Changes

A sticking point of the whole discussion has just been reached.  In the idyllic setting of a knowledge field the question of systematic inquiry takes on the following form:

A sticking point of the whole discussion has just been reached.  In the idyllic setting of a knowledge field the question of systematic inquiry takes on the following form:

: ''What piece of code should be followed in order to discover that code?''

| ''What piece of code should be followed in order to discover that code?''

It is a classic catch, whose pattern was traced out long ago in the paradox of Plato's ''Meno''.  Discussion of this dialogue and of the task it sets for AI, cognitive science, education, including the design of intelligent tutoring systems, can be found in (H. Gardner, 1985), (Chomsky, 1965, '72, '75, '80, '86), (Fodor, 1975, 1983), (Piattelli-Palmarini, 1980), and in (Collins & Stevens, 1991).  Though it appears to mask a legion of diversions, this text will present itself at least twice more in the current engagement, both on the horizon and at the gates of the project to fathom and to build intelligent systems.  Therefore, it is worth recalling how this inquiry begins.  The interlocutor Meno asks:

It is a classic catch, whose pattern was traced out long ago in the paradox of Plato's ''Meno''.  Discussion of this dialogue and of the task it sets for AI, cognitive science, education, including the design of intelligent tutoring systems, can be found in (H. Gardner, 1985), (Chomsky, 1965, '72, '75, '80, '86), (Fodor, 1975, 1983), (Piattelli-Palmarini, 1980), and in (Collins & Stevens, 1991).  Though it appears to mask a legion of diversions, this text will present itself at least twice more in the current engagement, both on the horizon and at the gates of the project to fathom and to build intelligent systems.  Therefore, it is worth recalling how this inquiry begins.  The interlocutor Meno asks:

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<p>Can you tell me, Socrates, whether virtue can be taught, or is acquired by practice, not teaching?  Or if neither by practice nor by learning, whether it comes to mankind by nature or in some other way?  (Plato, ''Meno'', p. 265).</p>

<p>Can you tell me, Socrates, whether virtue can be taught, or is acquired by practice, not teaching?  Or if neither by practice nor by learning, whether it comes to mankind by nature or in some other way?  (Plato, ''Meno'', p. 265).</p>

The primary factorization is typically only the first in a series of analytic decompositions that are needed to fully describe a complex domain of phenomena.  The question about proper factorization that this discussion has been at pains to point out becomes compounded into a question about the reality of all the various distinctions of analytic order.  Do the postulated levels really exist in nature, or do they arise only as the artifacts of our attempts to mine the ore of nature?  An early appreciation of the hypothetical character of these distinctions and the post hoc manner of their validation is recorded in (Chomsky, 1975, p. 100).

The primary factorization is typically only the first in a series of analytic decompositions that are needed to fully describe a complex domain of phenomena.  The question about proper factorization that this discussion has been at pains to point out becomes compounded into a question about the reality of all the various distinctions of analytic order.  Do the postulated levels really exist in nature, or do they arise only as the artifacts of our attempts to mine the ore of nature?  An early appreciation of the hypothetical character of these distinctions and the post hoc manner of their validation is recorded in (Chomsky, 1975, p. 100).

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<p>In linguistic theory, we face the problem of constructing this system of levels in an abstract manner, in such a way that a simple grammar will result when this complex of abstract structures is given an interpretation in actual linguistic material.</p>

<p>In linguistic theory, we face the problem of constructing this system of levels in an abstract manner, in such a way that a simple grammar will result when this complex of abstract structures is given an interpretation in actual linguistic material.</p>

One version of Peirce's sign definition is especially useful for the present purpose.  It establishes for signs a fundamental role in logic and is stated in terms of abstract relational properties that are flexible enough to be interpreted in the materials of dynamic systems.  Peirce gave this definition of signs in his 1902 "Application to the Carnegie Institution":

One version of Peirce's sign definition is especially useful for the present purpose.  It establishes for signs a fundamental role in logic and is stated in terms of abstract relational properties that are flexible enough to be interpreted in the materials of dynamic systems.  Peirce gave this definition of signs in his 1902 "Application to the Carnegie Institution":

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<p>Logic is ''formal semiotic''.  A sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, ''C'', its ''object'', as that in which itself stands to ''C''.  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  (Peirce, NEM 4, 54).</p>

<p>Logic is ''formal semiotic''.  A sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, ''C'', its ''object'', as that in which itself stands to ''C''.  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  (Peirce, NEM 4, 54).</p>

If the principles of systems theory are taken seriously in their application to AI, and if the tools that have been developed for dynamic systems are cast in with the array of techniques that are used in AI, a host of difficulties almost instantly arises.  One obstacle to integrating systems theory and artificial intelligence is the bifurcation of approaches that are severally specialized for qualitative and quantitative realms, the unavoidable differences between boolean-discrete and real-continuous domains.  My way of circumventing this obstruction will be to extend the compass of differential geometry and the rule of logic programming to what I see as a locus of natural contact.  Continuing the inquiry to naturalize intelligent systems as serious subjects of dynamic systems theory, a whole series of further questions comes up:

If the principles of systems theory are taken seriously in their application to AI, and if the tools that have been developed for dynamic systems are cast in with the array of techniques that are used in AI, a host of difficulties almost instantly arises.  One obstacle to integrating systems theory and artificial intelligence is the bifurcation of approaches that are severally specialized for qualitative and quantitative realms, the unavoidable differences between boolean-discrete and real-continuous domains.  My way of circumventing this obstruction will be to extend the compass of differential geometry and the rule of logic programming to what I see as a locus of natural contact.  Continuing the inquiry to naturalize intelligent systems as serious subjects of dynamic systems theory, a whole series of further questions comes up:

# What is the proper notion of state?

# How is the knowledge component or the "intellectual property" of this state to be characterized?

| What is the proper notion of state?

| How is the knowledge component or the "intellectual property" of this state to be characterized?

In accord with customary definitions, the knowledge component would need to be represented as a projection onto a knowledge subspace.  In those intelligences for whom not everything is knowledge, or at least for whom not everything is known at once, that is, the great majority of those we are likely to know, there must be an alternate projection onto another subspace.  Some real difficulties begin here which threaten to entangle our own resources intelligence of irretrievably.

In accord with customary definitions, the knowledge component would need to be represented as a projection onto a knowledge subspace.  In those intelligences for whom not everything is knowledge, or at least for whom not everything is known at once, that is, the great majority of those we are likely to know, there must be an alternate projection onto another subspace.  Some real difficulties begin here which threaten to entangle our own resources intelligence of irretrievably.

The project before me is simply to view intelligent systems as systems, to take the ostended substantive seriously.  To succeed at this it will be necessary to answer several questions:

The project before me is simply to view intelligent systems as systems, to take the ostended substantive seriously.  To succeed at this it will be necessary to answer several questions:

: What is the proper notion of a state vector?

| What is the proper notion of a state vector?

We need to analyze the state of the system into a knowledge component and a remaining or a sustaining component.  This "everything else" component may be called the physical component so long as this does not prejudice the issue of a naturalistic aim, which seeks to understand all components as ''physis'', that is, as coming under the original Greek idea of a natural process.  Even the ordinary notion of a state vector, though continuing to be useful as a basis of analogy, may have to be challenged:

We need to analyze the state of the system into a knowledge component and a remaining or a sustaining component.  This "everything else" component may be called the physical component so long as this does not prejudice the issue of a naturalistic aim, which seeks to understand all components as ''physis'', that is, as coming under the original Greek idea of a natural process.  Even the ordinary notion of a state vector, though continuing to be useful as a basis of analogy, may have to be challenged:

: Are the state elements, the moments of the system's experience, really vectors?

| Are the state elements, the moments of the system's experience, really vectors?

Consider the common frame of a venn diagram, overlapping pools of elements arrayed on a nondescript plain, an arena of conventional measure but not routinely examined significance.

Consider the common frame of a venn diagram, overlapping pools of elements arrayed on a nondescript plain, an arena of conventional measure but not routinely examined significance.

Why have I chosen differential geometry and logic programming to try jamming together?  A clue may be picked up in the quotation below.  When the foundations of that ingenious duplex, AI and cybernetics, were being poured, one who was present placed these words in a cornerstone of the structure (Ashby, 1956, p. 9).

Why have I chosen differential geometry and logic programming to try jamming together?  A clue may be picked up in the quotation below.  When the foundations of that ingenious duplex, AI and cybernetics, were being poured, one who was present placed these words in a cornerstone of the structure (Ashby, 1956, p. 9).

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<p>The most fundamental concept in cybernetics is that of "difference", either that two things are recognisably different or that one thing has changed with time.</p>

<p>The most fundamental concept in cybernetics is that of "difference", either that two things are recognisably different or that one thing has changed with time.</p>

Legend tells us that the primal twins of AI, the strife-born siblings of Goal-Seeking and Hill-Climbing, began to stumble and soon came to grief on certain notorious obstacles.  The typical scenario runs as follows.

Legend tells us that the primal twins of AI, the strife-born siblings of Goal-Seeking and Hill-Climbing, began to stumble and soon came to grief on certain notorious obstacles.  The typical scenario runs as follows.

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<p>At any moment in time the following question is posed:<br>

<p>At any moment in time the following question is posed:<br>

A symbolic calculus is needed to assist our reasoning and computation in the realm of propositions.  With an eye toward efficiency of computing and ease of human use, while preserving both functional and declarative properties of propositions, I have implemented an interpreter and assorted utilities for one such calculus.  The original form of this particular calculus goes back to the logician C.S. Peirce, who is my personal favorite candidate for the grand-uncle of AI.  Among other things, Peirce discovered the logical importance of NAND/NNOR operators (CP 4.12 ff, 4.264 f), (NE 4, ch. 5), inspired early ideas about logic machines (Peirce, 1883), is credited with "the first known effort to apply Boolean algebra to the design of switching circuits" (M. Gardner, p. 116 n), and even speculated on the nature of abstract interpreters and other "Quasi-Minds" (Peirce, CP 4.536, 4.550 ff).

A symbolic calculus is needed to assist our reasoning and computation in the realm of propositions.  With an eye toward efficiency of computing and ease of human use, while preserving both functional and declarative properties of propositions, I have implemented an interpreter and assorted utilities for one such calculus.  The original form of this particular calculus goes back to the logician C.S. Peirce, who is my personal favorite candidate for the grand-uncle of AI.  Among other things, Peirce discovered the logical importance of NAND/NNOR operators (CP 4.12 ff, 4.264 f), (NE 4, ch. 5), inspired early ideas about logic machines (Peirce, 1883), is credited with "the first known effort to apply Boolean algebra to the design of switching circuits" (M. Gardner, p. 116 n), and even speculated on the nature of abstract interpreters and other "Quasi-Minds" (Peirce, CP 4.536, 4.550 ff).

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<p>Thought is not necessarily connected with a brain.  It appears in the work of bees, of crystals, and throughout the purely physical world;  and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there.  (CP 4.551).</p>

<p>Thought is not necessarily connected with a brain.  It appears in the work of bees, of crystals, and throughout the purely physical world;  and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there.  (CP 4.551).</p>

Especially useful is the facility this notation provides for expressing partition constraints, or relations of mutual exclusion and exhaustion among logical features.  For example,

Especially useful is the facility this notation provides for expressing partition constraints, or relations of mutual exclusion and exhaustion among logical features.  For example,

: ((''p''₁),(''p''₂),(''p''₃))

| ((''p''₁),(''p''₂),(''p''₃))

says that the universe is partitioned among the three properties ''p''₁,&nbsp;''p''₂,&nbsp;''p''₃.  Finally,

says that the universe is partitioned among the three properties ''p''₁,&nbsp;''p''₂,&nbsp;''p''₃.  Finally,

: (''g'' , (''s''₁),(''s''₂),(''s''₃))

| (''g'' , (''s''₁),(''s''₂),(''s''₃))

says that the genus ''g'' is partitioned into the three species ''s''₁,&nbsp;''s''₂,&nbsp;''s''₃.  Its venn diagram looks like a pie chart.  This style of expression is also useful in representing the behavior of devices, for example:  finite state machines, which must occupy exactly one state at a time;  and Turing machines, whose tape head must engage just one tape cell at a time.

says that the genus ''g'' is partitioned into the three species ''s''₁,&nbsp;''s''₂,&nbsp;''s''₃.  Its venn diagram looks like a pie chart.  This style of expression is also useful in representing the behavior of devices, for example:  finite state machines, which must occupy exactly one state at a time;  and Turing machines, whose tape head must engage just one tape cell at a time.

In order to define a differential extension of a propositional universe of discourse ''U'', the alphabet <font face="lucida calligraphy">A</font> of ''U''’s defining features must be extended to include a set of symbols for differential features, or elementary "changes" in the universe of discourse.  Intuitively, these symbols may be construed as denoting primitive features of change, or propositions about how things or points in ''U'' change with respect to the features noted in the original alphabet <font face="lucida calligraphy">A</font>.  Hence, let d<font face="lucida calligraphy">A</font> = {da₁,&nbsp;…,&nbsp;da_''n''} and d''U'' = <font face="symbol">á</font>d<font face="lucida calligraphy">A</font><font face="symbol">ñ</font> = <font face="symbol">á</font>da₁,&nbsp;…,&nbsp;da_''n''<font face="symbol">ñ</font>.  As before, we may express d''U'' concretely as a product of distinct factors:

In order to define a differential extension of a propositional universe of discourse ''U'', the alphabet <font face="lucida calligraphy">A</font> of ''U''’s defining features must be extended to include a set of symbols for differential features, or elementary "changes" in the universe of discourse.  Intuitively, these symbols may be construed as denoting primitive features of change, or propositions about how things or points in ''U'' change with respect to the features noted in the original alphabet <font face="lucida calligraphy">A</font>.  Hence, let d<font face="lucida calligraphy">A</font> = {da₁,&nbsp;…,&nbsp;da_''n''} and d''U'' = <font face="symbol">á</font>d<font face="lucida calligraphy">A</font><font face="symbol">ñ</font> = <font face="symbol">á</font>da₁,&nbsp;…,&nbsp;da_''n''<font face="symbol">ñ</font>.  As before, we may express d''U'' concretely as a product of distinct factors:

: d''U'' = <font size="4">&times;</font>_''i'' d''A''_''i'' = d''A''₁ &times; &hellip; &times; d''A''_''n''.

| d''U'' = <font size="4">&times;</font>_''i'' d''A''_''i'' = d''A''₁ &times; &hellip; &times; d''A''_''n''.

Here, dA_''i'' is an alphabet of two symbols, dA_''i'' = {(da_''i''),&nbsp;da_''i''}, where (da_''i'') is a symbol with the logical value of "not da_''i''".  Each dA_''i'' has the type '''B''', under the ordered correspondence {(da_''i''),&nbsp;da_''i''} = {0,&nbsp;1}.  However, clarity is often served by acknowledging this differential usage with a distinct type '''D''', as follows:

Here, dA_''i'' is an alphabet of two symbols, dA_''i'' = {(da_''i''),&nbsp;da_''i''}, where (da_''i'') is a symbol with the logical value of "not da_''i''".  Each dA_''i'' has the type '''B''', under the ordered correspondence {(da_''i''),&nbsp;da_''i''} = {0,&nbsp;1}.  However, clarity is often served by acknowledging this differential usage with a distinct type '''D''', as follows:

: '''D''' = {(dx), dx} = {same, different} = {stay, change}.

| '''D''' = {(dx), dx} = {same, different} = {stay, change}.

Finally, let ''U''&prime; = ''U'' &times; d''U'' = <font face="symbol">á</font><font face="lucida calligraphy">A</font>&prime;&nbsp;<font face="symbol">ñ</font> = <font face="symbol">á</font><font face="lucida calligraphy">A</font>&nbsp;+&nbsp;d<font face="lucida calligraphy">A</font><font face="symbol">ñ</font> = <font face="symbol">á</font>a₁,&nbsp;…,&nbsp;a_''n'',&nbsp;da₁,&nbsp;…,&nbsp;da_''n''<font face="symbol">ñ</font>, giving ''U''&prime; the type '''B'''^''n''&nbsp;&times;&nbsp;'''D'''^''n''.

Finally, let ''U''&prime; = ''U'' &times; d''U'' = <font face="symbol">á</font><font face="lucida calligraphy">A</font>&prime;&nbsp;<font face="symbol">ñ</font> = <font face="symbol">á</font><font face="lucida calligraphy">A</font>&nbsp;+&nbsp;d<font face="lucida calligraphy">A</font><font face="symbol">ñ</font> = <font face="symbol">á</font>a₁,&nbsp;…,&nbsp;a_''n'',&nbsp;da₁,&nbsp;…,&nbsp;da_''n''<font face="symbol">ñ</font>, giving ''U''&prime; the type '''B'''^''n''&nbsp;&times;&nbsp;'''D'''^''n''.

Using the isomorphism between function spaces:

Using the isomorphism between function spaces:

: ('''B'''^''n''&nbsp;&times;&nbsp;'''D'''^''n''&nbsp;&rarr;&nbsp;'''B''') <math>\cong</math> ('''B'''^''n'' &rarr; ('''D'''^''n'' &rarr; '''B''')),

| ('''B'''^''n''&nbsp;&times;&nbsp;'''D'''^''n''&nbsp;&rarr;&nbsp;'''B''') <math>\cong</math> ('''B'''^''n'' &rarr; ('''D'''^''n'' &rarr; '''B''')),

each ''p'' : ''U''&prime; &rarr; '''B''' has a unique decomposition into a ''p''&prime; : '''B'''^''n'' &rarr; ('''D'''^''n'' &rarr; '''B''') and a set of ''p''&Prime; : '''D'''^''n'' &rarr; '''B''' such that:

each ''p'' : ''U''&prime; &rarr; '''B''' has a unique decomposition into a ''p''&prime; : '''B'''^''n'' &rarr; ('''D'''^''n'' &rarr; '''B''') and a set of ''p''&Prime; : '''D'''^''n'' &rarr; '''B''' such that:

: ''p'' : '''B'''^''n'' &times; '''D'''^''n'' &rarr; '''B''' <math>\cong</math> ''p''&prime; : '''B'''^''n'' &rarr; ''p''&Prime; : ('''D'''^''n'' &rarr; '''B''').

| ''p'' : '''B'''^''n'' &times; '''D'''^''n'' &rarr; '''B''' <math>\cong</math> ''p''&prime; : '''B'''^''n'' &rarr; ''p''&Prime; : ('''D'''^''n'' &rarr; '''B''').

For the sake of the visual intuition we may imagine that each cell ''x'' in the diagram of ''U'' has springing from it the diagram of the proposition ''p''&prime;(''x'') = ''p''&Prime; in d''U''.

For the sake of the visual intuition we may imagine that each cell ''x'' in the diagram of ''U'' has springing from it the diagram of the proposition ''p''&prime;(''x'') = ''p''&Prime; in d''U''.

To attempt a clarification let us now make one more pass.  Let ''x'' and ''y'' be variables ranging over ''U'' and d''U'', respectively.  Then each ''p''&nbsp;:&nbsp;''U''&prime; = ''U''&nbsp;&times;&nbsp;d''U''&nbsp;&rarr;&nbsp;'''B''' has a unique decomposition into a ''p''&prime;&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' and a set of ''p''(''x'')&prime;&nbsp;:&nbsp;d''U''&nbsp;&rarr;&nbsp;'''B''' such that

To attempt a clarification let us now make one more pass.  Let ''x'' and ''y'' be variables ranging over ''U'' and d''U'', respectively.  Then each ''p''&nbsp;:&nbsp;''U''&prime; = ''U''&nbsp;&times;&nbsp;d''U''&nbsp;&rarr;&nbsp;'''B''' has a unique decomposition into a ''p''&prime;&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' and a set of ''p''(''x'')&prime;&nbsp;:&nbsp;d''U''&nbsp;&rarr;&nbsp;'''B''' such that

: ''p''(''x'', ''y'') = ''p''&prime;(''x'')(''y'') = ''p''(''x'')&prime;(''y'').

| ''p''(''x'', ''y'') = ''p''&prime;(''x'')(''y'') = ''p''(''x'')&prime;(''y'').

The "''x''" in ''p''(''x'')&prime;(''y'') would ordinarily be subscripted as a parameter in the form ''p''_''x''&prime;&nbsp;, but this does not explain the difference between a parameter and a variable.  Here the difference is marked by the position of the prime (&prime;), which serves as a kind of "run-time marker".  The prime locates the point of inflexion in a piece of notation that is the boundary between local and global responsibilities of interpretation.  It tells the intended division between individual identity of functions (a name and a local habitation) and "socially" defined roles (signs falling to the duty of a global interpreter).  In the phrase ''p''&prime;(''x'') the ''p''&prime; names the function while the parenthetical (''x'') is part of the function notation, to be understood by a global interpreter.  In ''p''(''x'')&prime; the parenthetical (''x'') figures into the name of an individual function, having a local significance but only when ''x'' is specified.

The "''x''" in ''p''(''x'')&prime;(''y'') would ordinarily be subscripted as a parameter in the form ''p''_''x''&prime;&nbsp;, but this does not explain the difference between a parameter and a variable.  Here the difference is marked by the position of the prime (&prime;), which serves as a kind of "run-time marker".  The prime locates the point of inflexion in a piece of notation that is the boundary between local and global responsibilities of interpretation.  It tells the intended division between individual identity of functions (a name and a local habitation) and "socially" defined roles (signs falling to the duty of a global interpreter).  In the phrase ''p''&prime;(''x'') the ''p''&prime; names the function while the parenthetical (''x'') is part of the function notation, to be understood by a global interpreter.  In ''p''(''x'')&prime; the parenthetical (''x'') figures into the name of an individual function, having a local significance but only when ''x'' is specified.