state q_*, then at p_1 M is in state q_*, and
| |
| − | | If at p_1 M is in state q_#, then at p_2 M is in state q_#, and
| |
| − | | If at p_1 M is in state q_*, then at p_2 M is in state q_*.
| |
| − |
| |
| − | Terminal Conditions:
| |
| − |
| |
| − | (( p2_q# )( p2_q* ))
| |
| − |
| |
| − | In Ref Log, an expression of the form "(( X )( Y ))"
| |
| − | expresses a disjunction "X or Y" and it parses into:
| |
| − |
| |
| − | X Y
| |
| − | o o
| |
| − | \ /
| |
| − | o
| |
| − | |
| |
| − | @
| |
| − |
| |
| − | In effect, the Terminal Conditions state:
| |
| − |
| |
| − | | At p_2, M is in state q_#, or
| |
| − | | At p_2, M is in state q_*.
| |
| − |
| |
| − | State Partition:
| |
| − |
| |
| − | (( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* ))
| |
| − | (( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* ))
| |
| − | (( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* ))
| |
| − |
| |
| − | In Ref Log, an expression of the form "(( e_1 ),( e_2 ),( ... ),( e_k ))"
| |
| − | expresses the fact that "exactly one of the e_j is true, for j = 1 to k".
| |
| − | Expressions of this form are called "universal partition" expressions, and
| |
| − | they parse into a type of graph called a "painted and rooted cactus" (PARC):
| |
| − |
| |
| − | e_1 e_2 ... e_k
| |
| − | o o o
| |
| − | | | |
| |
| − | o-----o--- ... ---o
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | @
| |
| − |
| |
| − | The State Partition expresses the conditions that:
| |
| − |
| |
| − | | At each of the points-in-time p_i, for i = 0 to 2,
| |
| − | | M can be in exactly one state q_j, for j in the set {0, 1, #, *}.
| |
| − |
| |
| − | Register Partition:
| |
| − |
| |
| − | (( p0_r0 ),( p0_r1 ),( p0_r2 ))
| |
| − | (( p1_r0 ),( p1_r1 ),( p1_r2 ))
| |
| − | (( p2_r0 ),( p2_r1 ),( p2_r2 ))
| |
| − |
| |
| − | The Register Partition expresses the conditions that:
| |
| − |
| |
| − | | At each of the points-in-time p_i, for i = 0 to 2,
| |
| − | | H can be on exactly one cell r_j, for j = 0 to 2.
| |
| − |
| |
| − | Symbol Partition:
| |
| − |
| |
| − | (( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# ))
| |
| − | (( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# ))
| |
| − | (( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# ))
| |
| − |
| |
| − | (( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# ))
| |
| − | (( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# ))
| |
| − | (( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# ))
| |
| − |
| |
| − | (( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# ))
| |
| − | (( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# ))
| |
| − | (( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# ))
| |
| − |
| |
| − | The Symbol Partition expresses the conditions that:
| |
| − |
| |
| − | | At each of the points-in-time p_i, for i in {0, 1, 2},
| |
| − | | in each of the tape-registers r_j, for j in {0, 1, 2},
| |
| − | | there can be exactly one sign s_k, for k in {0, 1, #}.
| |
| − |
| |
| − | Interaction Conditions:
| |
| − |
| |
| − | (( p0_r0 ) p0_r0_s0 ( p1_r0_s0 ))
| |
| − | (( p0_r0 ) p0_r0_s1 ( p1_r0_s1 ))
| |
| − | (( p0_r0 ) p0_r0_s# ( p1_r0_s# ))
| |
| − |
| |
| − | (( p0_r1 ) p0_r1_s0 ( p1_r1_s0 ))
| |
| − | (( p0_r1 ) p0_r1_s1 ( p1_r1_s1 ))
| |
| − | (( p0_r1 ) p0_r1_s# ( p1_r1_s# ))
| |
| − |
| |
| − | (( p0_r2 ) p0_r2_s0 ( p1_r2_s0 ))
| |
| − | (( p0_r2 ) p0_r2_s1 ( p1_r2_s1 ))
| |
| − | (( p0_r2 ) p0_r2_s# ( p1_r2_s# ))
| |
| − |
| |
| − | (( p1_r0 ) p1_r0_s0 ( p2_r0_s0 ))
| |
| − | (( p1_r0 ) p1_r0_s1 ( p2_r0_s1 ))
| |
| − | (( p1_r0 ) p1_r0_s# ( p2_r0_s# ))
| |
| − |
| |
| − | (( p1_r1 ) p1_r1_s0 ( p2_r1_s0 ))
| |
| − | (( p1_r1 ) p1_r1_s1 ( p2_r1_s1 ))
| |
| − | (( p1_r1 ) p1_r1_s# ( p2_r1_s# ))
| |
| − |
| |
| − | (( p1_r2 ) p1_r2_s0 ( p2_r2_s0 ))
| |
| − | (( p1_r2 ) p1_r2_s1 ( p2_r2_s1 ))
| |
| − | (( p1_r2 ) p1_r2_s# ( p2_r2_s# ))
| |
| − |
| |
| − | In briefest terms, the Interaction Conditions merely express
| |
| − | the circumstance that the sign in a tape-cell cannot change
| |
| − | between two points-in-time unless the tape-head is over the
| |
| − | cell in question at the initial one of those points-in-time.
| |
| − | All that we have to do is to see how they manage to say this.
| |
| − |
| |
| − | In Ref Log, an expression of the following form:
| |
| − |
| |
| − | "(( p<i>_r<j> ) p<i>_r<j>_s<k> ( p<i+1>_r<j>_s<k> ))",
| |
| − |
| |
| − | and which parses as the graph:
| |
| − |
| |
| − | p<i>_r<j> o o p<i+1>_r<j>_s<k>
| |
| − | \ /
| |
| − | p<i>_r<j>_s<k> o
| |
| − | |
| |
| − | @
| |
| − |
| |
| − | can be read in the form of the following implication:
| |
| − |
| |
| − | | If
| |
| − | | at the point-in-time p<i>, the tape-cell r<j> bears the mark s<k>,
| |
| − | | but it is not the case that
| |
| − | | at the point-in-time p<i>, the tape-head is on the tape-cell r<j>.
| |
| − | | then
| |
| − | | at the point-in-time p<i+1>, the tape-cell r<j> bears the mark s<k>.
| |
| − |
| |
| − | Folks among us of a certain age and a peculiar manner of acculturation will
| |
| − | recognize these as the "Frame Conditions" for the change of state of the TM.
| |
| − |
| |
| − | Transition Relations:
| |
| − |
| |
| − | ( p0_q0 p0_r1 p0_r1_s0 ( p1_q0 p1_r2 p1_r1_s0 ))
| |
| − | ( p0_q0 p0_r1 p0_r1_s1 ( p1_q1 p1_r2 p1_r1_s1 ))
| |
| − | ( p0_q0 p0_r1 p0_r1_s# ( p1_q# p1_r0 p1_r1_s# ))
| |
| − | ( p0_q0 p0_r2 p0_r2_s# ( p1_q# p1_r1 p1_r2_s# ))
| |
| − |
| |
| − | ( p0_q1 p0_r1 p0_r1_s0 ( p1_q1 p1_r2 p1_r1_s0 ))
| |
| − | ( p0_q1 p0_r1 p0_r1_s1 ( p1_q0 p1_r2 p1_r1_s1 ))
| |
| − | ( p0_q1 p0_r1 p0_r1_s# ( p1_q* p1_r0 p1_r1_s# ))
| |
| − | ( p0_q1 p0_r2 p0_r2_s# ( p1_q* p1_r1 p1_r2_s# ))
| |
| − |
| |
| − | ( p1_q0 p1_r1 p1_r1_s0 ( p2_q0 p2_r2 p2_r1_s0 ))
| |
| − | ( p1_q0 p1_r1 p1_r1_s1 ( p2_q1 p2_r2 p2_r1_s1 ))
| |
| − | ( p1_q0 p1_r1 p1_r1_s# ( p2_q# p2_r0 p2_r1_s# ))
| |
| − | ( p1_q0 p1_r2 p1_r2_s# ( p2_q# p2_r1 p2_r2_s# ))
| |
| − |
| |
| − | ( p1_q1 p1_r1 p1_r1_s0 ( p2_q1 p2_r2 p2_r1_s0 ))
| |
| − | ( p1_q1 p1_r1 p1_r1_s1 ( p2_q0 p2_r2 p2_r1_s1 ))
| |
| − | ( p1_q1 p1_r1 p1_r1_s# ( p2_q* p2_r0 p2_r1_s# ))
| |
| − | ( p1_q1 p1_r2 p1_r2_s# ( p2_q* p2_r1 p2_r2_s# ))
| |
| − |
| |
| − | The Transition Conditions merely serve to express,
| |
| − | by means of 16 complex implication expressions,
| |
| − | the data of the TM table that was given above.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~OUTPUTS~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | And here are the outputs of the computation,
| |
| − | as emulated by its propositional rendition,
| |
| − | and as actually generated within that form
| |
| − | of transmogrification by the program that
| |
| − | I wrote for finding all of the satisfying
| |
| − | interpretations (truth-value assignments)
| |
| − | of propositional expressions in Ref Log:
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~OUTPUT~0~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Output Conditions:
| |
| − |
| |
| − | p0_q0
| |
| − | p0_r1
| |
| − | p0_r0_s#
| |
| − | p0_r1_s0
| |
| − | p0_r2_s#
| |
| − | p1_q0
| |
| − | p1_r2
| |
| − | p1_r2_s#
| |
| − | p1_r0_s#
| |
| − | p1_r1_s0
| |
| − | p2_q#
| |
| − | p2_r1
| |
| − | p2_r0_s#
| |
| − | p2_r1_s0
| |
| − | p2_r2_s#
| |
| − |
| |
| − | The Output Conditions amount to the sole satisfying interpretation,
| |
| − | that is, a "sequence of truth-value assignments" (SOTVA) that make
| |
| − | the entire proposition come out true, and they state the following:
| |
| − |
| |
| − | | At the point-in-time p_0, M is in the state q_0, and
| |
| − | | At the point-in-time p_0, H is on the cell r_1, and
| |
| − | | At the point-in-time p_0, cell r_0 bears the mark "#", and
| |
| − | | At the point-in-time p_0, cell r_1 bears the mark "0", and
| |
| − | | At the point-in-time p_0, cell r_2 bears the mark "#", and
| |
| − | |
| |
| − | | At the point-in-time p_1, M is in the state q_0, and
| |
| − | | At the point-in-time p_1, H is on the cell r_2, and
| |
| − | | At the point-in-time p_1, cell r_0 bears the mark "#", and
| |
| − | | At the point-in-time p_1, cell r_1 bears the mark "0", and
| |
| − | | At the point-in-time p_1, cell r_2 bears the mark "#", and
| |
| − | |
| |
| − | | At the point-in-time p_2, M is in the state q_#, and
| |
| − | | At the point-in-time p_2, H is on the cell r_1, and
| |
| − | | At the point-in-time p_2, cell r_0 bears the mark "#", and
| |
| − | | At the point-in-time p_2, cell r_1 bears the mark "0", and
| |
| − | | At the point-in-time p_2, cell r_2 bears the mark "#".
| |
| − |
| |
| − | In brief, the output for our sake being the symbol that rests
| |
| − | under the tape-head H when the machine M gets to a rest state,
| |
| − | we are now amazed by the remarkable result that Parity(0) = 0.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~OUTPUT~1~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Output Conditions:
| |
| − |
| |
| − | p0_q0
| |
| − | p0_r1
| |
| − | p0_r0_s#
| |
| − | p0_r1_s1
| |
| − | p0_r2_s#
| |
| − | p1_q1
| |
| − | p1_r2
| |
| − | p1_r2_s#
| |
| − | p1_r0_s#
| |
| − | p1_r1_s1
| |
| − | p2_q*
| |
| − | p2_r1
| |
| − | p2_r0_s#
| |
| − | p2_r1_s1
| |
| − | p2_r2_s#
| |
| − |
| |
| − | The Output Conditions amount to the sole satisfying interpretation,
| |
| − | that is, a "sequence of truth-value assignments" (SOTVA) that make
| |
| − | the entire proposition come out true, and they state the following:
| |
| − |
| |
| − | | At the point-in-time p_0, M is in the state q_0, and
| |
| − | | At the point-in-time p_0, H is on the cell r_1, and
| |
| − | | At the point-in-time p_0, cell r_0 bears the mark "#", and
| |
| − | | At the point-in-time p_0, cell r_1 bears the mark "1", and
| |
| − | | At the point-in-time p_0, cell r_2 bears the mark "#", and
| |
| − | |
| |
| − | | At the point-in-time p_1, M is in the state q_1, and
| |
| − | | At the point-in-time p_1, H is on the cell r_2, and
| |
| − | | At the point-in-time p_1, cell r_0 bears the mark "#", and
| |
| − | | At the point-in-time p_1, cell r_1 bears the mark "1", and
| |
| − | | At the point-in-time p_1, cell r_2 bears the mark "#", and
| |
| − | |
| |
| − | | At the point-in-time p_2, M is in the state q_*, and
| |
| − | | At the point-in-time p_2, H is on the cell r_1, and
| |
| − | | At the point-in-time p_2, cell r_0 bears the mark "#", and
| |
| − | | At the point-in-time p_2, cell r_1 bears the mark "1", and
| |
| − | | At the point-in-time p_2, cell r_2 bears the mark "#".
| |
| − |
| |
| − | In brief, the output for our sake being the symbol that rests
| |
| − | under the tape-head H when the machine M gets to a rest state,
| |
| − | we are now amazed by the remarkable result that Parity(1) = 1.
| |
| − |
| |
| − | I realized after sending that last bunch of bits that there is room
| |
| − | for confusion about what is the input/output of the Study module of
| |
| − | the Theme One program as opposed to what is the input/output of the
| |
| − | "finitely approximated turing automaton" (FATA). So here is better
| |
| − | delineation of what's what. The input to Study is a text file that
| |
| − | is known as LogFile(Whatever) and the output of Study is a sequence
| |
| − | of text files that summarize the various canonical and normal forms
| |
| − | that it generates. For short, let us call these NormFile(Whatelse).
| |
| − | With that in mind, here are the actual IO's of Study, excluding the
| |
| − | glosses in square brackets:
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~INPUT~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | [Input To Study = FATA Initial Conditions + FATA Program Conditions]
| |
| − |
| |
| − | [FATA Initial Conditions For Input 0]
| |
| − |
| |
| − | p0_q0
| |
| − |
| |
| − | p0_r1
| |
| − |
| |
| − | p0_r0_s#
| |
| − | p0_r1_s0
| |
| − | p0_r2_s#
| |
| − |
| |
| − | [FATA Program Conditions For Parity Machine]
| |
| − |
| |
| − | [Mediate Conditions]
| |
| − |
| |
| − | ( p0_q# ( p1_q# ))
| |
| − | ( p0_q* ( p1_q* ))
| |
| − |
| |
| − | ( p1_q# ( p2_q# ))
| |
| − | ( p1_q* ( p2_q* ))
| |
| − |
| |
| − | [Terminal Conditions]
| |
| − |
| |
| − | (( p2_q# )( p2_q* ))
| |
| − |
| |
| − | [State Partition]
| |
| − |
| |
| − | (( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* ))
| |
| − | (( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* ))
| |
| − | (( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* ))
| |
| − |
| |
| − | [Register Partition]
| |
| − |
| |
| − | (( p0_r0 ),( p0_r1 ),( p0_r2 ))
| |
| − | (( p1_r0 ),( p1_r1 ),( p1_r2 ))
| |
| − | (( p2_r0 ),( p2_r1 ),( p2_r2 ))
| |
| − |
| |
| − | [Symbol Partition]
| |
| − |
| |
| − | (( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# ))
| |
| − | (( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# ))
| |
| − | (( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# ))
| |
| − |
| |
| − | (( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# ))
| |
| − | (( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# ))
| |
| − | (( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# ))
| |
| − |
| |
| − | (( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# ))
| |
| − | (( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# ))
| |
| − | (( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# ))
| |
| − |
| |
| − | [Interaction Conditions]
| |
| − |
| |
| − | (( p0_r0 ) p0_r0_s0 ( p1_r0_s0 ))
| |
| − | (( p0_r0 ) p0_r0_s1 ( p1_r0_s1 ))
| |
| − | (( p0_r0 ) p0_r0_s# ( p1_r0_s# ))
| |
| − |
| |
| − | (( p0_r1 ) p0_r1_s0 ( p1_r1_s0 ))
| |
| − | (( p0_r1 ) p0_r1_s1 ( p1_r1_s1 ))
| |
| − | (( p0_r1 ) p0_r1_s# ( p1_r1_s# ))
| |
| − |
| |
| − | (( p0_r2 ) p0_r2_s0 ( p1_r2_s0 ))
| |
| − | (( p0_r2 ) p0_r2_s1 ( p1_r2_s1 ))
| |
| − | (( p0_r2 ) p0_r2_s# ( p1_r2_s# ))
| |
| − |
| |
| − | (( p1_r0 ) p1_r0_s0 ( p2_r0_s0 ))
| |
| − | (( p1_r0 ) p1_r0_s1 ( p2_r0_s1 ))
| |
| − | (( p1_r0 ) p1_r0_s# ( p2_r0_s# ))
| |
| − |
| |
| − | (( p1_r1 ) p1_r1_s0 ( p2_r1_s0 ))
| |
| − | (( p1_r1 ) p1_r1_s1 ( p2_r1_s1 ))
| |
| − | (( p1_r1 ) p1_r1_s# ( p2_r1_s# ))
| |
| − |
| |
| − | (( p1_r2 ) p1_r2_s0 ( p2_r2_s0 ))
| |
| − | (( p1_r2 ) p1_r2_s1 ( p2_r2_s1 ))
| |
| − | (( p1_r2 ) p1_r2_s# ( p2_r2_s# ))
| |
| − |
| |
| − | [Transition Relations]
| |
| − |
| |
| − | ( p0_q0 p0_r1 p0_r1_s0 ( p1_q0 p1_r2 p1_r1_s0 ))
| |
| − | ( p0_q0 p0_r1 p0_r1_s1 ( p1_q1 p1_r2 p1_r1_s1 ))
| |
| − | ( p0_q0 p0_r1 p0_r1_s# ( p1_q# p1_r0 p1_r1_s# ))
| |
| − | ( p0_q0 p0_r2 p0_r2_s# ( p1_q# p1_r1 p1_r2_s# ))
| |
| − |
| |
| − | ( p0_q1 p0_r1 p0_r1_s0 ( p1_q1 p1_r2 p1_r1_s0 ))
| |
| − | ( p0_q1 p0_r1 p0_r1_s1 ( p1_q0 p1_r2 p1_r1_s1 ))
| |
| − | ( p0_q1 p0_r1 p0_r1_s# ( p1_q* p1_r0 p1_r1_s# ))
| |
| − | ( p0_q1 p0_r2 p0_r2_s# ( p1_q* p1_r1 p1_r2_s# ))
| |
| − |
| |
| − | ( p1_q0 p1_r1 p1_r1_s0 ( p2_q0 p2_r2 p2_r1_s0 ))
| |
| − | ( p1_q0 p1_r1 p1_r1_s1 ( p2_q1 p2_r2 p2_r1_s1 ))
| |
| − | ( p1_q0 p1_r1 p1_r1_s# ( p2_q# p2_r0 p2_r1_s# ))
| |
| − | ( p1_q0 p1_r2 p1_r2_s# ( p2_q# p2_r1 p2_r2_s# ))
| |
| − |
| |
| − | ( p1_q1 p1_r1 p1_r1_s0 ( p2_q1 p2_r2 p2_r1_s0 ))
| |
| − | ( p1_q1 p1_r1 p1_r1_s1 ( p2_q0 p2_r2 p2_r1_s1 ))
| |
| − | ( p1_q1 p1_r1 p1_r1_s# ( p2_q* p2_r0 p2_r1_s# ))
| |
| − | ( p1_q1 p1_r2 p1_r2_s# ( p2_q* p2_r1 p2_r2_s# ))
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~OUTPUT~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | [Output Of Study = FATA Output For Input 0]
| |
| − |
| |
| − | p0_q0
| |
| − | p0_r1
| |
| − | p0_r0_s#
| |
| − | p0_r1_s0
| |
| − | p0_r2_s#
| |
| − | p1_q0
| |
| − | p1_r2
| |
| − | p1_r2_s#
| |
| − | p1_r0_s#
| |
| − | p1_r1_s0
| |
| − | p2_q#
| |
| − | p2_r1
| |
| − | p2_r0_s#
| |
| − | p2_r1_s0
| |
| − | p2_r2_s#
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Turing automata, finitely approximated or not, make my head spin and
| |
| − | my tape go loopy, and I still believe 'twere a far better thing I do
| |
| − | if I work up to that level of complexity in a more gracile graduated
| |
| − | manner. So let us return to our Example in this gradual progress to
| |
| − | that vastly more well-guarded grail of our long-term pilgrim's quest:
| |
| − |
| |
| − | | boy male girl female
| |
| − | | o---o child o---o child
| |
| − | | male female \ / \ / child human
| |
| − | | o---o o o o--o
| |
| − | | \ / | | |
| |
| − | | @ @ @ @
| |
| − | |
| |
| − | | (male , female)((boy , male child))((girl , female child))(child (human))
| |
| − |
| |
| − | One section of the Theme One program has a suite of utilities that fall
| |
| − | under the title "Theme One Study" ("To Study", or just "TOS" for short).
| |
| − | To Study is to read and to parse a so-called and a generally so-suffixed
| |
| − | "log" file, and then to conjoin what is called a "query", which is really
| |
| − | just an additional propositional expression that imposes a further logical
| |
| − | constraint on the input expression.
| |
| − |
| |
| − | The Figure roughly sketches the conjuncts of the graph-theoretic
| |
| − | data structure that the parser would commit to memory on reading
| |
| − | the appropriate log file that contains the text along the bottom.
| |
| − |
| |
| − | I will now explain the various sorts of things that the TOS utility
| |
| − | can do with the log file that describes the universe of discourse in
| |
| − | our present Example.
| |
| − |
| |
| − | Theme One Study is built around a suite of four successive generators
| |
| − | of "normal forms" for propositional expressions, just to use that term
| |
| − | in a very approximate way. The functions that compute these normal forms
| |
| − | are called "Model", "Tenor", "Canon", and "Sense", and so we may refer to
| |
| − | to their text-style outputs as the "mod", "ten", "can", and "sen" files.
| |
| − |
| |
| − | Though it could be any propositional expression on the same vocabulary
| |
| − | $A$ = {"boy", "child", "female", "girl", "human", "male"}, more usually
| |
| − | the query is a simple conjunction of one or more positive features that
| |
| − | we want to focus on or perhaps to filter out of the logical model space.
| |
| − | On our first run through this Example, we take the log file proposition
| |
| − | as it is, with no extra riders.
| |
| − |
| |
| − | | Procedural Note. TO Study Model displays a running tab of how much
| |
| − | | free memory space it has left. On some of the harder problems that
| |
| − | | you may think of to give it, Model may run out of free memory and
| |
| − | | terminate, abnormally exiting Theme One. Sometimes it helps to:
| |
| − | |
| |
| − | | 1. Rephrase the problem in logically equivalent
| |
| − | | but rhetorically increasedly felicitous ways.
| |
| − | |
| |
| − | | 2. Think of additional facts that are taken for granted but not
| |
| − | | made explicit and that cannot be logically inferred by Model.
| |
| − |
| |
| − | After Model has finished, it is ready to write out its mod file,
| |
| − | which you may choose to show on the screen or save to a named file.
| |
| − | Mod files are usually too long to see (or to care to see) all at once
| |
| − | on the screen, so it is very often best to save them for later replay.
| |
| − | In our Example the Model function yields a mod file that looks like so:
| |
| − |
| |
| − | Model Output and
| |
| − | Mod File Example
| |
| − | o-------------------o
| |
| − | | male |
| |
| − | | female - | 1
| |
| − | | (female ) |
| |
| − | | girl - | 2
| |
| − | | (girl ) |
| |
| − | | child |
| |
| − | | boy |
| |
| − | | human * | 3 *
| |
| − | | (human ) - | 4
| |
| − | | (boy ) - | 5
| |
| − | | (child ) |
| |
| − | | boy - | 6
| |
| − | | (boy ) * | 7 *
| |
| − | | (male ) |
| |
| − | | female |
| |
| − | | boy - | 8
| |
| − | | (boy ) |
| |
| − | | child |
| |
| − | | girl |
| |
| − | | human * | 9 *
| |
| − | | (human ) - | 10
| |
| − | | (girl ) - | 11
| |
| − | | (child ) |
| |
| − | | girl - | 12
| |
| − | | (girl ) * | 13 *
| |
| − | | (female ) - | 14
| |
| − | o-------------------o
| |
| − |
| |
| − | Counting the stars "*" that indicate true interpretations
| |
| − | and the bars "-" that indicate false interpretations of
| |
| − | the input formula, we can see that the Model function,
| |
| − | out of the 64 possible interpretations, has actually
| |
| − | gone through the work of making just 14 evaluations,
| |
| − | all in order to find the 4 models that are allowed
| |
| − | by the input definitions.
| |
| − |
| |
| − | To be clear about what this output means, the starred paths
| |
| − | indicate all of the complete specifications of objects in the
| |
| − | universe of discourse, that is, all of the consistent feature
| |
| − | conjunctions of maximum length, as permitted by the definitions
| |
| − | that are given in the log file.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Let's take a little break from the Example in progress
| |
| − | and look at where we are and what we have been doing from
| |
| − | computational, logical, and semiotic perspectives. Because,
| |
| − | after all, as is usually the case, we should not let our focus
| |
| − | and our fascination with this particular Example prevent us from
| |
| − | recognizing it, and all that we do with it, as just an Example of
| |
| − | much broader paradigms and predicaments and principles, not to say
| |
| − | but a glimmer of ultimately more concernful and fascinating objects.
| |
| − |
| |
| − | I chart the progression that we have just passed through in this way:
| |
| − |
| |
| − | | Parse
| |
| − | | Sign A o-------------->o Sign 1
| |
| − | | ^ |
| |
| − | | / |
| |
| − | | / |
| |
| − | | / |
| |
| − | | Object o | Transform
| |
| − | | ^ |
| |
| − | | \ |
| |
| − | | \ |
| |
| − | | \ v
| |
| − | | Sign B o<--------------o Sign 2
| |
| − | | Verse
| |
| − | |
| |
| − | | Figure. Computation As Sign Transformation
| |
| − |
| |
| − | In the present case, the Object is an objective situation
| |
| − | or a state of affairs, in effect, a particular pattern of
| |
| − | feature concurrences occurring to us in that world through
| |
| − | which we find ourselves most frequently faring, wily nily,
| |
| − | and the Signs are different tokens and different types of
| |
| − | data structures that we somehow or other find it useful
| |
| − | to devise or to discover for the sake of representing
| |
| − | current objects to ourselves on a recurring basis.
| |
| − |
| |
| − | But not all signs, not even signs of a single object, are alike
| |
| − | in every other respect that one might name, not even with respect
| |
| − | to their powers of relating, significantly, to that common object.
| |
| − |
| |
| − | And that is what our whole business of computation busies itself about,
| |
| − | when it minds its business best, that is, transmuting signs into signs
| |
| − | in ways that augment their powers of relating significantly to objects.
| |
| − |
| |
| − | We have seen how the Model function and the mod output format
| |
| − | indicate all of the complete specifications of objects in the
| |
| − | universe of discourse, that is, all of the consistent feature
| |
| − | conjunctions of maximal specificity that are permitted by the
| |
| − | constraints or the definitions that are given in the log file.
| |
| − |
| |
| − | To help identify these specifications of particular cells in
| |
| − | the universe of discourse, the next function and output format,
| |
| − | called "Tenor", edits the mod file to give only the true paths,
| |
| − | in effect, the "positive models", that are by default what we
| |
| − | usually mean when we say "models", and not the "anti-models"
| |
| − | or the "negative models" that fail to satisfy the formula
| |
| − | in question.
| |
| − |
| |
| − | In the present Example the Tenor function
| |
| − | generates a Ten file that looks like this:
| |
| − |
| |
| − | Tenor Output and
| |
| − | Ten File Example
| |
| − | o-------------------o
| |
| − | | male |
| |
| − | | (female ) |
| |
| − | | (girl ) |
| |
| − | | child |
| |
| − | | boy |
| |
| − | | human * | <1>
| |
| − | | (child ) |
| |
| − | | (boy ) * | <2>
| |
| − | | (male ) |
| |
| − | | female |
| |
| − | | (boy ) |
| |
| − | | child |
| |
| − | | girl |
| |
| − | | human * | <3>
| |
| − | | (child ) |
| |
| − | | (girl ) * | <4>
| |
| − | o-------------------o
| |
| − |
| |
| − | As I said, the Tenor function just abstracts a transcript of the models,
| |
| − | that is, the satisfying interpretations, that were already interspersed
| |
| − | throughout the complete Model output. These specifications, or feature
| |
| − | conjunctions, with the positive and the negative features listed in the
| |
| − | order of their actual budding on the "arboreal boolean expansion" twigs,
| |
| − | may be gathered and arranged in this antherypulogical flowering bouquet:
| |
| − |
| |
| − | 1. male (female ) (girl ) child boy human *
| |
| − | 2. male (female ) (girl ) (child ) (boy ) *
| |
| − | 3. (male ) female (boy ) child girl human *
| |
| − | 4. (male ) female (boy ) (child ) (girl ) *
| |
| − |
| |
| − | Notice that Model, as reflected in this abstract, did not consider
| |
| − | the six positive features in the same order along each path. This
| |
| − | is because the algorithm was designed to proceed opportunistically
| |
| − | in its attempt to reduce the original proposition through a series
| |
| − | of case-analytic considerations and the resulting simplifications.
| |
| − |
| |
| − | Notice, too, that Model is something of a lazy evaluator, quitting work
| |
| − | when and if a value is determined by less than the full set of variables.
| |
| − | This is the reason why paths <2> and <4> are not ostensibly of the maximum
| |
| − | length. According to this lazy mode of understanding, any path that is not
| |
| − | specified on a set of features really stands for the whole bundle of paths
| |
| − | that are derived by freely varying those features. Thus, specifications
| |
| − | <2> and <4> summarize four models altogether, with the logical choice
| |
| − | between "human" and "not human" being left open at the point where
| |
| − | they leave off their branches in the releavent deciduous tree.
| |
| − |
| |
| − | The last two functions in the Study section, "Canon" and "Sense",
| |
| − | extract further derivatives of the normal forms that are produced
| |
| − | by Model and Tenor. Both of these functions take the set of model
| |
| − | paths and simply throw away the negative labels. You may think of
| |
| − | these as the "rose colored glasses" or "job interview" normal forms,
| |
| − | in that they try to say everything that's true, so long as it can be
| |
| − | expressed in positive terms. Generally, this would mean losing a lot
| |
| − | of information, and the result could no longer be expected to have the
| |
| − | property of remaining logically equivalent to the original proposition.
| |
| − |
| |
| − | Fortunately, however, it seems that this type of positive projection of
| |
| − | the whole truth is just what is possible, most needed, and most clear in
| |
| − | many of the "natural" examples, that is, in examples that arise from the
| |
| − | domains of natural language and natural conceptual kinds. In these cases,
| |
| − | where most of the logical features are redundantly coded, for example, in
| |
| − | the way that "adult" = "not child" and "child" = "not adult", the positive
| |
| − | feature bearing redacts are often sufficiently expressive all by themselves.
| |
| − |
| |
| − | Canon merely censors its printing of the negative labels as it traverses the
| |
| − | model tree. This leaves the positive labels in their original columns of the
| |
| − | outline form, giving it a slightly skewed appearance. This can be misleading
| |
| − | unless you already know what you are looking for. However, this Canon format
| |
| − | is computationally quick, and frequently suffices, especially if you already
| |
| − | have a likely clue about what to expect in the way of a question's outcome.
| |
| − |
| |
| − | In the present Example the Canon function
| |
| − | generates a Can file that looks like this:
| |
| − |
| |
| − | Canon Output and
| |
| − | Can File Example
| |
| − | o-------------------o
| |
| − | | male |
| |
| − | | child |
| |
| − | | boy |
| |
| − | | human |
| |
| − | | female |
| |
| − | | child |
| |
| − | | girl |
| |
| − | | human |
| |
| − | o-------------------o
| |
| − |
| |
| − | The Sense function does the extra work that is required
| |
| − | to place the positive labels of the model tree at their
| |
| − | proper level in the outline.
| |
| − |
| |
| − | In the present Example the Sense function
| |
| − | generates a Sen file that looks like this:
| |
| − |
| |
| − | Sense Output and
| |
| − | Sen File Example
| |
| − | o-------------------o
| |
| − | | male |
| |
| − | | child |
| |
| − | | boy |
| |
| − | | human |
| |
| − | | female |
| |
| − | | child |
| |
| − | | girl |
| |
| − | | human |
| |
| − | o-------------------o
| |
| − |
| |
| − | The Canon and Sense outlines for this Example illustrate a certain
| |
| − | type of general circumstance that needs to be noted at this point.
| |
| − | Recall the model paths or the feature specifications that were
| |
| − | numbered <2> and <4> in the listing of the output for Tenor.
| |
| − | These paths, in effect, reflected Model's discovery that
| |
| − | the venn diagram cells for male or female non-children
| |
| − | and male or female non-humans were not excluded by
| |
| − | the definitions that were given in the Log file.
| |
| − | In the abstracts given by Canon and Sense, the
| |
| − | specifications <2> and <4> have been subsumed,
| |
| − | or absorbed unmarked, under the general topics
| |
| − | of their respective genders, male or female.
| |
| − | This happens because no purely positive
| |
| − | features were supplied to distinguish
| |
| − | the non-child and non-human cases.
| |
| − |
| |
| − | That completes the discussion of
| |
| − | this six-dimensional Example.
| |
| − |
| |
| − | Nota Bene, for possible future use. In the larger current of work
| |
| − | with respect to which this meander of a conduit was initially both
| |
| − | diversionary and tributary, before those high and dry regensquirm
| |
| − | years when it turned into an intellectual interglacial oxbow lake,
| |
| − | I once had in mind a scape in which expressions in a definitional
| |
| − | lattice were ordered according to their simplicity on some scale
| |
| − | or another, and in this setting the word "sense" was actually an
| |
| − | acronym for "semantically equivalent next-simplest expression".
| |
| − |
| |
| − | | If this is starting to sound a little bit familiar,
| |
| − | | it may be because the relationship between the two
| |
| − | | kinds of pictures of propositions, namely:
| |
| − | |
| |
| − | | 1. Propositions about things in general, here,
| |
| − | | about the times when certain facts are true,
| |
| − | | having the form of functions f : X -> B,
| |
| − | |
| |
| − | | 2. Propositions about binary codes, here, about
| |
| − | | the bit-vector labels on venn diagram cells,
| |
| − | | having the form of functions f' : B^k -> B,
| |
| − | |
| |
| − | | is an epically old story, one that I, myself,
| |
| − | | have related one or twice upon a time before,
| |
| − | | to wit, at least, at the following two cites:
| |
| − | |
| |
| − | | http://suo.ieee.org/email/msg01251.html
| |
| − | | http://suo.ieee.org/email/msg01293.html
| |
| − | |
| |
| − | | There, and now here, once more, and again, it may be observed
| |
| − | | that the relation is one whereby the proposition f : X -> B,
| |
| − | | the one about things and times and mores in general, factors
| |
| − | | into a coding function c : X -> B^k, followed by a derived
| |
| − | | proposition f' : B^k -> B that judges the resulting codes.
| |
| − | |
| |
| − | | f
| |
| − | | X o------>o B
| |
| − | | \ ^
| |
| − | | c = <x_1, ..., x_k> \ / f'
| |
| − | | v /
| |
| − | | o
| |
| − | | B^k
| |
| − | |
| |
| − | | You may remember that this was supposed to illustrate
| |
| − | | the "factoring" of a proposition f : X -> B = {0, 1}
| |
| − | | into the composition f'(c(x)), where c : X -> B^k is
| |
| − | | the "coding" of each x in X as an k-bit string in B^k,
| |
| − | | and where f' is the mapping of codes into a co-domain
| |
| − | | that we interpret as t-f-values, B = {0, 1} = {F, T}.
| |
| − |
| |
| − | In short, there is the standard equivocation ("systematic ambiguity"?) as to
| |
| − | whether we are talking about the "applied" and concretely typed proposition
| |
| − | f : X -> B or the "pure" and abstractly typed proposition f' : B^k -> B.
| |
| − | Or we can think of the latter object as the approximate code icon of
| |
| − | the former object.
| |
| − |
| |
| − | Anyway, these types of formal objects are the sorts of things that
| |
| − | I take to be the denotational objects of propositional expressions.
| |
| − | These objects, along with their invarious and insundry mathematical
| |
| − | properties, are the orders of things that I am talking about when
| |
| − | I refer to the "invariant structures in these objects themselves".
| |
| − |
| |
| − | "Invariant" means "invariant under a suitable set of transformations",
| |
| − | in this case the translations between various languages that preserve
| |
| − | the objects and the structures in question. In extremest generality,
| |
| − | this is what universal constructions in category theory are all about.
| |
| − |
| |
| − | In summation, the functions f : X -> B and f' : B* -> B have invariant, formal,
| |
| − | mathematical, objective properties that any adequate language might eventually
| |
| − | evolve to express, only some languages express them more obscurely than others.
| |
| − |
| |
| − | To be perfectly honest, I continue to be surprised that anybody in this group
| |
| − | has trouble with this. There are perfectly apt and familiar examples in the
| |
| − | contrast between roman numerals and arabic numerals, or the contrast between
| |
| − | redundant syntaxes, like those that use the pentalphabet {~, &, v, =>, <=>},
| |
| − | and trimmer syntaxes, like those used in existential and conceptual graphs.
| |
| − | Every time somebody says "Let's take {~, &, v, =>, <=>} as an operational
| |
| − | basis for logic" it's just like that old joke that mathematicians tell on
| |
| − | engineers where the ingenue in question says "1 is a prime, 2 is a prime,
| |
| − | 3 is a prime, 4 is a prime, ..." -- and I know you think that I'm being
| |
| − | hyperbolic, but I'm really only up to parabolas here ...
| |
| − |
| |
| − | I have already refined my criticism so that it does not apply to
| |
| − | the spirit of FOL or KIF or whatever, but only to the letters of
| |
| − | specific syntactic proposals. There is a fact of the matter as
| |
| − | to whether a concrete language provides a clean or a cluttered
| |
| − | basis for representing the identified set of formal objects.
| |
| − | And it shows up in pragmatic realities like the efficiency
| |
| − | of real time concept formation, concept use, learnability,
| |
| − | reasoning power, and just plain good use of real time.
| |
| − | These are the dire consequences that I learned in my
| |
| − | very first tries at mathematically oriented theorem
| |
| − | automation, and the only factor that has obscured
| |
| − | them in mainstream work since then is the speed
| |
| − | with which folks can now do all of the same
| |
| − | old dumb things that they used to do on
| |
| − | their way to kludging out the answers.
| |
| − |
| |
| − | It seems to be darn near impossible to explain to the
| |
| − | centurion all of the neat stuff that he's missing by
| |
| − | sticking to his old roman numerals. He just keeps
| |
| − | on reckoning that what he can't count must be of
| |
| − | no account at all. There is way too much stuff
| |
| − | that these original syntaxes keep us from even
| |
| − | beginning to discuss, like differential logic,
| |
| − | just for starters.
| |
| − |
| |
| − | Our next Example illustrates the use of the Cactus Language
| |
| − | for representing "absolute" and "relative" partitions, also
| |
| − | known as "complete" and "contingent" classifications of the
| |
| − | universe of discourse, all of which amounts to divvying it
| |
| − | up into mutually exclusive regions, exhaustive or not, as
| |
| − | one frequently needs in situations involving a genus and
| |
| − | its sundry species, and frequently pictures in the form
| |
| − | of a venn diagram that looks just like a "pie chart".
| |
| − |
| |
| − | Example. Partition, Genus & Species
| |
| − |
| |
| − | The idea that one needs for expressing partitions
| |
| − | in cactus expressions can be summed up like this:
| |
| − |
| |
| − | | If the propositional expression
| |
| − | |
| |
| − | | "( p , q , r , ... )"
| |
| − | |
| |
| − | | means that just one of
| |
| − | |
| |
| − | | p, q, r, ... is false,
| |
| − | |
| |
| − | | then the propositional expression
| |
| − | |
| |
| − | | "((p),(q),(r), ... )"
| |
| − | |
| |
| − | | must mean that just one of
| |
| − | |
| |
| − | | (p), (q), (r), ... is false,
| |
| − | |
| |
| − | | in other words, that just one of
| |
| − | |
| |
| − | | p, q, r, ... is true.
| |
| − |
| |
| − | Thus we have an efficient means to express and to enforce
| |
| − | a partition of the space of models, in effect, to maintain
| |
| − | the condition that a number of features or propositions are
| |
| − | to be held in mutually exclusive and exhaustive disjunction.
| |
| − | This supplies a much needed bridge between the binary domain
| |
| − | of two values and any other domain with a finite number of
| |
| − | feature values.
| |
| − |
| |
| − | Another variation on this theme allows one to maintain the
| |
| − | subsumption of many separate species under an explicit genus.
| |
| − | To see this, let us examine the following form of expression:
| |
| − |
| |
| − | ( q , ( q_1 ) , ( q_2 ) , ( q_3 ) ).
| |
| − |
| |
| − | Now consider what it would mean for this to be true. We see two cases:
| |
| − |
| |
| − | 1. If the proposition q is true, then exactly one of the
| |
| − | propositions (q_1), (q_2), (q_3) must be false, and so
| |
| − | just one of the propositions q_1, q_2, q_3 must be true.
| |
| − |
| |
| − | 2. If the proposition q is false, then every one of the
| |
| − | propositions (q_1), (q_2), (q_2) must be true, and so
| |
| − | each one of the propositions q_1, q_2, q_3 must be false.
| |
| − | In short, if q is false then all of the other q's are also.
| |
| − |
| |
| − | Figures 1 and 2 illustrate this type of situation.
| |
| − |
| |
| − | Figure 1 is the venn diagram of a 4-dimensional universe of discourse
| |
| − | X = [q, q_1, q_2, q_3], conventionally named after the gang of four
| |
| − | logical features that generate it. Strictly speaking, X is made up
| |
| − | of two layers, the position space X of abstract type %B%^4, and the
| |
| − | proposition space X^ = (X -> %B%) of abstract type %B%^4 -> %B%,
| |
| − | but it is commonly lawful enough to sign the signature of both
| |
| − | spaces with the same X, and thus to give the power of attorney
| |
| − | for the propositions to the so-indicted position space thereof.
| |
| − |
| |
| − | Figure 1 also makes use of the convention whereby the regions
| |
| − | or the subsets of the universe of discourse that correspond
| |
| − | to the basic features q, q_1, q_2, q_3 are labelled with
| |
| − | the parallel set of upper case letters Q, Q_1, Q_2, Q_3.
| |
| − |
| |
| − | | o
| |
| − | | / \
| |
| − | | / \
| |
| − | | / \
| |
| − | | / \
| |
| − | | o o
| |
| − | | /%\ /%\
| |
| − | | /%%%\ /%%%\
| |
| − | | /%%%%%\ /%%%%%\
| |
| − | | /%%%%%%%\ /%%%%%%%\
| |
| − | | o%%%%%%%%%o%%%%%%%%%o
| |
| − | | / \%%%%%%%/ \%%%%%%%/ \
| |
| − | | / \%%%%%/ \%%%%%/ \
| |
| − | | / \%%%/ \%%%/ \
| |
| − | | / \%/ \%/ \
| |
| − | | o o o o
| |
| − | | / \ /%\ / \ / \
| |
| − | | / \ /%%%\ / \ / \
| |
| − | | / \ /%%%%%\ / \ / \
| |
| − | | / \ /%%%%%%%\ / \ / \
| |
| − | | o o%%%%%%%%%o o o
| |
| − | | ·\ / \%%%%%%%/ \ / \ /·
| |
| − | | · \ / \%%%%%/ \ / \ / ·
| |
| − | | · \ / \%%%/ \ / \ / ·
| |
| − | | · \ / \%/ \ / \ / ·
| |
| − | | · o o o o ·
| |
| − | | · ·\ / \ / \ /· ·
| |
| − | | · · \ / \ / \ / · ·
| |
| − | | · · \ / \ / \ / · ·
| |
| − | | · Q · \ / \ / \ / ·Q_3 ·
| |
| − | | ··········o o o··········
| |
| − | | · \ /%\ / ·
| |
| − | | · \ /%%%\ / ·
| |
| − | | · \ /%%%%%\ / ·
| |
| − | | · Q_1 \ /%%%%%%%\ / Q_2 ·
| |
| − | | ··········o%%%%%%%%%o··········
| |
| − | | \%%%%%%%/
| |
| − | | \%%%%%/
| |
| − | | \%%%/
| |
| − | | \%/
| |
| − | | o
| |
| − | |
| |
| − | | Figure 1. Genus Q and Species Q_1, Q_2, Q_3
| |
| − |
| |
| − | Figure 2 is another form of venn diagram that one often uses,
| |
| − | where one collapses the unindited cells and leaves only the
| |
| − | models of the proposition in question. Some people would
| |
| − | call the transformation that changes from the first form
| |
| − | to the next form an operation of "taking the quotient",
| |
| − | but I tend to think of it as the "soap bubble picture"
| |
| − | or more exactly the "wire & thread & soap film" model
| |
| − | of the universe of discourse, where one pops out of
| |
| − | consideration the sections of the soap film that
| |
| − | stretch across the anti-model regions of space.
| |
| − |
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | X |
| |
| − | | |
| |
| − | | o |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | o Q_1 o |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / Q \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / Q_2 | Q_3 \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | o-----------------o-----------------o |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − |
| |
| − | Figure 2. Genus Q and Species Q_1, Q_2, Q_3
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Example. Partition, Genus & Species (cont.)
| |
| − |
| |
| − | Last time we considered in general terms how the forms
| |
| − | of complete partition and contingent partition operate
| |
| − | to maintain mutually disjoint and possibly exhaustive
| |
| − | categories of positions in a universe of discourse.
| |
| − |
| |
| − | This time we contemplate another concrete Example of
| |
| − | near minimal complexity, designed to demonstrate how
| |
| − | the forms of partition and subsumption can interact
| |
| − | in structuring a space of feature specifications.
| |
| − |
| |
| − | In this Example, we describe a universe of discourse
| |
| − | in terms of the following vocabulary of five features:
| |
| − |
| |
| − | | L. living_thing
| |
| − | |
| |
| − | | N. non_living
| |
| − | |
| |
| − | | A. animal
| |
| − | |
| |
| − | | V. vegetable
| |
| − | |
| |
| − | | M. mineral
| |
| − |
| |
| − | Let us construe these features as being subject to four constraints:
| |
| − |
| |
| − | | 1. Everything is either a living_thing or non_living, but not both.
| |
| − | |
| |
| − | | 2. Everything is either animal, vegetable, or mineral,
| |
| − | | but no two of these together.
| |
| − | |
| |
| − | | 3. A living_thing is either animal or vegetable, but not both,
| |
| − | | and everything animal or vegetable is a living_thing.
| |
| − | |
| |
| − | | 4. Everything mineral is non_living.
| |
| − |
| |
| − | These notions and constructions are expressed in the Log file shown below:
| |
| − |
| |
| − | Logical Input File
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | ( living_thing , non_living ) |
| |
| − | | |
| |
| − | | (( animal ),( vegetable ),( mineral )) |
| |
| − | | |
| |
| − | | ( living_thing ,( animal ),( vegetable )) |
| |
| − | | |
| |
| − | | ( mineral ( non_living )) |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − |
| |
| − | The cactus expression in this file is the expression
| |
| − | of a "zeroth order theory" (ZOT), one that can be
| |
| − | paraphrased in more ordinary language to say:
| |
| − |
| |
| − | Translation
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | living_thing =/= non_living |
| |
| − | | |
| |
| − | | par : all -> {animal, vegetable, mineral} |
| |
| − | | |
| |
| − | | par : living_thing -> {animal, vegetable} |
| |
| − | | |
| |
| − | | mineral => non_living |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − |
| |
| − | Here, "par : all -> {p, q, r}" is short for an assertion
| |
| − | that the universe as a whole is partitioned into subsets
| |
| − | that correspond to the features p, q, r.
| |
| − |
| |
| − | Also, "par : q -> {r, s}" asserts that "Q partitions into R and S.
| |
| − |
| |
| − | It is probably enough just to list the outputs of Model, Tenor, and Sense
| |
| − | when run on the preceding Log file. Using the same format and labeling as
| |
| − | before, we may note that Model has, from 2^5 = 32 possible interpretations,
| |
| − | made 11 evaluations, and found 3 models answering the generic descriptions
| |
| − | that were imposed by the logical input file.
| |
| − |
| |
| − | Model Outline
| |
| − | o------------------------o
| |
| − | | living_thing |
| |
| − | | non_living - | 1
| |
| − | | (non_living ) |
| |
| − | | mineral - | 2
| |
| − | | (mineral ) |
| |
| − | | animal |
| |
| − | | vegetable - | 3
| |
| − | | (vegetable ) * | 4 *
| |
| − | | (animal ) |
| |
| − | | vegetable * | 5 *
| |
| − | | (vegetable ) - | 6
| |
| − | | (living_thing ) |
| |
| − | | non_living |
| |
| − | | animal - | 7
| |
| − | | (animal ) |
| |
| − | | vegetable - | 8
| |
| − | | (vegetable ) |
| |
| − | | mineral * | 9 *
| |
| − | | (mineral ) - | 10
| |
| − | | (non_living ) - | 11
| |
| − | o------------------------o
| |
| − |
| |
| − | Tenor Outline
| |
| − | o------------------------o
| |
| − | | living_thing |
| |
| − | | (non_living ) |
| |
| − | | (mineral ) |
| |
| − | | animal |
| |
| − | | (vegetable ) * | <1>
| |
| − | | (animal ) |
| |
| − | | vegetable * | <2>
| |
| − | | (living_thing ) |
| |
| − | | non_living |
| |
| − | | (animal ) |
| |
| − | | (vegetable ) |
| |
| − | | mineral * | <3>
| |
| − | o------------------------o
| |
| − |
| |
| − | Sense Outline
| |
| − | o------------------------o
| |
| − | | living_thing |
| |
| − | | animal |
| |
| − | | vegetable |
| |
| − | | non_living |
| |
| − | | mineral |
| |
| − | o------------------------o
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Example. Molly's World
| |
| − |
| |
| − | I think that we are finally ready to tackle a more respectable example.
| |
| − | The Example known as "Molly's World" is borrowed from the literature on
| |
| − | computational learning theory, adapted with a few changes from the example
| |
| − | called "Molly’s Problem" in the paper "Learning With Hints" by Dana Angluin.
| |
| − | By way of setting up the problem, I quote Angluin's motivational description:
| |
| − |
| |
| − | | Imagine that you have become acquainted with an alien named Molly from the
| |
| − | | planet Ornot, who is currently employed in a day-care center. She is quite
| |
| − | | good at propositional logic, but a bit weak on knowledge of Earth. So you
| |
| − | | decide to formulate the beginnings of a propositional theory to help her
| |
| − | | label things in her immediate environment.
| |
| − | |
| |
| − | | Angluin, Dana, "Learning With Hints", pages 167-181, in:
| |
| − | | David Haussler & Leonard Pitt (eds.), 'Proceedings of the 1988 Workshop
| |
| − | | on Computational Learning Theory', Morgan Kaufmann, San Mateo, CA, 1989.
| |
| − |
| |
| − | The purpose of this quaint pretext is, of course, to make sure that the
| |
| − | reader appreciates the constraints of the problem: that no extra savvy
| |
| − | is fair, all facts must be presumed or deduced on the immediate premises.
| |
| − |
| |
| − | My use of this example is not directly relevant to the purposes of the
| |
| − | discussion from which it is taken, so I simply give my version of it
| |
| − | without comment on those issues.
| |
| − |
| |
| − | Here is my rendition of the initial knowledge base delimiting Molly’s World:
| |
| − |
| |
| − | Logical Input File: Molly.Log
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | ( object ,( toy ),( vehicle )) |
| |
| − | | (( small_size ),( medium_size ),( large_size )) |
| |
| − | | (( two_wheels ),( three_wheels ),( four_wheels )) |
| |
| − | | (( no_seat ),( one_seat ),( few_seats ),( many_seats )) |
| |
| − | | ( object ,( scooter ),( bike ),( trike ),( car ),( bus ),( wagon )) |
| |
| − | | ( two_wheels no_seat ,( scooter )) |
| |
| − | | ( two_wheels one_seat pedals ,( bike )) |
| |
| − | | ( three_wheels one_seat pedals ,( trike )) |
| |
| − | | ( four_wheels few_seats doors ,( car )) |
| |
| − | | ( four_wheels many_seats doors ,( bus )) |
| |
| − | | ( four_wheels no_seat handle ,( wagon )) |
| |
| − | | ( scooter ( toy small_size )) |
| |
| − | | ( wagon ( toy small_size )) |
| |
| − | | ( trike ( toy small_size )) |
| |
| − | | ( bike small_size ( toy )) |
| |
| − | | ( bike medium_size ( vehicle )) |
| |
| − | | ( bike large_size ) |
| |
| − | | ( car ( vehicle large_size )) |
| |
| − | | ( bus ( vehicle large_size )) |
| |
| − | | ( toy ( object )) |
| |
| − | | ( vehicle ( object )) |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − |
| |
| − | All of the logical forms that are used in the preceding Log file
| |
| − | will probably be familiar from earlier discussions. The purpose
| |
| − | of one or two constructions may, however, be a little obscure,
| |
| − | so I will insert a few words of additional explanation here:
| |
| − |
| |
| − | The rule "( bike large_size )", for example, merely
| |
| − | says that nothing can be both a bike and large_size.
| |
| − |
| |
| − | The rule "( three_wheels one_seat pedals ,( trike ))" says that anything
| |
| − | with all the features of three_wheels, one_seat, and pedals is excluded
| |
| − | from being anything but a trike. In short, anything with just those
| |
| − | three features is equivalent to a trike.
| |
| − |
| |
| − | Recall that the form "( p , q )" may be interpreted to assert either
| |
| − | the exclusive disjunction or the logical inequivalence of p and q.
| |
| − |
| |
| − | The rules have been stated in this particular way simply
| |
| − | to imitate the style of rules in the reference example.
| |
| − |
| |
| − | This last point does bring up an important issue, the question
| |
| − | of "rhetorical" differences in expression and their potential
| |
| − | impact on the "pragmatics" of computation. Unfortunately,
| |
| − | I will have to abbreviate my discussion of this topic for
| |
| − | now, and only mention in passing the following facts.
| |
| − |
| |
| − | Logically equivalent expressions, even though they must lead
| |
| − | to logically equivalent normal forms, may have very different
| |
| − | characteristics when it comes to the efficiency of processing.
| |
| − |
| |
| − | For instance, consider the following four forms:
| |
| − |
| |
| − | | 1. (( p , q ))
| |
| − | |
| |
| − | | 2. ( p ,( q ))
| |
| − | |
| |
| − | | 3. (( p ), q )
| |
| − | |
| |
| − | | 4. (( p , q ))
| |
| − |
| |
| − | All of these are equally succinct ways of maintaining that
| |
| − | p is logically equivalent to q, yet each can have different
| |
| − | effects on the route that Model takes to arrive at an answer.
| |
| − | Apparently, some equalities are more equal than others.
| |
| − |
| |
| − | These effects occur partly because the algorithm chooses to make cases
| |
| − | of variables on a basis of leftmost shallowest first, but their impact
| |
| − | can be complicated by the interactions that each expression has with
| |
| − | the context that it occupies. The main lesson to take away from all
| |
| − | of this, at least, for the time being, is that it is probably better
| |
| − | not to bother too much about these problems, but just to experiment
| |
| − | with different ways of expressing equivalent pieces of information
| |
| − | until you get a sense of what works best in various situations.
| |
| − |
| |
| − | I think that you will be happy to see only the
| |
| − | ultimate Sense of Molly’s World, so here it is:
| |
| − |
| |
| − | Sense Outline: Molly.Sen
| |
| − | o------------------------o
| |
| − | | object |
| |
| − | | two_wheels |
| |
| − | | no_seat |
| |
| − | | scooter |
| |
| − | | toy |
| |
| − | | small_size |
| |
| − | | one_seat |
| |
| − | | pedals |
| |
| − | | bike |
| |
| − | | small_size |
| |
| − | | toy |
| |
| − | | medium_size |
| |
| − | | vehicle |
| |
| − | | three_wheels |
| |
| − | | one_seat |
| |
| − | | pedals |
| |
| − | | trike |
| |
| − | | toy |
| |
| − | | small_size |
| |
| − | | four_wheels |
| |
| − | | few_seats |
| |
| − | | doors |
| |
| − | | car |
| |
| − | | vehicle |
| |
| − | | large_size |
| |
| − | | many_seats |
| |
| − | | doors |
| |
| − | | bus |
| |
| − | | vehicle |
| |
| − | | large_size |
| |
| − | | no_seat |
| |
| − | | handle |
| |
| − | | wagon |
| |
| − | | toy |
| |
| − | | small_size |
| |
| − | o------------------------o
| |
| − |
| |
| − | This outline is not the Sense of the unconstrained Log file,
| |
| − | but the result of running Model with a query on the single
| |
| − | feature "object". Using this focus helps the Modeler
| |
| − | to make more relevant Sense of Molly’s World.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | DM = Douglas McDavid
| |
| − |
| |
| − | DM: This, again, is an example of how real issues of ontology are
| |
| − | so often trivialized at the expense of technicalities. I just
| |
| − | had a burger, some fries, and a Coke. I would say all that was
| |
| − | non-living and non-mineral. A virus, I believe is non-animal,
| |
| − | non-vegetable, but living (and non-mineral). Teeth, shells,
| |
| − | and bones are virtually pure mineral, but living. These are
| |
| − | the kinds of issues that are truly "ontological," in my
| |
| − | opinion. You are not the only one to push them into
| |
| − | the background as of lesser importance. See the
| |
| − | discussion of "18-wheelers" in John Sowa's book.
| |
| − |
| |
| − | it's not my example, and from you say, it's not your example either.
| |
| − | copied it out of a book or a paper somewhere, too long ago to remember.
| |
| − | i am assuming that the author or tardition from which it came must have
| |
| − | seen some kind of sense in it. tell you what, write out your own theory
| |
| − | of "what is" in so many variables, more or less, publish it in a book or
| |
| − | a paper, and then folks will tell you that they dispute each and every
| |
| − | thing that you have just said, and it won't really matter all that much
| |
| − | how complex it is or how subtle you are. that has been the way of all
| |
| − | ontology for about as long as anybody can remember or even read about.
| |
| − | me? i don't have sufficient arrogance to be an ontologist, and you
| |
| − | know that's saying a lot, as i can't even imagine a way to convince
| |
| − | myself that i believe i know "what is", really and truly for sure
| |
| − | like some folks just seem to do. so i am working to improve our
| |
| − | technical ability to do logic, which is mostly a job of shooting
| |
| − | down the more serious delusions that we often get ourselves into.
| |
| − | can i be of any use to ontologists? i dunno. i guess it depends
| |
| − | on how badly they are attached to some of the delusions of knowing
| |
| − | what their "common" sense tells them everybody ought to already know,
| |
| − | but that every attempt to check that out in detail tells them it just
| |
| − | ain't so. a problem for which denial was just begging to be invented,
| |
| − | and so it was.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Example. Molly's World (cont.)
| |
| − |
| |
| − | In preparation for a contingently possible future discussion,
| |
| − | I need to attach a few parting thoughts to the case workup
| |
| − | of Molly's World that may not seem terribly relevant to
| |
| − | the present setting, but whose pertinence I hope will
| |
| − | become clearer in time.
| |
| − |
| |
| − | The logical paradigm from which this Example was derived is that
| |
| − | of "Zeroth Order Horn Clause Theories". The clauses at issue
| |
| − | in these theories are allowed to be of just three kinds:
| |
| − |
| |
| − | | 1. p & q & r & ... => z
| |
| − | |
| |
| − | | 2. z
| |
| − | |
| |
| − | | 3. ~[p & q & r & ...]
| |
| − |
| |
| − | Here, the proposition letters "p", "q", "r", ..., "z"
| |
| − | are restricted to being single positive features, not
| |
| − | themselves negated or otherwise complex expressions.
| |
| − |
| |
| − | In the Cactus Language or Existential Graph syntax
| |
| − | these forms would take on the following appearances:
| |
| − |
| |
| − | | 1. ( p q r ... ( z ))
| |
| − | |
| |
| − | | 2. z
| |
| − | |
| |
| − | | 3. ( p q r ... )
| |
| − |
| |
| − | The style of deduction in Horn clause logics is essentially
| |
| − | proof-theoretic in character, with the main burden of proof
| |
| − | falling on implication relations ("=>") and on "projective"
| |
| − | forms of inference, that is, information-losing inferences
| |
| − | like modus ponens and resolution. Cf. [Llo], [MaW].
| |
| − |
| |
| − | In contrast, the method used here is substantially model-theoretic,
| |
| − | the stress being to start from more general forms of expression for
| |
| − | laying out facts (for example, distinctions, equations, partitions)
| |
| − | and to work toward results that maintain logical equivalence with
| |
| − | their origins.
| |
| − |
| |
| − | What all of this has to do with the output above is this:
| |
| − | >From the perspective that is adopted in the present work,
| |
| − | almost any theory, for example, the one that is founded
| |
| − | on the postulates of Molly's World, will have far more
| |
| − | models than the implicational and inferential mode of
| |
| − | reasoning is designed to discover. We will be forced
| |
| − | to confront them, however, if we try to run Model on
| |
| − | a large set of implications.
| |
| − |
| |
| − | The typical Horn clause interpreter gets around this
| |
| − | difficulty only by a stratagem that takes clauses to
| |
| − | mean something other than what they say, that is, by
| |
| − | distorting the principles of semantics in practice.
| |
| − | Our Model, on the other hand, has no such finesse.
| |
| − |
| |
| − | This explains why it was necessary to impose the
| |
| − | prerequisite "object" constraint on the Log file
| |
| − | for Molly's World. It supplied no more than what
| |
| − | we usually take for granted, in order to obtain
| |
| − | a set of models that we would normally think of
| |
| − | as being the intended import of the definitions.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Example. Jets & Sharks
| |
| − |
| |
| − | The propositional calculus based on the boundary operator, that is,
| |
| − | the multigrade logical connective of the form "( , , , ... )" can be
| |
| − | interpreted in a way that resembles the logic of activation states and
| |
| − | competition constraints in certain neural network models. One way to do
| |
| − | this is by interpreting the blank or unmarked state as the resting state
| |
| − | of a neural pool, the bound or marked state as its activated state, and
| |
| − | by representing a mutually inhibitory pool of neurons p, q, r by means
| |
| − | of the expression "( p , q , r )". To illustrate this possibility,
| |
| − | I transcribe into cactus language expressions a notorious example
| |
| − | from the "parallel distributed processing" (PDP) paradigm [McR]
| |
| − | and work through two of the associated exercises as portrayed
| |
| − | in this format.
| |
| − |
| |
| − | Logical Input File: JAS = ZOT(Jets And Sharks)
| |
| − | o----------------------------------------------------------------o
| |
| − | | |
| |
| − | | (( art ),( al ),( sam ),( clyde ),( mike ), |
| |
| − | | ( jim ),( greg ),( john ),( doug ),( lance ), |
| |
| − | | ( george ),( pete ),( fred ),( gene ),( ralph ), |
| |
| − | | ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), |
| |
| − | | ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) |
| |
| − | | |
| |
| − | | ( jets , sharks ) |
| |
| − | | |
| |
| − | | ( jets , |
| |
| − | | ( art ),( al ),( sam ),( clyde ),( mike ), |
| |
| − | | ( jim ),( greg ),( john ),( doug ),( lance ), |
| |
| − | | ( george ),( pete ),( fred ),( gene ),( ralph )) |
| |
| − | | |
| |
| − | | ( sharks , |
| |
| − | | ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), |
| |
| − | | ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) |
| |
| − | | |
| |
| − | | (( 20's ),( 30's ),( 40's )) |
| |
| − | | |
| |
| − | | ( 20's , |
| |
| − | | ( sam ),( jim ),( greg ),( john ),( lance ), |
| |
| − | | ( george ),( pete ),( fred ),( gene ),( ken )) |
| |
| − | | |
| |
| − | | ( 30's , |
| |
| − | | ( al ),( mike ),( doug ),( ralph ), |
| |
| − | | ( phil ),( ike ),( nick ),( don ), |
| |
| − | | ( ned ),( rick ),( ol ),( neal ),( dave )) |
| |
| − | | |
| |
| − | | ( 40's , |
| |
| − | | ( art ),( clyde ),( karl ),( earl )) |
| |
| − | | |
| |
| − | | (( junior_high ),( high_school ),( college )) |
| |
| − | | |
| |
| − | | ( junior_high , |
| |
| − | | ( art ),( al ),( clyde ),( mike ),( jim ), |
| |
| − | | ( john ),( lance ),( george ),( ralph ),( ike )) |
| |
| − | | |
| |
| − | | ( high_school , |
| |
| − | | ( greg ),( doug ),( pete ),( fred ),( nick ), |
| |
| − | | ( karl ),( ken ),( earl ),( rick ),( neal ),( dave )) |
| |
| − | | |
| |
| − | | ( college , |
| |
| − | | ( sam ),( gene ),( phil ),( don ),( ned ),( ol )) |
| |
| − | | |
| |
| − | | (( single ),( married ),( divorced )) |
| |
| − | | |
| |
| − | | ( single , |
| |
| − | | ( art ),( sam ),( clyde ),( mike ), |
| |
| − | | ( doug ),( pete ),( fred ),( gene ), |
| |
| − | | ( ralph ),( ike ),( nick ),( ken ),( neal )) |
| |
| − | | |
| |
| − | | ( married , |
| |
| − | | ( al ),( greg ),( john ),( lance ),( phil ), |
| |
| − | | ( don ),( ned ),( karl ),( earl ),( ol )) |
| |
| − | | |
| |
| − | | ( divorced , |
| |
| − | | ( jim ),( george ),( rick ),( dave )) |
| |
| − | | |
| |
| − | | (( bookie ),( burglar ),( pusher )) |
| |
| − | | |
| |
| − | | ( bookie , |
| |
| − | | ( sam ),( clyde ),( mike ),( doug ), |
| |
| − | | ( pete ),( ike ),( ned ),( karl ),( neal )) |
| |
| − | | |
| |
| − | | ( burglar , |
| |
| − | | ( al ),( jim ),( john ),( lance ), |
| |
| − | | ( george ),( don ),( ken ),( earl ),( rick )) |
| |
| − | | |
| |
| − | | ( pusher , |
| |
| − | | ( art ),( greg ),( fred ),( gene ), |
| |
| − | | ( ralph ),( phil ),( nick ),( ol ),( dave )) |
| |
| − | | |
| |
| − | o----------------------------------------------------------------o
| |
| − |
| |
| − | We now apply Study to the proposition that
| |
| − | defines the Jets and Sharks knowledge base,
| |
| − | that is to say, the knowledge that we are
| |
| − | given about the Jets and Sharks, not the
| |
| − | knowledge that the Jets and Sharks have.
| |
| − |
| |
| − | With a query on the name "ken" we obtain the following
| |
| − | output, giving all of the features associated with Ken:
| |
| − |
| |
| − | Sense Outline: JAS & Ken
| |
| − | o---------------------------------------o
| |
| − | | ken |
| |
| − | | sharks |
| |
| − | | 20's |
| |
| − | | high_school |
| |
| − | | single |
| |
| − | | burglar |
| |
| − | o---------------------------------------o
| |
| − |
| |
| − | With a query on the two features "college" and "sharks"
| |
| − | we obtain the following outline of all of the features
| |
| − | that satisfy these constraints:
| |
| − |
| |
| − | Sense Outline: JAS & College & Sharks
| |
| − | o---------------------------------------o
| |
| − | | college |
| |
| − | | sharks |
| |
| − | | 30's |
| |
| − | | married |
| |
| − | | bookie |
| |
| − | | ned |
| |
| − | | burglar |
| |
| − | | don |
| |
| − | | pusher |
| |
| − | | phil |
| |
| − | | ol |
| |
| − | o---------------------------------------o
| |
| − |
| |
| − | >From this we discover that all college Sharks
| |
| − | are 30-something and married. Furthermore,
| |
| − | we have a complete listing of their names
| |
| − | broken down by occupation, as I have no
| |
| − | doubt that all of them will be in time.
| |
| − |
| |
| − | | Reference:
| |
| − | |
| |
| − | | McClelland, James L. & Rumelhart, David E.,
| |
| − | |'Explorations in Parallel Distributed Processing:
| |
| − | | A Handbook of Models, Programs, and Exercises',
| |
| − | | MIT Press, Cambridge, MA, 1988.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | One of the issues that my pondering weak and weary over
| |
| − | has caused me to burn not a few barrels of midnight oil
| |
| − | over the past elventeen years or so is the relationship
| |
| − | among divers and sundry "styles of inference", by which
| |
| − | I mean particular choices of inference paradigms, rules,
| |
| − | or schemata. The chief breakpoint seems to lie between
| |
| − | information-losing and information-maintaining modes of
| |
| − | inference, also called "implicational" and "equational",
| |
| − | or "projective" and "preservative" brands, respectively.
| |
| − |
| |
| − | Since it appears to be mostly the implicational and projective
| |
| − | styles of inference that are more familiar to folks hereabouts,
| |
| − | I will start off this subdiscussion by introducing a number of
| |
| − | risibly simple but reasonably manageable examples of the other
| |
| − | brand of inference, treated as equational reasoning approaches
| |
| − | to problems about satisfying "zeroth order constraints" (ZOC's).
| |
| − |
| |
| − | Applications of a Propositional Calculator:
| |
| − | Constraint Satisfaction Problems.
| |
| − | Jon Awbrey, April 24, 1995.
| |
| − |
| |
| − | The Four Houses Puzzle
| |
| − |
| |
| − | Constructed on the model of the "Five Houses Puzzle" in [VaH, 132-136].
| |
| − |
| |
| − | Problem Statement. Four people with different nationalities live in the
| |
| − | first four houses of a street. They practice four distinct professions,
| |
| − | and each of them has a favorite animal, all of them different. The four
| |
| − | houses are painted different colors. The following facts are known:
| |
| − |
| |
| − | | 1. The Englander lives in the first house on the left.
| |
| − | | 2. The doctor lives in the second house.
| |
| − | | 3. The third house is painted red.
| |
| − | | 4. The zebra is a favorite in the fourth house.
| |
| − | | 5. The person in the first house has a dog.
| |
| − | | 6. The Japanese lives in the third house.
| |
| − | | 7. The red house is on the left of the yellow one.
| |
| − | | 8. They breed snails in the house to right of the doctor.
| |
| − | | 9. The Englander lives next to the green house.
| |
| − | | 10. The fox is in the house next to to the diplomat.
| |
| − | | 11. The Spaniard likes zebras.
| |
| − | | 12. The Japanese is a painter.
| |
| − | | 13. The Italian lives in the green house.
| |
| − | | 14. The violinist lives in the yellow house.
| |
| − | | 15. The dog is a pet in the blue house.
| |
| − | | 16. The doctor keeps a fox.
| |
| − |
| |
| − | The problem is to find all of the assignments of
| |
| − | features to houses that satisfy these requirements.
| |
| − |
| |
| − | Logical Input File: House^4.Log
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | eng_1 doc_2 red_3 zeb_4 dog_1 jap_3 |
| |
| − | | |
| |
| − | | (( red_1 yel_2 ),( red_2 yel_3 ),( red_3 yel_4 )) |
| |
| − | | (( doc_1 sna_2 ),( doc_2 sna_3 ),( doc_3 sna_4 )) |
| |
| − | | |
| |
| − | | (( eng_1 gre_2 ), |
| |
| − | | ( eng_2 gre_3 ),( eng_2 gre_1 ), |
| |
| − | | ( eng_3 gre_4 ),( eng_3 gre_2 ), |
| |
| − | | ( eng_4 gre_3 )) |
| |
| − | | |
| |
| − | | (( dip_1 fox_2 ), |
| |
| − | | ( dip_2 fox_3 ),( dip_2 fox_1 ), |
| |
| − | | ( dip_3 fox_4 ),( dip_3 fox_2 ), |
| |
| − | | ( dip_4 fox_3 )) |
| |
| − | | |
| |
| − | | (( spa_1 zeb_1 ),( spa_2 zeb_2 ),( spa_3 zeb_3 ),( spa_4 zeb_4 )) |
| |
| − | | (( jap_1 pai_1 ),( jap_2 pai_2 ),( jap_3 pai_3 ),( jap_4 pai_4 )) |
| |
| − | | (( ita_1 gre_1 ),( ita_2 gre_2 ),( ita_3 gre_3 ),( ita_4 gre_4 )) |
| |
| − | | |
| |
| − | | (( yel_1 vio_1 ),( yel_2 vio_2 ),( yel_3 vio_3 ),( yel_4 vio_4 )) |
| |
| − | | (( blu_1 dog_1 ),( blu_2 dog_2 ),( blu_3 dog_3 ),( blu_4 dog_4 )) |
| |
| − | | |
| |
| − | | (( doc_1 fox_1 ),( doc_2 fox_2 ),( doc_3 fox_3 ),( doc_4 fox_4 )) |
| |
| − | | |
| |
| − | | (( |
| |
| − | | |
| |
| − | | (( eng_1 ),( eng_2 ),( eng_3 ),( eng_4 )) |
| |
| − | | (( spa_1 ),( spa_2 ),( spa_3 ),( spa_4 )) |
| |
| − | | (( jap_1 ),( jap_2 ),( jap_3 ),( jap_4 )) |
| |
| − | | (( ita_1 ),( ita_2 ),( ita_3 ),( ita_4 )) |
| |
| − | | |
| |
| − | | (( eng_1 ),( spa_1 ),( jap_1 ),( ita_1 )) |
| |
| − | | (( eng_2 ),( spa_2 ),( jap_2 ),( ita_2 )) |
| |
| − | | (( eng_3 ),( spa_3 ),( jap_3 ),( ita_3 )) |
| |
| − | | (( eng_4 ),( spa_4 ),( jap_4 ),( ita_4 )) |
| |
| − | | |
| |
| − | | (( gre_1 ),( gre_2 ),( gre_3 ),( gre_4 )) |
| |
| − | | (( red_1 ),( red_2 ),( red_3 ),( red_4 )) |
| |
| − | | (( yel_1 ),( yel_2 ),( yel_3 ),( yel_4 )) |
| |
| − | | (( blu_1 ),( blu_2 ),( blu_3 ),( blu_4 )) |
| |
| − | | |
| |
| − | | (( gre_1 ),( red_1 ),( yel_1 ),( blu_1 )) |
| |
| − | | (( gre_2 ),( red_2 ),( yel_2 ),( blu_2 )) |
| |
| − | | (( gre_3 ),( red_3 ),( yel_3 ),( blu_3 )) |
| |
| − | | (( gre_4 ),( red_4 ),( yel_4 ),( blu_4 )) |
| |
| − | | |
| |
| − | | (( pai_1 ),( pai_2 ),( pai_3 ),( pai_4 )) |
| |
| − | | (( dip_1 ),( dip_2 ),( dip_3 ),( dip_4 )) |
| |
| − | | (( vio_1 ),( vio_2 ),( vio_3 ),( vio_4 )) |
| |
| − | | (( doc_1 ),( doc_2 ),( doc_3 ),( doc_4 )) |
| |
| − | | |
| |
| − | | (( pai_1 ),( dip_1 ),( vio_1 ),( doc_1 )) |
| |
| − | | (( pai_2 ),( dip_2 ),( vio_2 ),( doc_2 )) |
| |
| − | | (( pai_3 ),( dip_3 ),( vio_3 ),( doc_3 )) |
| |
| − | | (( pai_4 ),( dip_4 ),( vio_4 ),( doc_4 )) |
| |
| − | | |
| |
| − | | (( dog_1 ),( dog_2 ),( dog_3 ),( dog_4 )) |
| |
| − | | (( zeb_1 ),( zeb_2 ),( zeb_3 ),( zeb_4 )) |
| |
| − | | (( fox_1 ),( fox_2 ),( fox_3 ),( fox_4 )) |
| |
| − | | (( sna_1 ),( sna_2 ),( sna_3 ),( sna_4 )) |
| |
| − | | |
| |
| − | | (( dog_1 ),( zeb_1 ),( fox_1 ),( sna_1 )) |
| |
| − | | (( dog_2 ),( zeb_2 ),( fox_2 ),( sna_2 )) |
| |
| − | | (( dog_3 ),( zeb_3 ),( fox_3 ),( sna_3 )) |
| |
| − | | (( dog_4 ),( zeb_4 ),( fox_4 ),( sna_4 )) |
| |
| − | | |
| |
| − | | )) |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − |
| |
| − | Sense Outline: House^4.Sen
| |
| − | o-----------------------------o
| |
| − | | eng_1 |
| |
| − | | doc_2 |
| |
| − | | red_3 |
| |
| − | | zeb_4 |
| |
| − | | dog_1 |
| |
| − | | jap_3 |
| |
| − | | yel_4 |
| |
| − | | sna_3 |
| |
| − | | gre_2 |
| |
| − | | dip_1 |
| |
| − | | fox_2 |
| |
| − | | spa_4 |
| |
| − | | pai_3 |
| |
| − | | ita_2 |
| |
| − | | vio_4 |
| |
| − | | blu_1 |
| |
| − | o-----------------------------o
| |
| − |
| |
| − | Table 1. Solution to the Four Houses Puzzle
| |
| − | o------------o------------o------------o------------o------------o
| |
| − | | | House 1 | House 2 | House 3 | House 4 |
| |
| − | o------------o------------o------------o------------o------------o
| |
| − | | Nation | England | Italy | Japan | Spain |
| |
| − | | Color | blue | green | red | yellow |
| |
| − | | Profession | diplomat | doctor | painter | violinist |
| |
| − | | Animal | dog | fox | snails | zebra |
| |
| − | o------------o------------o------------o------------o------------o
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | First off, I do not trivialize the "real issues of ontology", indeed,
| |
| − | it is precisely my estimate of the non-trivial difficulty of this task,
| |
| − | of formulating the types of "generic ontology" that we propose to do here,
| |
| − | that forces me to choose and to point out the inescapability of the approach
| |
| − | that I am currently taking, which is to enter on the necessary preliminary of
| |
| − | building up the logical tools that we need to tackle the ontology task proper.
| |
| − | And I would say, to the contrary, that it is those who think we can arrive at
| |
| − | a working general ontology by sitting on the porch shooting the breeze about
| |
| − | "what it is" until the cows come home -- that is, the method for which it
| |
| − | has become cliche to indict the Ancient Greeks, though, if truth be told,
| |
| − | we'd have to look to the pre-socratics and the pre-stoics to find a good
| |
| − | match for the kinds of revelation that are common hereabouts -- I would
| |
| − | say that it's those folks who trivialize the "real issues of ontology".
| |
| − |
| |
| − | A person, living in our times, who is serious about knowing the being of things,
| |
| − | really only has one choice -- to pick what tiny domain of things he or she just
| |
| − | has to know about the most, thence to hie away to the adept gurus of the matter
| |
| − | in question, forgeting the rest, cause "general ontology" is a no-go these days.
| |
| − | It is presently in a state like astronomy before telescopes, and that means not
| |
| − | entirely able to discern itself from astrology and other psychically projective
| |
| − | exercises of wishful and dreadful thinking like that.
| |
| − |
| |
| − | So I am busy grinding lenses ...
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | DM = Douglas McDavid
| |
| − |
| |
| − | DM: Thanks for both the original and additional response. I'm not trying to
| |
| − | single you out, as I have been picking on various postings in a similar
| |
| − | manner ever since I started contributing to this discussion. I agree with
| |
| − | you that the task of this working group is non-trivially difficult. In fact,
| |
| − | I believe we are still a long way from a clear and useful agreement about what
| |
| − | constitutes "upper" ontology, and what it would mean to standardize it. However,
| |
| − | I don't agree that the only place to make progress is in tiny domains of things.
| |
| − | I've contributed the thought that a fundamental, upper-level concept is the
| |
| − | concept of system, and that that would be a good place to begin. And I'll
| |
| − | never be able to refrain from evaluating the content as well as the form
| |
| − | of any examples presented for consideration here. Probably should
| |
| − | accompany these comments with a ;-)
| |
| − |
| |
| − | There will never be a standard universal ontology
| |
| − | of the absolute essential impertubable monolithic
| |
| − | variety that some people still dream of in their
| |
| − | fantasies of spectating on and speculating about
| |
| − | a pre-relativistically non-participatory universe
| |
| − | from their singular but isolated gods'eye'views.
| |
| − | The bells tolled for that one many years ago,
| |
| − | but some of the more blithe of the blissful
| |
| − | islanders have just not gotten the news yet.
| |
| − |
| |
| − | But there is still a lot to do that would be useful
| |
| − | under the banner of a "standard upper ontology",
| |
| − | if only we stay loose in our interpretation
| |
| − | of what that implies in practical terms.
| |
| − |
| |
| − | One likely approach to the problem would be to take
| |
| − | a hint from the afore-allusioned history of physics --
| |
| − | to inquire for whom, else, the bell tolls -- and to
| |
| − | see if there are any bits of wisdom from that prior
| |
| − | round of collective experience that can be adapted
| |
| − | by dint of analogy to our present predicament.
| |
| − | I happen to think that there are.
| |
| − |
| |
| − | And there the answer was, not to try and force a return,
| |
| − | though lord knows they all gave it their very best shot,
| |
| − | to an absolute and imperturbable framework of existence,
| |
| − | but to see the reciprocal participant relation that all
| |
| − | partakers have to the constitution of that framing, yes,
| |
| − | even unto those who would abdictators and abstainees be.
| |
| − |
| |
| − | But what does that imply about some shred of a standard?
| |
| − | It means that we are better off seeking, not a standard,
| |
| − | one-size-fits-all ontology, but more standard resources
| |
| − | for trying to interrelate diverse points of view and to
| |
| − | transform the data that's gathered from one perspective
| |
| − | in ways that it can most appropriately be compared with
| |
| − | the data that is gathered from other standpoints on the
| |
| − | splendorous observational scenes and theorematic stages.
| |
| − |
| |
| − | That is what I am working on.
| |
| − | And it hasn't been merely
| |
| − | for a couple of years.
| |
| − |
| |
| − | As to this bit:
| |
| − |
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | ( living_thing , non_living ) |
| |
| − | | |
| |
| − | | (( animal ),( vegetable ),( mineral )) |
| |
| − | | |
| |
| − | | ( living_thing ,( animal ),( vegetable )) |
| |
| − | | |
| |
| − | | ( mineral ( non_living )) |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − |
| |
| − | My 5-dimensional Example, that I borrowed from some indifferent source
| |
| − | of what is commonly recognized as "common sense" -- and I think rather
| |
| − | obviously designed more for the classification of pre-modern species
| |
| − | of whole critters and pure matters of natural substance than the
| |
| − | motley mixture of un/natural and in/organic conglouterites that
| |
| − | we find served up on the menu of modernity -- was not intended
| |
| − | even so much as a toy ontology, but simply as an expository
| |
| − | example, concocted for the sake of illustrating the sorts
| |
| − | of logical interaction that occur among four different
| |
| − | patterns of logical constraint, all of which types
| |
| − | arise all the time no matter what the domain, and
| |
| − | which I believe that my novel forms of expression,
| |
| − | syntactically speaking, express quite succinctly,
| |
| − | especially when you contemplate the complexities
| |
| − | of the computation that may flow and must follow
| |
| − | from even these meagre propositional expressions.
| |
| − |
| |
| − | Yes, systems -- but -- even here usage differs in significant ways.
| |
| − | I have spent ten years now trying to integrate my earlier efforts
| |
| − | under an explicit systems banner, but even within the bounds of
| |
| − | a systems engineering programme at one site there is a wide
| |
| − | semantic dispersion that issues from this word "system".
| |
| − | I am committed, and in writing, to taking what we so
| |
| − | glibly and prospectively call "intelligent systems"
| |
| − | seriously as dynamical systems. That has many
| |
| − | consequences, and I have to pick and choose
| |
| − | which of those I may be suited to follow.
| |
| − |
| |
| − | But that is too long a story for now ...
| |
| − |
| |
| − | ";-)"?
| |
| − |
| |
| − | Somehow that has always looked like
| |
| − | the Chesshire Cat's grin to me ...
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | By way of catering to popular demand, I have decided to
| |
| − | render this symposium a bit more à la carte, and thus to
| |
| − | serve up as faster food than heretofore a choice selection
| |
| − | of the more sumptuous bits that I have in my logical larder,
| |
| − | not yet full fare, by any means, but a sample of what might
| |
| − | one day approach to being an abundantly moveable feast of
| |
| − | ontological contents and general metaphysical delights.
| |
| − | I'll leave it to you to name your poison, as it were.
| |
| − |
| |
| − | Applications of a Propositional Calculator:
| |
| − | Constraint Satisfaction Problems.
| |
| − | Jon Awbrey, April 24, 1995.
| |
| − |
| |
| − | Fabric Knowledge Base
| |
| − | Based on the example in [MaW, pages 8-16].
| |
| − |
| |
| − | Logical Input File: Fab.Log
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | (has_floats , plain_weave ) |
| |
| − | | (has_floats ,(twill_weave ),(satin_weave )) |
| |
| − | | |
| |
| − | | (plain_weave , |
| |
| − | | (plain_weave one_color ), |
| |
| − | | (color_groups ), |
| |
| − | | (grouped_warps ), |
| |
| − | | (some_thicker ), |
| |
| − | | (crossed_warps ), |
| |
| − | | (loop_threads ), |
| |
| − | | (plain_weave flannel )) |
| |
| − | | |
| |
| − | | (plain_weave one_color cotton balanced smooth ,(percale )) |
| |
| − | | (plain_weave one_color cotton sheer ,(organdy )) |
| |
| − | | (plain_weave one_color silk sheer ,(organza )) |
| |
| − | | |
| |
| − | | (plain_weave color_groups warp_stripe fill_stripe ,(plaid )) |
| |
| − | | (plaid equal_stripe ,(gingham )) |
| |
| − | | |
| |
| − | | (plain_weave grouped_warps ,(basket_weave )) |
| |
| − | | |
| |
| − | | (basket_weave typed , |
| |
| − | | (type_2_to_1 ), |
| |
| − | | (type_2_to_2 ), |
| |
| − | | (type_4_to_4 )) |
| |
| − | | |
| |
| − | | (basket_weave typed type_2_to_1 thicker_fill ,(oxford )) |
| |
| − | | (basket_weave typed (type_2_to_2 , |
| |
| − | | type_4_to_4 ) same_thickness ,(monks_cloth )) |
| |
| − | | (basket_weave (typed ) rough open ,(hopsacking )) |
| |
| − | | |
| |
| − | | (typed (basket_weave )) |
| |
| − | | |
| |
| − | | (basket_weave ,(oxford ),(monks_cloth ),(hopsacking )) |
| |
| − | | |
| |
| − | | (plain_weave some_thicker ,(ribbed_weave )) |
| |
| − | | |
| |
| − | | (ribbed_weave ,(small_rib ),(medium_rib ),(heavy_rib )) |
| |
| − | | (ribbed_weave ,(flat_rib ),(round_rib )) |
| |
| − | | |
| |
| − | | (ribbed_weave thicker_fill ,(cross_ribbed )) |
| |
| − | | (cross_ribbed small_rib flat_rib ,(faille )) |
| |
| − | | (cross_ribbed small_rib round_rib ,(grosgrain )) |
| |
| − | | (cross_ribbed medium_rib round_rib ,(bengaline )) |
| |
| − | | (cross_ribbed heavy_rib round_rib ,(ottoman )) |
| |
| − | | |
| |
| − | | (cross_ribbed ,(faille ),(grosgrain ),(bengaline ),(ottoman )) |
| |
| − | | |
| |
| − | | (plain_weave crossed_warps ,(leno_weave )) |
| |
| − | | (leno_weave open ,(marquisette )) |
| |
| − | | (plain_weave loop_threads ,(pile_weave )) |
| |
| − | | |
| |
| − | | (pile_weave ,(fill_pile ),(warp_pile )) |
| |
| − | | (pile_weave ,(cut ),(uncut )) |
| |
| − | | |
| |
| − | | (pile_weave warp_pile cut ,(velvet )) |
| |
| − | | (pile_weave fill_pile cut aligned_pile ,(corduroy )) |
| |
| − | | (pile_weave fill_pile cut staggered_pile ,(velveteen )) |
| |
| − | | (pile_weave fill_pile uncut reversible ,(terry )) |
| |
| − | | |
| |
| − | | (pile_weave fill_pile cut ( (aligned_pile , staggered_pile ) )) |
| |
| − | | |
| |
| − | | (pile_weave ,(velvet ),(corduroy ),(velveteen ),(terry )) |
| |
| − | | |
| |
| − | | (plain_weave , |
| |
| − | | (percale ),(organdy ),(organza ),(plaid ), |
| |
| − | | (oxford ),(monks_cloth ),(hopsacking ), |
| |
| − | | (faille ),(grosgrain ),(bengaline ),(ottoman ), |
| |
| − | | (leno_weave ),(pile_weave ),(plain_weave flannel )) |
| |
| − | | |
| |
| − | | (twill_weave , |
| |
| − | | (warp_faced ), |
| |
| − | | (filling_faced ), |
| |
| − | | (even_twill ), |
| |
| − | | (twill_weave flannel )) |
| |
| − | | |
| |
| − | | (twill_weave warp_faced colored_warp white_fill ,(denim )) |
| |
| − | | (twill_weave warp_faced one_color ,(drill )) |
| |
| − | | (twill_weave even_twill diagonal_rib ,(serge )) |
| |
| − | | |
| |
| − | | (twill_weave warp_faced ( |
| |
| − | | (one_color , |
| |
| − | | ((colored_warp )(white_fill )) ) |
| |
| − | | )) |
| |
| − | | |
| |
| − | | (twill_weave warp_faced ,(denim ),(drill )) |
| |
| − | | (twill_weave even_twill ,(serge )) |
| |
| − | | |
| |
| − | | (( |
| |
| − | | ( ((plain_weave )(twill_weave )) |
| |
| − | | ((cotton )(wool )) napped ,(flannel )) |
| |
| − | | )) |
| |
| − | | |
| |
| − | | (satin_weave ,(warp_floats ),(fill_floats )) |
| |
| − | | |
| |
| − | | (satin_weave ,(satin_weave smooth ),(satin_weave napped )) |
| |
| − | | (satin_weave ,(satin_weave cotton ),(satin_weave silk )) |
| |
| − | | |
| |
| − | | (satin_weave warp_floats smooth ,(satin )) |
| |
| − | | (satin_weave fill_floats smooth ,(sateen )) |
| |
| − | | (satin_weave napped cotton ,(moleskin )) |
| |
| − | | |
| |
| − | | (satin_weave ,(satin ),(sateen ),(moleskin )) |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − |
| |
| − | | Reference [MaW]
| |
| − | |
| |
| − | | Maier, David & Warren, David S.,
| |
| − | |'Computing with Logic: Logic Programming with Prolog',
| |
| − | | Benjamin/Cummings, Menlo Park, CA, 1988.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | I think that it might be a good idea to go back to a simpler example
| |
| − | of a constraint satisfaction problem, and to discuss the elements of
| |
| − | its expression as a ZOT in a less cluttered setting before advancing
| |
| − | onward once again to problems on the order of the Four Houses Puzzle.
| |
| − |
| |
| − | | Applications of a Propositional Calculator:
| |
| − | | Constraint Satisfaction Problems.
| |
| − | | Jon Awbrey, April 24, 1995.
| |
| − |
| |
| − | Graph Coloring
| |
| − |
| |
| − | Based on the discussion in [Wil, page 196].
| |
| − |
| |
| − | One is given three colors, say, orange, silver, indigo,
| |
| − | and a graph on four nodes that has the following shape:
| |
| − |
| |
| − | | 1
| |
| − | | o
| |
| − | | / \
| |
| − | | / \
| |
| − | | 4 o-----o 2
| |
| − | | \ /
| |
| − | | \ /
| |
| − | | o
| |
| − | | 3
| |
| − |
| |
| − | The problem is to color the nodes of the graph
| |
| − | in such a way that no pair of nodes that are
| |
| − | adjacent in the graph, that is, linked by
| |
| − | an edge, get the same color.
| |
| − |
| |
| − | The objective situation that is to be achieved can be represented
| |
| − | in a so-called "declarative" fashion, in effect, by employing the
| |
| − | cactus language as a very simple sort of declarative programming
| |
| − | language, and depicting the prospective solution to the problem
| |
| − | as a ZOT.
| |
| − |
| |
| − | To do this, begin by declaring the following set of
| |
| − | twelve boolean variables or "zeroth order features":
| |
| − |
| |
| − | {1_orange, 1_silver, 1_indigo,
| |
| − | 2_orange, 2_silver, 2_indigo,
| |
| − | 3_orange, 3_silver, 3_indigo,
| |
| − | 4_orange, 4_silver, 4_indigo}
| |
| − |
| |
| − | The interpretation to keep in mind will be such that
| |
| − | the feature name of the form "<node i>_<color j>"
| |
| − | says that the node i is assigned the color j.
| |
| − |
| |
| − | Logical Input File: Color.Log
| |
| − | o----------------------------------------------------------------------o
| |
| − | | |
| |
| − | | (( 1_orange ),( 1_silver ),( 1_indigo )) |
| |
| − | | (( 2_orange ),( 2_silver ),( 2_indigo )) |
| |
| − | | (( 3_orange ),( 3_silver ),( 3_indigo )) |
| |
| − | | (( 4_orange ),( 4_silver ),( 4_indigo )) |
| |
| − | | |
| |
| − | | ( 1_orange 2_orange )( 1_silver 2_silver )( 1_indigo 2_indigo ) |
| |
| − | | ( 1_orange 4_orange )( 1_silver 4_silver )( 1_indigo 4_indigo ) |
| |
| − | | ( 2_orange 3_orange )( 2_silver 3_silver )( 2_indigo 3_indigo ) |
| |
| − | | ( 2_orange 4_orange )( 2_silver 4_silver )( 2_indigo 4_indigo ) |
| |
| − | | ( 3_orange 4_orange )( 3_silver 4_silver )( 3_indigo 4_indigo ) |
| |
| − | | |
| |
| − | o----------------------------------------------------------------------o
| |
| − |
| |
| − | The first stanza of verses declares that
| |
| − | every node is assigned exactly one color.
| |
| − |
| |
| − | The second stanza of verses declares that
| |
| − | no adjacent nodes get the very same color.
| |
| − |
| |
| − | Each satisfying interpretation of this ZOT
| |
| − | that is also a program corresponds to what
| |
| − | graffitists call a "coloring" of the graph.
| |
| − |
| |
| − | Theme One's Model interpreter, when we set
| |
| − | it to work on this ZOT, will array before
| |
| − | our eyes all of the colorings of the graph.
| |
| − |
| |
| − | Sense Outline: Color.Sen
| |
| − | o-----------------------------o
| |
| − | | 1_orange |
| |
| − | | 2_silver |
| |
| − | | 3_orange |
| |
| − | | 4_indigo |
| |
| − | | 2_indigo |
| |
| − | | 3_orange |
| |
| − | | 4_silver |
| |
| − | | 1_silver |
| |
| − | | 2_orange |
| |
| − | | 3_silver |
| |
| − | | 4_indigo |
| |
| − | | 2_indigo |
| |
| − | | 3_silver |
| |
| − | | 4_orange |
| |
| − | | 1_indigo |
| |
| − | | 2_orange |
| |
| − | | 3_indigo |
| |
| − | | 4_silver |
| |
| − | | 2_silver |
| |
| − | | 3_indigo |
| |
| − | | 4_orange |
| |
| − | o-----------------------------o
| |
| − |
| |
| − | | Reference [Wil]
| |
| − | |
| |
| − | | Wilf, Herbert S.,
| |
| − | |'Algorithms and Complexity',
| |
| − | | Prentice-Hall, Englewood Cliffs, NJ, 1986.
| |
| − | |
| |
| − | | Nota Bene. There is a wrong Figure in some
| |
| − | | printings of the book, that does not match
| |
| − | | the description of the Example that is
| |
| − | | given in the text.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Let us continue to examine the properties of the cactus language
| |
| − | as a minimal style of declarative programming language. Even in
| |
| − | the likes of this zeroth order microcosm one can observe, and on
| |
| − | a good day still more clearly for the lack of other distractions,
| |
| − | many of the buzz worlds that will spring into full bloom, almost
| |
| − | as if from nowhere, to become the first order of business in the
| |
| − | latter day logical organa, plus combinators, plus lambda calculi.
| |
| − |
| |
| − | By way of homage to the classics of the art, I can hardly pass
| |
| − | this way without paying my dues to the next sample of examples.
| |
| − |
| |
| − | N Queens Problem
| |
| − |
| |
| − | I will give the ZOT that describes the N Queens Problem for N = 5,
| |
| − | since that is the most that I and my old 286 could do when last I
| |
| − | wrote up this Example.
| |
| − |
| |
| − | The problem is now to write a "zeroth order program" (ZOP) that
| |
| − | describes the following objective: To place 5 chess queens on
| |
| − | a 5 by 5 chessboard so that no queen attacks any other queen.
| |
| − |
| |
| − | It is clear that there can be at most one queen on each row
| |
| − | of the board and so by dint of regal necessity, exactly one
| |
| − | queen in each row of the desired array. This gambit allows
| |
| − | us to reduce the problem to one of picking a permutation of
| |
| − | five things in fives places, and this affords us sufficient
| |
| − | clue to begin down a likely path toward the intended object,
| |
| − | by recruiting the following phalanx of 25 logical variables:
| |
| − |
| |
| − | Literal Input File: Q5.Lit
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | q1_r1, q1_r2, q1_r3, q1_r4, q1_r5, |
| |
| − | | q2_r1, q2_r2, q2_r3, q2_r4, q2_r5, |
| |
| − | | q3_r1, q3_r2, q3_r3, q3_r4, q3_r5, |
| |
| − | | q4_r1, q4_r2, q4_r3, q4_r4, q4_r5, |
| |
| − | | q5_r1, q5_r2, q5_r3, q5_r4, q5_r5. |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − |
| |
| − | Thus we seek to define a function, of abstract type f : %B%^25 -> %B%,
| |
| − | whose fibre of truth f^(-1)(%1%) is a set of interpretations, each of
| |
| − | whose elements bears the abstract type of a point in the space %B%^25,
| |
| − | and whose reading will inform us of our desired set of configurations.
| |
| − |
| |
| − | Logical Input File: Q5.Log
| |
| − | o------------------------------------------------------------o
| |
| − | | |
| |
| − | | ((q1_r1 ),(q1_r2 ),(q1_r3 ),(q1_r4 ),(q1_r5 )) |
| |
| − | | ((q2_r1 ),(q2_r2 ),(q2_r3 ),(q2_r4 ),(q2_r5 )) |
| |
| − | | ((q3_r1 ),(q3_r2 ),(q3_r3 ),(q3_r4 ),(q3_r5 )) |
| |
| − | | ((q4_r1 ),(q4_r2 ),(q4_r3 ),(q4_r4 ),(q4_r5 )) |
| |
| − | | ((q5_r1 ),(q5_r2 ),(q5_r3 ),(q5_r4 ),(q5_r5 )) |
| |
| − | | |
| |
| − | | ((q1_r1 ),(q2_r1 ),(q3_r1 ),(q4_r1 ),(q5_r1 )) |
| |
| − | | ((q1_r2 ),(q2_r2 ),(q3_r2 ),(q4_r2 ),(q5_r2 )) |
| |
| − | | ((q1_r3 ),(q2_r3 ),(q3_r3 ),(q4_r3 ),(q5_r3 )) |
| |
| − | | ((q1_r4 ),(q2_r4 ),(q3_r4 ),(q4_r4 ),(q5_r4 )) |
| |
| − | | ((q1_r5 ),(q2_r5 ),(q3_r5 ),(q4_r5 ),(q5_r5 )) |
| |
| − | | |
| |
| − | | (( |
| |
| − | | |
| |
| − | | (q1_r1 q2_r2 )(q1_r1 q3_r3 )(q1_r1 q4_r4 )(q1_r1 q5_r5 ) |
| |
| − | | (q2_r2 q3_r3 )(q2_r2 q4_r4 )(q2_r2 q5_r5 ) |
| |
| − | | (q3_r3 q4_r4 )(q3_r3 q5_r5 ) |
| |
| − | | (q4_r4 q5_r5 ) |
| |
| − | | |
| |
| − | | (q1_r2 q2_r3 )(q1_r2 q3_r4 )(q1_r2 q4_r5 ) |
| |
| − | | (q2_r3 q3_r4 )(q2_r3 q4_r5 ) |
| |
| − | | (q3_r4 q4_r5 ) |
| |
| − | | |
| |
| − | | (q1_r3 q2_r4 )(q1_r3 q3_r5 ) |
| |
| − | | (q2_r4 q3_r5 ) |
| |
| − | | |
| |
| − | | (q1_r4 q2_r5 ) |
| |
| − | | |
| |
| − | | (q2_r1 q3_r2 )(q2_r1 q4_r3 )(q2_r1 q5_r4 ) |
| |
| − | | (q3_r2 q4_r3 )(q3_r2 q5_r4 ) |
| |
| − | | (q4_r3 q5_r4 ) |
| |
| − | | |
| |
| − | | (q3_r1 q4_r2 )(q3_r1 q5_r3 ) |
| |
| − | | (q4_r2 q5_r3 ) |
| |
| − | | |
| |
| − | | (q4_r1 q5_r2 ) |
| |
| − | | |
| |
| − | | (q1_r5 q2_r4 )(q1_r5 q3_r3 )(q1_r5 q4_r2 )(q1_r5 q5_r1 ) |
| |
| − | | (q2_r4 q3_r3 )(q2_r4 q4_r2 )(q2_r4 q5_r1 ) |
| |
| − | | (q3_r3 q4_r2 )(q3_r3 q5_r1 ) |
| |
| − | | (q4_r2 q5_r1 ) |
| |
| − | | |
| |
| − | | (q2_r5 q3_r4 )(q2_r5 q4_r3 )(q2_r5 q5_r2 ) |
| |
| − | | (q3_r4 q4_r3 )(q3_r4 q5_r2 ) |
| |
| − | | (q4_r3 q5_r2 ) |
| |
| − | | |
| |
| − | | (q3_r5 q4_r4 )(q3_r5 q5_r3 ) |
| |
| − | | (q4_r4 q5_r3 ) |
| |
| − | | |
| |
| − | | (q4_r5 q5_r4 ) |
| |
| − | | |
| |
| − | | (q1_r4 q2_r3 )(q1_r4 q3_r2 )(q1_r4 q4_r1 ) |
| |
| − | | (q2_r3 q3_r2 )(q2_r3 q4_r1 ) |
| |
| − | | (q3_r2 q4_r1 ) |
| |
| − | | |
| |
| − | | (q1_r3 q2_r2 )(q1_r3 q3_r1 ) |
| |
| − | | (q2_r2 q3_r1 ) |
| |
| − | | |
| |
| − | | (q1_r2 q2_r1 ) |
| |
| − | | |
| |
| − | | )) |
| |
| − | | |
| |
| − | o------------------------------------------------------------o
| |
| − |
| |
| − | The vanguard of this logical regiment consists of two
| |
| − | stock'a'block platoons, the pattern of whose features
| |
| − | is the usual sort of array for conveying permutations.
| |
| − | Between the stations of their respective offices they
| |
| − | serve to warrant that all of the interpretations that
| |
| − | are left standing on the field of valor at the end of
| |
| − | the day will be ones that tell of permutations 5 by 5.
| |
| − | The rest of the ruck and the runt of the mill in this
| |
| − | regimental logos are there to cover the diagonal bias
| |
| − | against attacking queens that is our protocol to suit.
| |
| − |
| |
| − | And here is the issue of the day:
| |
| − |
| |
| − | Sense Output: Q5.Sen
| |
| − | o-------------------o
| |
| − | | q1_r1 |
| |
| − | | q2_r3 |
| |
| − | | q3_r5 |
| |
| − | | q4_r2 |
| |
| − | | q5_r4 | <1>
| |
| − | | q2_r4 |
| |
| − | | q3_r2 |
| |
| − | | q4_r5 |
| |
| − | | q5_r3 | <2>
| |
| − | | q1_r2 |
| |
| − | | q2_r4 |
| |
| − | | q3_r1 |
| |
| − | | q4_r3 |
| |
| − | | q5_r5 | <3>
| |
| − | | q2_r5 |
| |
| − | | q3_r3 |
| |
| − | | q4_r1 |
| |
| − | | q5_r4 | <4>
| |
| − | | q1_r3 |
| |
| − | | q2_r1 |
| |
| − | | q3_r4 |
| |
| − | | q4_r2 |
| |
| − | | q5_r5 | <5>
| |
| − | | q2_r5 |
| |
| − | | q3_r2 |
| |
| − | | q4_r4 |
| |
| − | | q5_r1 | <6>
| |
| − | | q1_r4 |
| |
| − | | q2_r1 |
| |
| − | | q3_r3 |
| |
| − | | q4_r5 |
| |
| − | | q5_r2 | <7>
| |
| − | | q2_r2 |
| |
| − | | q3_r5 |
| |
| − | | q4_r3 |
| |
| − | | q5_r1 | <8>
| |
| − | | q1_r5 |
| |
| − | | q2_r2 |
| |
| − | | q3_r4 |
| |
| − | | q4_r1 |
| |
| − | | q5_r3 | <9>
| |
| − | | q2_r3 |
| |
| − | | q3_r1 |
| |
| − | | q4_r4 |
| |
| − | | q5_r2 | <A>
| |
| − | o-------------------o
| |
| − |
| |
| − | The number at least checks with all of the best authorities,
| |
| − | so I can breathe a sigh of relief on that account, at least.
| |
| − | I am sure that there just has to be a more clever way to do
| |
| − | this, that is to say, within the bounds of ZOT reason alone,
| |
| − | but the above is the best that I could figure out with the
| |
| − | time that I had at the time.
| |
| − |
| |
| − | References: [BaC, 166], [VaH, 122], [Wir, 143].
| |
| − |
| |
| − | [BaC] Ball, W.W. Rouse, & Coxeter, H.S.M.,
| |
| − | 'Mathematical Recreations and Essays',
| |
| − | 13th ed., Dover, New York, NY, 1987.
| |
| − |
| |
| − | [VaH] Van Hentenryck, Pascal,
| |
| − | 'Constraint Satisfaction in Logic Programming,
| |
| − | MIT Press, Cambridge, MA, 1989.
| |
| − |
| |
| − | [Wir] Wirth, Niklaus,
| |
| − | 'Algorithms + Data Structures = Programs',
| |
| − | Prentice-Hall, Englewood Cliffs, NJ, 1976.
| |
| − |
| |
| − | http://mathworld.wolfram.com/QueensProblem.html
| |
| − | http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000170
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | I turn now to another golden oldie of a constraint satisfaction problem
| |
| − | that I would like to give here a slightly new spin, but not so much for
| |
| − | the sake of these trifling novelties as from a sense of old time's ache
| |
| − | and a duty to -- well, what's the opposite of novelty?
| |
| − |
| |
| − | Phobic Apollo
| |
| − |
| |
| − | | Suppose Peter, Paul, and Jane are musicians. One of them plays
| |
| − | | saxophone, another plays guitar, and the third plays drums. As
| |
| − | | it happens, one of them is afraid of things associated with the
| |
| − | | number 13, another of them is afraid of cats, and the third is
| |
| − | | afraid of heights. You also know that Peter and the guitarist
| |
| − | | skydive, that Paul and the saxophone player enjoy cats, and
| |
| − | | that the drummer lives in apartment 13 on the 13th floor.
| |
| − | |
| |
| − | | Soon we will want to use these facts to reason
| |
| − | | about whether or not certain identity relations
| |
| − | | hold or are excluded. Assume X(Peter, Guitarist)
| |
| − | | means "the person who is Peter is not the person who
| |
| − | | plays the guitar". In this notation, the facts become:
| |
| − | |
| |
| − | | 1. X(Peter, Guitarist)
| |
| − | | 2. X(Peter, Fears Heights)
| |
| − | | 3. X(Guitarist, Fears Heights)
| |
| − | | 4. X(Paul, Fears Cats)
| |
| − | | 5. X(Paul, Saxophonist)
| |
| − | | 6. X(Saxophonist, Fears Cats)
| |
| − | | 7. X(Drummer, Fears 13)
| |
| − | | 8. X(Drummer, Fears Heights)
| |
| − | |
| |
| − | | Exercise attributed to Kenneth D. Forbus, pages 449-450 in:
| |
| − | | Patrick Henry Winston, 'Artificial Intelligence', 2nd ed.,
| |
| − | | Addison-Wesley, Reading, MA, 1984.
| |
| − |
| |
| − | Here is one way to represent these facts in the form of a ZOT
| |
| − | and use it as a logical program to draw a succinct conclusion:
| |
| − |
| |
| − | Logical Input File: ConSat.Log
| |
| − | o-----------------------------------------------------------------------o
| |
| − | | |
| |
| − | | (( pete_plays_guitar ),( pete_plays_sax ),( pete_plays_drums )) |
| |
| − | | (( paul_plays_guitar ),( paul_plays_sax ),( paul_plays_drums )) |
| |
| − | | (( jane_plays_guitar ),( jane_plays_sax ),( jane_plays_drums )) |
| |
| − | | |
| |
| − | | (( pete_plays_guitar ),( paul_plays_guitar ),( jane_plays_guitar )) |
| |
| − | | (( pete_plays_sax ),( paul_plays_sax ),( jane_plays_sax )) |
| |
| − | | (( pete_plays_drums ),( paul_plays_drums ),( jane_plays_drums )) |
| |
| − | | |
| |
| − | | (( pete_fears_13 ),( pete_fears_cats ),( pete_fears_height )) |
| |
| − | | (( paul_fears_13 ),( paul_fears_cats ),( paul_fears_height )) |
| |
| − | | (( jane_fears_13 ),( jane_fears_cats ),( jane_fears_height )) |
| |
| − | | |
| |
| − | | (( pete_fears_13 ),( paul_fears_13 ),( jane_fears_13 )) |
| |
| − | | (( pete_fears_cats ),( paul_fears_cats ),( jane_fears_cats )) |
| |
| − | | (( pete_fears_height ),( paul_fears_height ),( jane_fears_height )) |
| |
| − | | |
| |
| − | | (( |
| |
| − | | |
| |
| − | | ( pete_plays_guitar ) |
| |
| − | | ( pete_fears_height ) |
| |
| − | | |
| |
| − | | ( pete_plays_guitar pete_fears_height ) |
| |
| − | | ( paul_plays_guitar paul_fears_height ) |
| |
| − | | ( jane_plays_guitar jane_fears_height ) |
| |
| − | | |
| |
| − | | ( paul_fears_cats ) |
| |
| − | | ( paul_plays_sax ) |
| |
| − | | |
| |
| − | | ( pete_plays_sax pete_fears_cats ) |
| |
| − | | ( paul_plays_sax paul_fears_cats ) |
| |
| − | | ( jane_plays_sax jane_fears_cats ) |
| |
| − | | |
| |
| − | | ( pete_plays_drums pete_fears_13 ) |
| |
| − | | ( paul_plays_drums paul_fears_13 ) |
| |
| − | | ( jane_plays_drums jane_fears_13 ) |
| |
| − | | |
| |
| − | | ( pete_plays_drums pete_fears_height ) |
| |
| − | | ( paul_plays_drums paul_fears_height ) |
| |
| − | | ( jane_plays_drums jane_fears_height ) |
| |
| − | | |
| |
| − | | )) |
| |
| − | | |
| |
| − | o-----------------------------------------------------------------------o
| |
| − |
| |
| − | Sense Outline: ConSat.Sen
| |
| − | o-----------------------------o
| |
| − | | pete_plays_drums |
| |
| − | | paul_plays_guitar |
| |
| − | | jane_plays_sax |
| |
| − | | pete_fears_cats |
| |
| − | | paul_fears_13 |
| |
| − | | jane_fears_height |
| |
| − | o-----------------------------o
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Phobic Apollo (cont.)
| |
| − |
| |
| − | It might be instructive to review various aspects
| |
| − | of how the Theme One Study function actually went
| |
| − | about arriving at its answer to that last problem.
| |
| − | Just to prove that my program and I really did do
| |
| − | our homework on that Phobic Apollo ConSat problem,
| |
| − | and didn't just provoke some Oracle or other data
| |
| − | base server to give it away, here is the middling
| |
| − | output of the Model function as run on ConSat.Log:
| |
| − |
| |
| − | Model Outline: ConSat.Mod
| |
| − | o-------------------------------------------------o
| |
| − | | pete_plays_guitar - |
| |
| − | | (pete_plays_guitar ) |
| |
| − | | pete_plays_sax |
| |
| − | | pete_plays_drums - |
| |
| − | | (pete_plays_drums ) |
| |
| − | | paul_plays_sax - |
| |
| − | | (paul_plays_sax ) |
| |
| − | | jane_plays_sax - |
| |
| − | | (jane_plays_sax ) |
| |
| − | | paul_plays_guitar |
| |
| − | | paul_plays_drums - |
| |
| − | | (paul_plays_drums ) |
| |
| − | | jane_plays_guitar - |
| |
| − | | (jane_plays_guitar ) |
| |
| − | | jane_plays_drums |
| |
| − | | pete_fears_13 |
| |
| − | | pete_fears_cats - |
| |
| − | | (pete_fears_cats ) |
| |
| − | | pete_fears_height - |
| |
| − | | (pete_fears_height ) |
| |
| − | | paul_fears_13 - |
| |
| − | | (paul_fears_13 ) |
| |
| − | | jane_fears_13 - |
| |
| − | | (jane_fears_13 ) |
| |
| − | | paul_fears_cats - |
| |
| − | | (paul_fears_cats ) |
| |
| − | | paul_fears_height - |
| |
| − | | (paul_fears_height ) - |
| |
| − | | (pete_fears_13 ) |
| |
| − | | pete_fears_cats - |
| |
| − | | (pete_fears_cats ) |
| |
| − | | pete_fears_height - |
| |
| − | | (pete_fears_height ) - |
| |
| − | | (jane_plays_drums ) - |
| |
| − | | (paul_plays_guitar ) |
| |
| − | | paul_plays_drums |
| |
| − | | jane_plays_drums - |
| |
| − | | (jane_plays_drums ) |
| |
| − | | jane_plays_guitar |
| |
| − | | pete_fears_13 |
| |
| − | | pete_fears_cats - |
| |
| − | | (pete_fears_cats ) |
| |
| − | | pete_fears_height - |
| |
| − | | (pete_fears_height ) |
| |
| − | | paul_fears_13 - |
| |
| − | | (paul_fears_13 ) |
| |
| − | | jane_fears_13 - |
| |
| − | | (jane_fears_13 ) |
| |
| − | | paul_fears_cats - |
| |
| − | | (paul_fears_cats ) |
| |
| − | | paul_fears_height - |
| |
| − | | (paul_fears_height ) - |
| |
| − | | (pete_fears_13 ) |
| |
| − | | pete_fears_cats - |
| |
| − | | (pete_fears_cats ) |
| |
| − | | pete_fears_height - |
| |
| − | | (pete_fears_height ) - |
| |
| − | | (jane_plays_guitar ) - |
| |
| − | | (paul_plays_drums ) - |
| |
| − | | (pete_plays_sax ) |
| |
| − | | pete_plays_drums |
| |
| − | | paul_plays_drums - |
| |
| − | | (paul_plays_drums ) |
| |
| − | | jane_plays_drums - |
| |
| − | | (jane_plays_drums ) |
| |
| − | | paul_plays_guitar |
| |
| − | | paul_plays_sax - |
| |
| − | | (paul_plays_sax ) |
| |
| − | | jane_plays_guitar - |
| |
| − | | (jane_plays_guitar ) |
| |
| − | | jane_plays_sax |
| |
| − | | pete_fears_13 - |
| |
| − | | (pete_fears_13 ) |
| |
| − | | pete_fears_cats |
| |
| − | | pete_fears_height - |
| |
| − | | (pete_fears_height ) |
| |
| − | | paul_fears_cats - |
| |
| − | | (paul_fears_cats ) |
| |
| − | | jane_fears_cats - |
| |
| − | | (jane_fears_cats ) |
| |
| − | | paul_fears_13 |
| |
| − | | paul_fears_height - |
| |
| − | | (paul_fears_height ) |
| |
| − | | jane_fears_13 - |
| |
| − | | (jane_fears_13 ) |
| |
| − | | jane_fears_height * |
| |
| − | | (jane_fears_height ) - |
| |
| − | | (paul_fears_13 ) |
| |
| − | | paul_fears_height - |
| |
| − | | (paul_fears_height ) - |
| |
| − | | (pete_fears_cats ) |
| |
| − | | pete_fears_height - |
| |
| − | | (pete_fears_height ) - |
| |
| − | | (jane_plays_sax ) - |
| |
| − | | (paul_plays_guitar ) |
| |
| − | | paul_plays_sax - |
| |
| − | | (paul_plays_sax ) - |
| |
| − | | (pete_plays_drums ) - |
| |
| − | o-------------------------------------------------o
| |
| − |
| |
| − | This is just the traverse of the "arboreal boolean expansion" (ABE) tree
| |
| − | that Model function germinates from the propositional expression that we
| |
| − | planted in the file Consat.Log, which works to describe the facts of the
| |
| − | situation in question. Since there are 18 logical feature names in this
| |
| − | propositional expression, we are literally talking about a function that
| |
| − | enjoys the abstract type f : %B%^18 -> %B%. If I had wanted to evaluate
| |
| − | this function by expressly writing out its truth table, then it would've
| |
| − | required 2^18 = 262144 rows. Now I didn't bother to count, but I'm sure
| |
| − | that the above output does not have anywhere near that many lines, so it
| |
| − | must be that my program, and maybe even its author, has done a couple of
| |
| − | things along the way that are moderately intelligent. At least, we hope.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | AK = Antti Karttunen
| |
| − | JA = Jon Awbrey
| |
| − |
| |
| − | AK: Am I (and other SeqFanaticians) missing something from this thread?
| |
| − |
| |
| − | AK: Your previous message on seqfan (headers below) is a bit of the same topic,
| |
| − | but does it belong to the same thread? Where I could obtain the other
| |
| − | messages belonging to those two threads? (I'm just now starting to
| |
| − | study "mathematical logic", and its relations to combinatorics are
| |
| − | very interesting.) Is this "cactus" language documented anywhere?
| |
| − |
| |
| − | here i was just following a courtesy of copying people
| |
| − | when i reference their works, in this case neil's site:
| |
| − |
| |
| − | http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000170
| |
| − |
| |
| − | but then i thought that the seqfantasians might be amused, too.
| |
| − |
| |
| − | the bit on higher order propositions, in particular,
| |
| − | those of type h : (B^2 -> B) -> B, i sent because
| |
| − | of the significance that 2^2^2^2 = 65536 took on
| |
| − | for us around that time. & the ho, ho, ho joke.
| |
| − |
| |
| − | "zeroth order logic" (zol) is just another name for
| |
| − | the propositional calculus or the sentential logic
| |
| − | that comes before "first order logic" (fol), aka
| |
| − | first intens/tional logic, quantificational logic,
| |
| − | or predicate calculus, depending on who you talk to.
| |
| − |
| |
| − | the line of work that i have been doing derives from
| |
| − | the ideas of c.s. peirce (1839-1914), who developed
| |
| − | a couple of systems of "logical graphs", actually,
| |
| − | two variant interpretations of the same abstract
| |
| − | structures, called "entitative" and "existential"
| |
| − | graphs. he organized his system into "alpha",
| |
| − | "beta", and "gamma" layers, roughly equivalent
| |
| − | to our propositional, quantificational, and
| |
| − | modal levels of logic today.
| |
| − |
| |
| − | on the more contemporary scene, peirce's entitative interpretation
| |
| − | of logical graphs was revived and extended by george spencer brown
| |
| − | in his book 'laws of form', while the existential interpretation
| |
| − | has flourished in the development of "conceptual graphs" by
| |
| − | john f sowa and a community of growing multitudes.
| |
| − |
| |
| − | a passel of links:
| |
| − |
| |
| − | http://members.door.net/arisbe/
| |
| − | http://www.enolagaia.com/GSB.html
| |
| − | http://www.cs.uah.edu/~delugach/CG/
| |
| − | http://www.jfsowa.com/
| |
| − | http://www.jfsowa.com/cg/
| |
| − | http://www.jfsowa.com/peirce/ms514w.htm
| |
| − | http://users.bestweb.net/~sowa/
| |
| − | http://users.bestweb.net/~sowa/peirce/ms514.htm
| |
| − |
| |
| − | i have mostly focused on "alpha" (prop calc or zol) --
| |
| − | though the "func conception of quant logic" thread was
| |
| − | a beginning try at saying how the same line of thought
| |
| − | might be extended to 1st, 2nd, & higher order logics --
| |
| − | and i devised a particular graph & string syntax that
| |
| − | is based on a species of cacti, officially described as
| |
| − | the "reflective extension of logical graphs" (ref log),
| |
| − | but more lately just referred to as "cactus language".
| |
| − |
| |
| − | it turns out that one can do many interesting things
| |
| − | with prop calc if one has an efficient enough syntax
| |
| − | and a powerful enough interpreter for it, even using
| |
| − | it as a very minimal sort of declarative programming
| |
| − | language, hence, the current thread was directed to
| |
| − | applying "zeroth order theories" (zot's) as brands
| |
| − | of "zeroth order programs" (zop's) to a set of old
| |
| − | constraint satisfaction and knowledge rep examples.
| |
| − |
| |
| − | more recent expositions of the cactus language have been directed
| |
| − | toward what some people call "ontology engineering" -- it sounds
| |
| − | so much cooler than "taxonomy" -- and so these are found in the
| |
| − | ieee standard upper ontology working group discussion archives.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Let's now pause and reflect on the mix of abstract and concrete material
| |
| − | that we have cobbled together in spectacle of this "World Of Zero" (WOZ),
| |
| − | since I believe that we may have seen enough, if we look at it right, to
| |
| − | illustrate a few of the more salient phenomena that would normally begin
| |
| − | to weigh in as a major force only on a much larger scale. Now, it's not
| |
| − | exactly like this impoverished sample, all by itself, could determine us
| |
| − | to draw just the right generalizations, or force us to see the shape and
| |
| − | flow of its immanent law -- it is much too sparse a scattering of points
| |
| − | to tease out the lines of its up and coming generations quite so clearly --
| |
| − | but it can be seen to exemplify many of the more significant themes that
| |
| − | we know evolve in more substantial environments, that is, On Beyond Zero,
| |
| − | since we have already seen them, "tho' obscur'd", in these higher realms.
| |
| − |
| |
| − | One the the themes that I want to to keep an eye on as this discussion
| |
| − | develops is the subject that might be called "computation as semiosis".
| |
| − |
| |
| − | In this light, any calculus worth its salt must be capable of helping
| |
| − | us do two things, calculation, of course, but also analysis. This is
| |
| − | probably one of the reasons why the ordinary sort of differential and
| |
| − | integral calculus over quantitative domains is frequently referred to
| |
| − | as "real analysis", or even just "analysis". It seems quite clear to
| |
| − | me that any adequate logical calculus, in many ways expected to serve
| |
| − | as a qualitative analogue of analytic geometry in the way that it can
| |
| − | be used to describe configurations in logically circumscribed domains,
| |
| − | ought to qualify in both dimensions, namely, analysis and computation.
| |
| − |
| |
| − | With all of these various features of the situation in mind, then, we come
| |
| − | to the point of viewing analysis and computation as just so many different
| |
| − | kinds of "sign transformations in respect of pragmata" (STIROP's). Taking
| |
| − | this insight to heart, let us next work to assemble a comprehension of our
| |
| − | concrete examples, set in the medium of the abstract calculi that allow us
| |
| − | to express their qualitative patterns, that may hope to be an increment or
| |
| − | two less inchoate than we have seen so far, and that may even permit us to
| |
| − | catch the action of these fading fleeting sign transformations on the wing.
| |
| − |
| |
| − | Here is how I picture our latest round of examples
| |
| − | as filling out the framework of this investigation:
| |
| − |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | Objective Framework | Interpretive Framework |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | |
| |
| − | | s_1 = Logue(o) | |
| |
| − | | / | |
| |
| − | | / | |
| |
| − | | @ | |
| |
| − | | · \ | |
| |
| − | | · \ | |
| |
| − | | · i_1 = Model(o) v |
| |
| − | | · s_2 = Model(o) | |
| |
| − | | · / | |
| |
| − | | · / | |
| |
| − | | Object = o · · · · · · @ | |
| |
| − | | · \ | |
| |
| − | | · \ | |
| |
| − | | · i_2 = Tenor(o) v |
| |
| − | | · s_3 = Tenor(o) | |
| |
| − | | · / | |
| |
| − | | · / | |
| |
| − | | @ | |
| |
| − | | \ | |
| |
| − | | \ | |
| |
| − | | i_3 = Sense(o) v |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure. Computation As Semiotic Transformation
| |
| − |
| |
| − | The Figure shows three distinct sign triples of the form <o, s, i>, where
| |
| − | o = ostensible objective = the observed, indicated, or intended situation.
| |
| − |
| |
| − | | A. <o, Logue(o), Model(o)>
| |
| − | |
| |
| − | | B. <o, Model(o), Tenor(o)>
| |
| − | |
| |
| − | | C. <o, Tenor(o), Sense(o)>
| |
| − |
| |
| − | Let us bring these several signs together in one place,
| |
| − | to compare and contrast their common and their diverse
| |
| − | characters, and to think about why we make such a fuss
| |
| − | about passing from one to the other in the first place.
| |
| − |
| |
| − | 1. Logue(o) = Consat.Log
| |
| − | o-----------------------------------------------------------------------o
| |
| − | | |
| |
| − | | (( pete_plays_guitar ),( pete_plays_sax ),( pete_plays_drums )) |
| |
| − | | (( paul_plays_guitar ),( paul_plays_sax ),( paul_plays_drums )) |
| |
| − | | (( jane_plays_guitar ),( jane_plays_sax ),( jane_plays_drums )) |
| |
| − | | |
| |
| − | | (( pete_plays_guitar ),( paul_plays_guitar ),( jane_plays_guitar )) |
| |
| − | | (( pete_plays_sax ),( paul_plays_sax ),( jane_plays_sax )) |
| |
| − | | (( pete_plays_drums ),( paul_plays_drums ),( jane_plays_drums )) |
| |
| − | | |
| |
| − | | (( pete_fears_13 ),( pete_fears_cats ),( pete_fears_height )) |
| |
| − | | (( paul_fears_13 ),( paul_fears_cats ),( paul_fears_height )) |
| |
| − | | (( jane_fears_13 ),( jane_fears_cats ),( jane_fears_height )) |
| |
| − | | |
| |
| − | | (( pete_fears_13 ),( paul_fears_13 ),( jane_fears_13 )) |
| |
| − | | (( pete_fears_cats ),( paul_fears_cats ),( jane_fears_cats )) |
| |
| − | | (( pete_fears_height ),( paul_fears_height ),( jane_fears_height )) |
| |
| − | | |
| |
| − | | (( |
| |
| − | | |
| |
| − | | ( pete_plays_guitar ) |
| |
| − | | ( pete_fears_height ) |
| |
| − | | |
| |
| − | | ( pete_plays_guitar pete_fears_height ) |
| |
| − | | ( paul_plays_guitar paul_fears_height ) |
| |
| − | | ( jane_plays_guitar jane_fears_height ) |
| |
| − | | |
| |
| − | | ( paul_fears_cats ) |
| |
| − | | ( paul_plays_sax ) |
| |
| − | | |
| |
| − | | ( pete_plays_sax pete_fears_cats ) |
| |
| − | | ( paul_plays_sax paul_fears_cats ) |
| |
| − | | ( jane_plays_sax jane_fears_cats ) |
| |
| − | | |
| |
| − | | ( pete_plays_drums pete_fears_13 ) |
| |
| − | | ( paul_plays_drums paul_fears_13 ) |
| |
| − | | ( jane_plays_drums jane_fears_13 ) |
| |
| − | | |
| |
| − | | ( pete_plays_drums pete_fears_height ) |
| |
| − | | ( paul_plays_drums paul_fears_height ) |
| |
| − | | ( jane_plays_drums jane_fears_height ) |
| |
| − | | |
| |
| − | | )) |
| |
| − | | |
| |
| − | o-----------------------------------------------------------------------o
| |
| − |
| |
| − | 2. Model(o) = Consat.Mod ><> http://suo.ieee.org/ontology/msg03718.html
| |
| − |
| |
| − | 3. Tenor(o) = Consat.Ten (Just The Gist Of It)
| |
| − | o-------------------------------------------------o
| |
| − | | (pete_plays_guitar ) | <01> -
| |
| − | | (pete_plays_sax ) | <02> -
| |
| − | | pete_plays_drums | <03> +
| |
| − | | (paul_plays_drums ) | <04> -
| |
| − | | (jane_plays_drums ) | <05> -
| |
| − | | paul_plays_guitar | <06> +
| |
| − | | (paul_plays_sax ) | <07> -
| |
| − | | (jane_plays_guitar ) | <08> -
| |
| − | | jane_plays_sax | <09> +
| |
| − | | (pete_fears_13 ) | <10> -
| |
| − | | pete_fears_cats | <11> +
| |
| − | | (pete_fears_height ) | <12> -
| |
| − | | (paul_fears_cats ) | <13> -
| |
| − | | (jane_fears_cats ) | <14> -
| |
| − | | paul_fears_13 | <15> +
| |
| − | | (paul_fears_height ) | <16> -
| |
| − | | (jane_fears_13 ) | <17> -
| |
| − | | jane_fears_height * | <18> +
| |
| − | o-------------------------------------------------o
| |
| − |
| |
| − | 4. Sense(o) = Consat.Sen
| |
| − | o-------------------------------------------------o
| |
| − | | pete_plays_drums | <03>
| |
| − | | paul_plays_guitar | <06>
| |
| − | | jane_plays_sax | <09>
| |
| − | | pete_fears_cats | <11>
| |
| − | | paul_fears_13 | <15>
| |
| − | | jane_fears_height | <18>
| |
| − | o-------------------------------------------------o
| |
| − |
| |
| − | As one proceeds through the subsessions of the Theme One Study session,
| |
| − | the computation transforms its larger "signs", in this case text files,
| |
| − | from one to the next, in the sequence: Logue, Model, Tenor, and Sense.
| |
| − |
| |
| − | Let us see if we can pin down, on sign-theoretic grounds,
| |
| − | why this very sort of exercise is so routinely necessary.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | We were in the middle of pursuing several questions about
| |
| − | sign relational transformations in general, in particular,
| |
| − | the following Example of a sign transformation that arose
| |
| − | in the process of setting up and solving a classical sort
| |
| − | of constraint satisfaction problem.
| |
| − |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | Objective Framework | Interpretive Framework |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | |
| |
| − | | s_1 = Logue(o) | |
| |
| − | | / | |
| |
| − | | / | |
| |
| − | | @ | |
| |
| − | | · \ | |
| |
| − | | · \ | |
| |
| − | | · i_1 = Model(o) v |
| |
| − | | · s_2 = Model(o) | |
| |
| − | | · / | |
| |
| − | | · / | |
| |
| − | | Object = o · · · · · · @ | |
| |
| − | | · \ | |
| |
| − | | · \ | |
| |
| − | | · i_2 = Tenor(o) v |
| |
| − | | · s_3 = Tenor(o) | |
| |
| − | | · / | |
| |
| − | | · / | |
| |
| − | | @ | |
| |
| − | | \ | |
| |
| − | | \ | |
| |
| − | | i_3 = Sense(o) v |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure. Computation As Semiotic Transformation
| |
| − |
| |
| − | 1. Logue(o) = Consat.Log
| |
| − | o-----------------------------------------------------------------------o
| |
| − | | |
| |
| − | | (( pete_plays_guitar ),( pete_plays_sax ),( pete_plays_drums )) |
| |
| − | | (( paul_plays_guitar ),( paul_plays_sax ),( paul_plays_drums )) |
| |
| − | | (( jane_plays_guitar ),( jane_plays_sax ),( jane_plays_drums )) |
| |
| − | | |
| |
| − | | (( pete_plays_guitar ),( paul_plays_guitar ),( jane_plays_guitar )) |
| |
| − | | (( pete_plays_sax ),( paul_plays_sax ),( jane_plays_sax )) |
| |
| − | | (( pete_plays_drums ),( paul_plays_drums ),( jane_plays_drums )) |
| |
| − | | |
| |
| − | | (( pete_fears_13 ),( pete_fears_cats ),( pete_fears_height )) |
| |
| − | | (( paul_fears_13 ),( paul_fears_cats ),( paul_fears_height )) |
| |
| − | | (( jane_fears_13 ),( jane_fears_cats ),( jane_fears_height )) |
| |
| − | | |
| |
| − | | (( pete_fears_13 ),( paul_fears_13 ),( jane_fears_13 )) |
| |
| − | | (( pete_fears_cats ),( paul_fears_cats ),( jane_fears_cats )) |
| |
| − | | (( pete_fears_height ),( paul_fears_height ),( jane_fears_height )) |
| |
| − | | |
| |
| − | | (( |
| |
| − | | |
| |
| − | | ( pete_plays_guitar ) |
| |
| − | | ( pete_fears_height ) |
| |
| − | | |
| |
| − | | ( pete_plays_guitar pete_fears_height ) |
| |
| − | | ( paul_plays_guitar paul_fears_height ) |
| |
| − | | ( jane_plays_guitar jane_fears_height ) |
| |
| − | | |
| |
| − | | ( paul_fears_cats ) |
| |
| − | | ( paul_plays_sax ) |
| |
| − | | |
| |
| − | | ( pete_plays_sax pete_fears_cats ) |
| |
| − | | ( paul_plays_sax paul_fears_cats ) |
| |
| − | | ( jane_plays_sax jane_fears_cats ) |
| |
| − | | |
| |
| − | | ( pete_plays_drums pete_fears_13 ) |
| |
| − | | ( paul_plays_drums paul_fears_13 ) |
| |
| − | | ( jane_plays_drums jane_fears_13 ) |
| |
| − | | |
| |
| − | | ( pete_plays_drums pete_fears_height ) |
| |
| − | | ( paul_plays_drums paul_fears_height ) |
| |
| − | | ( jane_plays_drums jane_fears_height ) |
| |
| − | | |
| |
| − | | )) |
| |
| − | | |
| |
| − | o-----------------------------------------------------------------------o
| |
| − |
| |
| − | 2. Model(o) = Consat.Mod ><> http://suo.ieee.org/ontology/msg03718.html
| |
| − |
| |
| − | 3. Tenor(o) = Consat.Ten (Just The Gist Of It)
| |
| − | o-------------------------------------------------o
| |
| − | | (pete_plays_guitar ) | <01> -
| |
| − | | (pete_plays_sax ) | <02> -
| |
| − | | pete_plays_drums | <03> +
| |
| − | | (paul_plays_drums ) | <04> -
| |
| − | | (jane_plays_drums ) | <05> -
| |
| − | | paul_plays_guitar | <06> +
| |
| − | | (paul_plays_sax ) | <07> -
| |
| − | | (jane_plays_guitar ) | <08> -
| |
| − | | jane_plays_sax | <09> +
| |
| − | | (pete_fears_13 ) | <10> -
| |
| − | | pete_fears_cats | <11> +
| |
| − | | (pete_fears_height ) | <12> -
| |
| − | | (paul_fears_cats ) | <13> -
| |
| − | | (jane_fears_cats ) | <14> -
| |
| − | | paul_fears_13 | <15> +
| |
| − | | (paul_fears_height ) | <16> -
| |
| − | | (jane_fears_13 ) | <17> -
| |
| − | | jane_fears_height * | <18> +
| |
| − | o-------------------------------------------------o
| |
| − |
| |
| − | 4. Sense(o) = Consat.Sen
| |
| − | o-------------------------------------------------o
| |
| − | | pete_plays_drums | <03>
| |
| − | | paul_plays_guitar | <06>
| |
| − | | jane_plays_sax | <09>
| |
| − | | pete_fears_cats | <11>
| |
| − | | paul_fears_13 | <15>
| |
| − | | jane_fears_height | <18>
| |
| − | o-------------------------------------------------o
| |
| − |
| |
| − | We can worry later about the proper use of quotation marks
| |
| − | in discussing such a case, where the file name "Yada.Yak"
| |
| − | denotes a piece of text that expresses a proposition that
| |
| − | describes an objective situation or an intentional object,
| |
| − | but whatever the case it is clear that we are knee & neck
| |
| − | deep in a sign relational situation of a modest complexity.
| |
| − |
| |
| − | I think that the right sort of analogy might help us
| |
| − | to sort it out, or at least to tell what's important
| |
| − | from the things that are less so. The paradigm that
| |
| − | comes to mind for me is the type of context in maths
| |
| − | where we talk about the "locus" or the "solution set"
| |
| − | of an equation, and here we think of the equation as
| |
| − | denoting its solution set or describing a locus, say,
| |
| − | a point or a curve or a surface or so on up the scale.
| |
| − |
| |
| − | In this figure of speech, we might say for instance:
| |
| − |
| |
| − | | o is
| |
| − | | what "x^3 - 3x^2 + 3x - 1 = 0" denotes is
| |
| − | | what "(x-1)^3 = 0" denotes is
| |
| − | | what "1" denotes
| |
| − | | is 1.
| |
| − |
| |
| − | Making explicit the assumptive interpretations
| |
| − | that the context probably enfolds in this case,
| |
| − | we assume this description of the solution set:
| |
| − |
| |
| − | {x in the Reals : x^3 - 3x^2 + 3x -1 = 0} = {1}.
| |
| − |
| |
| − | In sign relational terms, we have the 3-tuples:
| |
| − |
| |
| − | | <o, "x^3 - 3x^2 + 3x - 1 = 0", "(x-1)^3 = 0">
| |
| − | |
| |
| − | | <o, "(x-1)^3 = 0", "1">
| |
| − | |
| |
| − | | <o, "1", "1">
| |
| − |
| |
| − | As it turns out we discover that the
| |
| − | object o was really just 1 all along.
| |
| − |
| |
| − | But why do we put ourselves through the rigors of these
| |
| − | transformations at all? If 1 is what we mean, why not
| |
| − | just say "1" in the first place and be done with it?
| |
| − | A person who asks a question like that has forgetten
| |
| − | how we keep getting ourselves into these quandaries,
| |
| − | and who it is that assigns the problems, for it is
| |
| − | Nature herself who is the taskmistress here and the
| |
| − | problems are set in the manner that she determines,
| |
| − | not in the style to which we would like to become
| |
| − | accustomed. The best that we can demand of our
| |
| − | various and sundry calculi is that they afford
| |
| − | us with the nets and the snares more readily
| |
| − | to catch the shape of the problematic game
| |
| − | as it flies up before us on its own wings,
| |
| − | and only then to tame it to the amenable
| |
| − | demeanors that we find to our liking.
| |
| − |
| |
| − | In sum, the first place is not ours to take.
| |
| − | We are but poor second players in this game.
| |
| − |
| |
| − | That understood, I can now lay out our present Example
| |
| − | along the lines of this familiar mathematical exercise.
| |
| − |
| |
| − | | o is
| |
| − | | what Consat.Log denotes is
| |
| − | | what Consat.Mod denotes is
| |
| − | | what Consat.Ten denotes is
| |
| − | | what Consat.Sen denotes.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | It will be good to keep this picture before us a while longer:
| |
| − |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | Objective Framework | Interpretive Framework |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | |
| |
| − | | s_1 = Logue(o) | |
| |
| − | | / | |
| |
| − | | / | |
| |
| − | | @ | |
| |
| − | | · \ | |
| |
| − | | · \ | |
| |
| − | | · i_1 = Model(o) v |
| |
| − | | · s_2 = Model(o) | |
| |
| − | | · / | |
| |
| − | | · / | |
| |
| − | | Object = o · · · · · · @ | |
| |
| − | | · \ | |
| |
| − | | · \ | |
| |
| − | | · i_2 = Tenor(o) v |
| |
| − | | · s_3 = Tenor(o) | |
| |
| − | | · / | |
| |
| − | | · / | |
| |
| − | | @ | |
| |
| − | | \ | |
| |
| − | | \ | |
| |
| − | | i_3 = Sense(o) v |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure. Computation As Semiotic Transformation
| |
| − |
| |
| − | The labels that decorate the syntactic plane and indicate
| |
| − | the semiotic transitions in the interpretive panel of the
| |
| − | framework point us to text files whose contents rest here:
| |
| − |
| |
| − | http://suo.ieee.org/ontology/msg03722.html
| |
| − |
| |
| − | The reason that I am troubling myself -- and no doubt you --
| |
| − | with the details of this Example is because it highlights
| |
| − | a number of the thistles that we will have to grasp if we
| |
| − | ever want to escape from the traps of YARNBOL and YARWARS
| |
| − | in which so many of our fairweather fiends are seeking to
| |
| − | ensnare us, and not just us -- the whole web of the world.
| |
| − |
| |
| − | YARNBOL = Yet Another Roman Numeral Based Ontology Language.
| |
| − | YARWARS = Yet Another Representation Without A Reasoning System.
| |
| − |
| |
| − | In order to avoid this, or to reverse the trend once it gets started,
| |
| − | we just have to remember what a dynamic living process a computation
| |
| − | really is, precisely because it is meant to serve as an iconic image
| |
| − | of dynamic, deliberate, purposeful transformations that we are bound
| |
| − | to go through and to carry out in a hopeful pursuit of the solutions
| |
| − | to the many real live problems that life and society place before us.
| |
| − | So I take it rather seriously.
| |
| − |
| |
| − | Okay, back to the grindstone.
| |
| − |
| |
| − | The question is: "Why are these trips necessary?"
| |
| − |
| |
| − | How come we don't just have one proper expression
| |
| − | for each situation under the sun, or all possible
| |
| − | suns, I guess, for some, and just use that on any
| |
| − | appearance, instance, occasion of that situation?
| |
| − |
| |
| − | Why is it ever necessary to begin with an obscure description
| |
| − | of a situation? -- for that is exactly what the propositional
| |
| − | expression caled "Logue(o)", for Example, the Consat.Log file,
| |
| − | really is.
| |
| − |
| |
| − | Maybe I need to explain that first.
| |
| − |
| |
| − | The first three items of syntax -- Logue(o), Model(o), Tenor(o) --
| |
| − | are all just so many different propositional expressions that
| |
| − | denote one and the same logical-valued function p : X -> %B%,
| |
| − | and one whose abstract image we may well enough describe as
| |
| − | a boolean function of the abstract type q : %B%^k -> %B%,
| |
| − | where k happens to be 18 in the present Consat Example.
| |
| − |
| |
| − | If we were to write out the truth table for q : %B%^18 -> %B%
| |
| − | it would take 2^18 = 262144 rows. Using the bold letter #x#
| |
| − | for a coordinate tuple, writing #x# = <x_1, ..., x_18>, each
| |
| − | row of the table would have the form <x_1, ..., x_18, q(#x#)>.
| |
| − | And the function q is such that all rows evalue to %0% save 1.
| |
| − |
| |
| − | Each of the four different formats expresses this fact about q
| |
| − | in its own way. The first three are logically equivalent, and
| |
| − | the last one is the maximally determinate positive implication
| |
| − | of what the others all say.
| |
| − |
| |
| − | From this point of view, the logical computation that we went through,
| |
| − | in the sequence Logue, Model, Tenor, Sense, was a process of changing
| |
| − | from an obscure sign of the objective proposition to a more organized
| |
| − | arrangement of its satisfying or unsatisfying interpretations, to the
| |
| − | most succinct possible expression of the same meaning, to an adequate
| |
| − | positive projection of it that is useful enough in the proper context.
| |
| − |
| |
| − | This is the sort of mill -- it's called "computation" -- that we have
| |
| − | to be able to put our representations through on a recurrent, regular,
| |
| − | routine basis, that is, if we expect them to have any utility at all.
| |
| − | And it is only when we have started to do that in genuinely effective
| |
| − | and efficient ways, that we can even begin to think about facilitating
| |
| − | any bit of qualitative conceptual analysis through computational means.
| |
| − |
| |
| − | And as far as the qualitative side of logical computation
| |
| − | and conceptual analysis goes, we have barely even started.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | We are contemplating the sequence of initial and normal forms
| |
| − | for the Consat problem and we have noted the following system
| |
| − | of logical relations, taking the enchained expressions of the
| |
| − | objective situation o in a pairwise associated way, of course:
| |
| − |
| |
| − | Logue(o) <=> Model(o) <=> Tenor(o) => Sense(o).
| |
| − |
| |
| − | The specifics of the propositional expressions are cited here:
| |
| − |
| |
| − | http://suo.ieee.org/ontology/msg03722.html
| |
| − |
| |
| − | If we continue to pursue the analogy that we made with the form
| |
| − | of mathematical activity commonly known as "solving equations",
| |
| − | then there are many salient features of this type of logical
| |
| − | problem solving endeavor that suddenly leap into the light.
| |
| − |
| |
| − | First of all, we notice the importance of "equational reasoning"
| |
| − | in mathematics, by which I mean, not just the quantitative type
| |
| − | of equation that forms the matter of the process, but also the
| |
| − | qualitative type of equation, or the "logical equivalence",
| |
| − | that connects each expression along the way, right up to
| |
| − | the penultimate stage, when we are satisfied in a given
| |
| − | context to take a projective implication of the total
| |
| − | knowledge of the situation that we have been taking
| |
| − | some pains to preserve at every intermediate stage
| |
| − | of the game.
| |
| − |
| |
| − | This general pattern or strategy of inference, working its way through
| |
| − | phases of "equational" or "total information preserving" inference and
| |
| − | phases of "implicational" or "selective information losing" inference,
| |
| − | is actually very common throughout mathematics, and I have in mind to
| |
| − | examine its character in greater detail and in a more general setting.
| |
| − |
| |
| − | Just as the barest hint of things to come along these lines, you might
| |
| − | consider the question of what would constitute the equational analogue
| |
| − | of modus ponens, in other words the scheme of inference that goes from
| |
| − | x and x=>y to y. Well the answer is a scheme of inference that passes
| |
| − | from x and x=>y to x&y, and then being reversible, back again. I will
| |
| − | explore the rationale and the utility of this gambit in future reports.
| |
| − |
| |
| − | One observation that we can make already at this point,
| |
| − | however, is that these schemes of equational reasoning,
| |
| − | or reversible inference, remain poorly developed among
| |
| − | our currently prevailing styles of inference in logic,
| |
| − | their potentials for applied logical software hardly
| |
| − | being broached in our presently available systems.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − |
| |
| − | Extra Examples
| |
| − |
| |
| − | 1. Propositional logic example.
| |
| − | Files: Alpha.lex + Prop.log
| |
| − | Ref: [Cha, 20, Example 2.12]
| |
| − |
| |
| − | 2. Chemical synthesis problem.
| |
| − | Files: Chem.*
| |
| − | Ref: [Cha, 21, Example 2.13]
| |
| − |
| |
| − | 3. N Queens problem.
| |
| − | Files: Queen*.*, Q8.*, Q5.*
| |
| − | Refs: [BaC, 166], [VaH, 122], [Wir, 143].
| |
| − | Notes: Only the 5 Queens example will run in 640K memory.
| |
| − | Use the "Queen.lex" file to load the "Q5.eg*" log files.
| |
| − |
| |
| − | 4. Five Houses puzzle.
| |
| − | Files: House.*
| |
| − | Ref: [VaH, 132].
| |
| − | Notes: Will not run in 640K memory.
| |
| − |
| |
| − | 5. Graph coloring example.
| |
| − | Files: Color.*
| |
| − | Ref: [Wil, 196].
| |
| − |
| |
| − | 6. Examples of Cook's Theorem in computational complexity,
| |
| − | that propositional satisfiability is NP-complete.
| |
| − |
| |
| − | Files: StiltN.* = "Space and Time Limited Turing Machine",
| |
| − | with N units of space and N units of time.
| |
| − | StuntN.* = "Space and Time Limited Turing Machine",
| |
| − | for computing the parity of a bit string,
| |
| − | with Number of Tape cells of input equal to N.
| |
| − | Ref: [Wil, 188-201].
| |
| − | Notes: Can only run Turing machine example for input of size 2.
| |
| − | Since the last tape cell is used for an end-of-file marker,
| |
| − | this amounts to only one significant digit of computation.
| |
| − | Use the "Stilt3.lex" file to load the "Stunt2.egN" files.
| |
| − | Their Sense file outputs appear on the "Stunt2.seN" files.
| |
| − |
| |
| − | 7. Fabric knowledge base.
| |
| − | Files: Fabric.*, Fab.*
| |
| − | Ref: [MaW, 8-16].
| |
| − |
| |
| − | 8. Constraint Satisfaction example.
| |
| − | Files: Consat1.*, Consat2.*
| |
| − | Ref: [Win, 449, Exercise 3-9].
| |
| − | Notes: Attributed to Kenneth D. Forbus.
| |
| − |
| |
| − | References
| |
| − |
| |
| − | | Angluin, Dana,
| |
| − | |"Learning with Hints", in
| |
| − | |'Proceedings of the 1988 Workshop on Computational Learning Theory',
| |
| − | | edited by D. Haussler & L. Pitt, Morgan Kaufmann, San Mateo, CA, 1989.
| |
| − |
| |
| − | | Ball, W.W. Rouse, & Coxeter, H.S.M.,
| |
| − | |'Mathematical Recreations and Essays', 13th ed.,
| |
| − | | Dover, New York, NY, 1987.
| |
| − |
| |
| − | | Chang, Chin-Liang & Lee, Richard Char-Tung,
| |
| − | |'Symbolic Logic and Mechanical Theorem Proving',
| |
| − | | Academic Press, New York, NY, 1973.
| |
| − |
| |
| − | | Denning, Peter J., Dennis, Jack B., and Qualitz, Joseph E.,
| |
| − | |'Machines, Languages, and Computation',
| |
| − | | Prentice-Hall, Englewood Cliffs, NJ, 1978.
| |
| − |
| |
| − | | Edelman, Gerald M.,
| |
| − | |'Topobiology: An Introduction to Molecular Embryology',
| |
| − | | Basic Books, New York, NY, 1988.
| |
| − |
| |
| − | | Lloyd, J.W.,
| |
| − | |'Foundations of Logic Programming',
| |
| − | | Springer-Verlag, Berlin, 1984.
| |
| − |
| |
| − | | Maier, David & Warren, David S.,
| |
| − | |'Computing with Logic: Logic Programming with Prolog',
| |
| − | | Benjamin/Cummings, Menlo Park, CA, 1988.
| |
| − |
| |
| − | | McClelland, James L. and Rumelhart, David E.,
| |
| − | |'Explorations in Parallel Distributed Processing:
| |
| − | | A Handbook of Models, Programs, and Exercises',
| |
| − | | MIT Press, Cambridge, MA, 1988.
| |
| − |
| |
| − | | Peirce, Charles Sanders,
| |
| − | |'Collected Papers of Charles Sanders Peirce',
| |
| − | | edited by Charles Hartshorne, Paul Weiss, & Arthur W. Burks,
| |
| − | | Harvard University Press, Cambridge, MA, 1931-1960.
| |
| − |
| |
| − | | Peirce, Charles Sanders,
| |
| − | |'The New Elements of Mathematics',
| |
| − | | edited by Carolyn Eisele, Mouton, The Hague, 1976.
| |
| − |
| |
| − | |'Charles S. Peirce: Selected Writings; Values in a Universe of Chance',
| |
| − | | edited by Philip P. Wiener, Dover, New York, NY, 1966.
| |
| − |
| |
| − | | Spencer Brown, George,
| |
| − | |'Laws of Form',
| |
| − | | George Allen & Unwin, London, UK, 1969.
| |
| − |
| |
| − | | Van Hentenryck, Pascal,
| |
| − | |'Constraint Satisfaction in Logic Programming',
| |
| − | | MIT Press, Cambridge, MA, 1989.
| |
| − |
| |
| − | | Wilf, Herbert S.,
| |
| − | |'Algorithms and Complexity',
| |
| − | | Prentice-Hall, Englewood Cliffs, NJ, 1986.
| |
| − |
| |
| − | | Winston, Patrick Henry,
| |
| − | |'Artificial Intelligence, 2nd ed.,
| |
| − | | Addison-Wesley, Reading, MA, 1984.
| |
| − |
| |
| − | | Wirth, Niklaus,
| |
| − | |'Algorithms + Data Structures = Programs',
| |
| − | | Prentice-Hall, Englewood Cliffs, NJ, 1976.
| |
| − |
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | </pre>
| |
| | | | |
| | ==Document History== | | ==Document History== |
