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| {{DISPLAYTITLE:Differential Analytic Turing Automata}} | | {{DISPLAYTITLE:Differential Analytic Turing Automata}} |
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| + | ==DATA. Note 1== |
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| <pre> | | <pre> |
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− | IDS -- DATA
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− | DATA. Differential Analytic Turing Automata
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− | DATA. Note 1
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| For the purposes of the NKS Forum my aim is to chart a course from | | For the purposes of the NKS Forum my aim is to chart a course from |
| general ideas about "transformational equivalence classes of graphs" | | general ideas about "transformational equivalence classes of graphs" |
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| That should be enough to get started. | | That should be enough to get started. |
| + | </pre> |
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| + | ==DATA. Note 2== |
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− | DATA. Note 2 | |
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| + | <pre> |
| I will be making use of the "cactus language" extension of | | I will be making use of the "cactus language" extension of |
| Peirce's Alpha Graphs, so called because it uses a species | | Peirce's Alpha Graphs, so called because it uses a species |
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| in hopes of making them as clear as they can be, | | in hopes of making them as clear as they can be, |
| so please let me know if you have any questions. | | so please let me know if you have any questions. |
| + | </pre> |
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| + | ==DATA. Note 3== |
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− | DATA. Note 3 | |
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| + | <pre> |
| I will draw on those previously advertized resources | | I will draw on those previously advertized resources |
| of notation and theory as needed, but right now | | of notation and theory as needed, but right now |
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| I'll leave you to muse on | | I'll leave you to muse on |
| the possibilities of that. | | the possibilities of that. |
| + | </pre> |
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| + | ==DATA. Note 4== |
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− | DATA. Note 4 | |
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| + | <pre> |
| I am preparing a more fleshed-out 1-variable example, | | I am preparing a more fleshed-out 1-variable example, |
| but in the mean time, for anybody who's finished all | | but in the mean time, for anybody who's finished all |
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| DLOG D11. http://stderr.org/pipermail/inquiry/2003-May/000489.html | | DLOG D11. http://stderr.org/pipermail/inquiry/2003-May/000489.html |
| + | </pre> |
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| + | ==DATA. Note 5== |
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− | DATA. Note 5 | |
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| + | <pre> |
| For a slightly more interesting example, let's suppose that | | For a slightly more interesting example, let's suppose that |
| we have a dynamic system that is known by its state space X, | | we have a dynamic system that is known by its state space X, |
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| DLOG D20. http://stderr.org/pipermail/inquiry/2003-May/000498.html | | DLOG D20. http://stderr.org/pipermail/inquiry/2003-May/000498.html |
| + | </pre> |
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| + | ==DATA. Note 6== |
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− | DATA. Note 6 | |
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| + | <pre> |
| One more example may serve to suggest just how much dynamic | | One more example may serve to suggest just how much dynamic |
| complexity can be built on a universe of discourse that has | | complexity can be built on a universe of discourse that has |
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| Zoom^4 ... | | Zoom^4 ... |
| + | </pre> |
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| + | ==DATA. Note 7== |
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− | DATA. Note 7 | |
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| + | <pre> |
| Here are the 4^th gear curves over the 1-feature universe | | Here are the 4^th gear curves over the 1-feature universe |
| X = <|x|> arranged in the form of tabular arrays, listing | | X = <|x|> arranged in the form of tabular arrays, listing |
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| DLOG D24. http://stderr.org/pipermail/inquiry/2003-May/000503.html | | DLOG D24. http://stderr.org/pipermail/inquiry/2003-May/000503.html |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 8== |
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− | DATA. Note 8 | |
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| + | <pre> |
| I am going to tip-toe in silence/consilience past many | | I am going to tip-toe in silence/consilience past many |
| questions of a philosophical nature/nurture that might | | questions of a philosophical nature/nurture that might |
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| DLOG D73. http://stderr.org/pipermail/inquiry/2003-June/000557.html | | DLOG D73. http://stderr.org/pipermail/inquiry/2003-June/000557.html |
| + | </pre> |
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| + | ==DATA. Note 9== |
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− | DATA. Note 9 | |
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| + | <pre> |
| By virtue of Zipf's law -- yes, there is a dynamics to its economics -- | | By virtue of Zipf's law -- yes, there is a dynamics to its economics -- |
| I have found myself forced, when it comes to matters that I've been | | I have found myself forced, when it comes to matters that I've been |
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| No more angels on pinheads, | | No more angels on pinheads, |
| the brass tacks next time. | | the brass tacks next time. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 10== |
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− | DATA. Note 10 | |
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| + | <pre> |
| It is time to formulate the differential analysis of | | It is time to formulate the differential analysis of |
| a logical transformation, or a "mapping of discourse". | | a logical transformation, or a "mapping of discourse". |
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| let's return to particular cases, and carry out the first order | | let's return to particular cases, and carry out the first order |
| analysis of the transformation F<u, v> = <((u)(v)), ((u, v))>. | | analysis of the transformation F<u, v> = <((u)(v)), ((u, v))>. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 11== |
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− | DATA. Note 11 | |
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| + | <pre> |
| By way of getting our feet back on solid ground, let's crank up | | By way of getting our feet back on solid ground, let's crank up |
| our current case of a transformation of discourse, F : U% -> X%, | | our current case of a transformation of discourse, F : U% -> X%, |
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| the first basin and the orbit <1, 1> making | | the first basin and the orbit <1, 1> making |
| up an isolated basin. | | up an isolated basin. |
| + | </pre> |
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| + | ==DATA. Note 12== |
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− | DATA. Note 12 | |
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| + | <pre> |
| Way back in DATA Note 3, we guessed, or "abduced", | | Way back in DATA Note 3, we guessed, or "abduced", |
| as a line of logicians from Aristotle to Peirce | | as a line of logicians from Aristotle to Peirce |
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| I am going to take a nap to clear my head. | | I am going to take a nap to clear my head. |
| + | </pre> |
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| + | ==DATA. Note 13== |
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− | DATA. Note 13 | |
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| + | <pre> |
| I think that it ought to be clear at this point that we | | I think that it ought to be clear at this point that we |
| need a more systematic symbolic method for computing the | | need a more systematic symbolic method for computing the |
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| I'll break this here in case anyone wants | | I'll break this here in case anyone wants |
| to try and do the work for g on their own. | | to try and do the work for g on their own. |
| + | </pre> |
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| + | ==DATA. Note 14== |
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− | DATA. Note 14 | |
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| + | <pre> |
| No doubt everybody who's still awake sacrificed | | No doubt everybody who's still awake sacrificed |
| the few spare moments of their sleep last night | | the few spare moments of their sleep last night |
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| o---------------------------------------o | | o---------------------------------------o |
| Figure 2.3. Dg = g + Eg | | Figure 2.3. Dg = g + Eg |
| + | </pre> |
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| + | ==DATA. Note 15== |
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− | DATA. Note 15 | |
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| + | <pre> |
| | 'Tis a derivative from me to mine, | | | 'Tis a derivative from me to mine, |
| | And only that I stand for. | | | And only that I stand for. |
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| and these are the propositions that are commonly denoted: | | and these are the propositions that are commonly denoted: |
| 0, du, dv, du + dv, in other words, (), du, dv, (du, dv). | | 0, du, dv, du + dv, in other words, (), du, dv, (du, dv). |
| + | </pre> |
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| + | ==DATA. Note 16== |
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− | DATA. Note 16 | |
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| + | <pre> |
| | for equalities are so weighed | | | for equalities are so weighed |
| | that curiosity in neither can | | | that curiosity in neither can |
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| Well, g, that was easy, seeing as how Dg | | Well, g, that was easy, seeing as how Dg |
| is already linear at each locus, dg = Dg. | | is already linear at each locus, dg = Dg. |
| + | </pre> |
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| + | ==DATA. Note 17== |
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− | DATA. Note 17 | |
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| + | <pre> |
| We have been conducting the differential analysis | | We have been conducting the differential analysis |
| of the logical transformation F : [u, v] -> [u, v] | | of the logical transformation F : [u, v] -> [u, v] |
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| Take your pick, Gimli ... | | Take your pick, Gimli ... |
| + | </pre> |
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| + | ==DATA. Note 18== |
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− | DATA. Note 18 | |
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| + | <pre> |
| Let's push on with the analysis of the transformation: | | Let's push on with the analysis of the transformation: |
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| Exercise for the Reader. | | Exercise for the Reader. |
| + | </pre> |
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| + | ==DATA. Note 19== |
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− | DATA. Note 19 | |
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| + | <pre> |
| I'd never rob readers of exercise ... | | I'd never rob readers of exercise ... |
| but for my ain sense of an ending --- | | but for my ain sense of an ending --- |
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| | Leonard Cohen, "Teachers" (1967) | | | Leonard Cohen, "Teachers" (1967) |
| + | </pre> |
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| + | ==DATA. Note 20== |
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− | DATA. Note 20 | |
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| + | <pre> |
| In my work on "Differential Logic and Dynamic Systems", | | In my work on "Differential Logic and Dynamic Systems", |
| I found it useful to develop several different ways of | | I found it useful to develop several different ways of |
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| that u and v are not both true at the same time, while | | that u and v are not both true at the same time, while |
| du is equal in value to v, and dv is the opposite of u. | | du is equal in value to v, and dv is the opposite of u. |
| + | </pre> |
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| + | ==DATA. Note 21== |
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− | DATA. Note 21 | |
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| + | <pre> |
| By way of providing a simple illustration of Cook's Theorem, | | By way of providing a simple illustration of Cook's Theorem, |
| namely, that "Propositional Satisfiability is NP-Complete", | | namely, that "Propositional Satisfiability is NP-Complete", |
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| The "tape-head" (that is, the "read-unit") will be called "H". | | The "tape-head" (that is, the "read-unit") will be called "H". |
| The "registers" are also called "tape-cells" or "tape-squares". | | The "registers" are also called "tape-cells" or "tape-squares". |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 22== |
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− | DATA. Note 22 | |
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| + | <pre> |
| To see how each finite approximation to a given turing machine | | To see how each finite approximation to a given turing machine |
| can be given a purely propositional description, one fixes the | | can be given a purely propositional description, one fixes the |
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| At the point-in-time p_i, | | At the point-in-time p_i, |
| the tape-cell r_j bears the mark s_k. | | the tape-cell r_j bears the mark s_k. |
| + | </pre> |
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| + | ==DATA. Note 23== |
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− | DATA. Note 23 | |
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| + | <pre> |
| Given but a single free square on the tape, there are just | | Given but a single free square on the tape, there are just |
| two different sets of initial conditions for Stunt(2), the | | two different sets of initial conditions for Stunt(2), the |
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| At time p_0, cell r_1 contains "1", and | | At time p_0, cell r_1 contains "1", and |
| At time p_0, cell r_2 contains "#". | | At time p_0, cell r_2 contains "#". |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 24== |
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− | DATA. Note 24 | |
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| + | <pre> |
| A complete description of Stunt(2) in propositional form is obtained by | | A complete description of Stunt(2) in propositional form is obtained by |
| conjoining one of the above choices for initial conditions with all of | | conjoining one of the above choices for initial conditions with all of |
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| ( p1_q1 p1_r1 p1_r1_s# ( p2_q* p2_r0 p2_r1_s# )) | | ( 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# )) | | ( p1_q1 p1_r2 p1_r2_s# ( p2_q* p2_r1 p2_r2_s# )) |
| + | </pre> |
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| + | ==DATA. Note 25== |
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− | DATA. Note 25 | |
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| + | <pre> |
| Interpretation of the Propositional Program | | Interpretation of the Propositional Program |
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| If M at p_1 is in state q_#, then M at p_2 is in state q_#, and | | If M at p_1 is in state q_#, then M at p_2 is in state q_#, and |
| If M at p_1 is in state q_*, then M at p_2 is in state q_*. | | If M at p_1 is in state q_*, then M at p_2 is in state q_*. |
| + | </pre> |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 26== |
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− | DATA. Note 26 | |
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| + | <pre> |
| Interpretation of the Propositional Program (cont.) | | Interpretation of the Propositional Program (cont.) |
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| At time p_2, M is in state q_#, or | | At time p_2, M is in state q_#, or |
| At time p_2, M is in state q_*. | | At time p_2, M is in state q_*. |
| + | </pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 27== |
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− | DATA. Note 27 | |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Interpretation of the Propositional Program (cont.) | | Interpretation of the Propositional Program (cont.) |
| | | |
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| At each of the points in time p_i, for i in the set {0, 1, 2} | | At each of the points in time p_i, for i in the set {0, 1, 2} |
| M can be in exactly one state q_j, for j in the set {0, 1, #, *}. | | M can be in exactly one state q_j, for j in the set {0, 1, #, *}. |
| + | </pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 28== |
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− | DATA. Note 28 | |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| | | |
| + | <pre> |
| Interpretation of the Propositional Program (cont.) | | Interpretation of the Propositional Program (cont.) |
| | | |
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| At each of the points in time p_i, for i = 0, 1, 2, | | At each of the points in time p_i, for i = 0, 1, 2, |
| H is reading exactly one cell r_j, for j = 0, 1, 2. | | H is reading exactly one cell r_j, for j = 0, 1, 2. |
| + | </pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 29== |
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− | DATA. Note 29 | |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| | | |
| + | <pre> |
| Interpretation of the Propositional Program (cont.) | | Interpretation of the Propositional Program (cont.) |
| | | |
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| in each of the tape registers r_j, for j 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, #}. | | there can be exactly one sign s_k, for k in {0, 1, #}. |
| + | </pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 30== |
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− | DATA. Note 30 | |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| | | |
| + | <pre> |
| Interpretation of the Propositional Program (cont.) | | Interpretation of the Propositional Program (cont.) |
| | | |
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| one such constraint on symbol changes for each combination of the | | one such constraint on symbol changes for each combination of the |
| times p_0, p_1, registers r_0, r_1, r_2, and symbols s_0, s_1, s_#. | | times p_0, p_1, registers r_0, r_1, r_2, and symbols s_0, s_1, s_#. |
| + | </pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 31== |
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− | DATA. Note 31 | |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| + | <pre> |
| Interpretation of the Propositional Program (cont.) | | Interpretation of the Propositional Program (cont.) |
| | | |
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| At time p_1, H is reading cell r_2, and | | At time p_1, H is reading cell r_2, and |
| At time p_1, cell r_1 contains "1". | | At time p_1, cell r_1 contains "1". |
| + | </pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 32== |
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− | DATA. Note 32 | |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| | | |
| + | <pre> |
| Interpretation of the Propositional Program (cont.) | | Interpretation of the Propositional Program (cont.) |
| | | |
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Line 2,406: |
| to get the output of the program P is to read off the proper | | to get the output of the program P is to read off the proper |
| part of the data from the expression of this interpretation. | | part of the data from the expression of this interpretation. |
| + | </pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Note 33== |
− | | |
− | DATA. Note 33 | |
− | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| | | |
| + | <pre> |
| Interpretation of the Propositional Program (concl.) | | Interpretation of the Propositional Program (concl.) |
| | | |
Line 2,603: |
Line 2,527: |
| the tape head H when and if the machine M reaches one of | | the tape head H when and if the machine M reaches one of |
| its resting states, we get the result that Parity(1) = 1. | | its resting states, we get the result that Parity(1) = 1. |
| + | </pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Work Area== |
− | | |
− | DATA. Note 34
| |
− | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
− | | |
− | | |
− | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
− | | |
− | DATA. Work Area 2 | |
− | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| | | |
| + | <pre> |
| DATA 20. http://forum.wolframscience.com/showthread.php?postid=791#post791 | | DATA 20. http://forum.wolframscience.com/showthread.php?postid=791#post791 |
| | | |
Line 2,696: |
Line 2,610: |
| | | | | | | |
| o---------------------------------------o | | o---------------------------------------o |
| + | </pre> |
| | | |
| + | ==DATA. Discussion== |
| | | |
− | | + | <pre> |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
− | | |
− | DATA. Differential Analytic Turing Automata -- Discussion
| |
− | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
− | | |
− | DATA. Discussion Note 1
| |
− | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
− | | |
| PD = Philip Dutton | | PD = Philip Dutton |
| | | |
Line 2,765: |
Line 2,671: |
| last month without Google, is to archive a copy at one of the | | last month without Google, is to archive a copy at one of the |
| other Google-visible discussion lists that I'm on at present. | | other Google-visible discussion lists that I'm on at present. |
| + | </pre> |
| | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | ==DATA. Document History== |
− | | |
− | DATA. Discussion Note 2 | |
− | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
− | | |
− | | |
− | | |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| | | |
| + | <pre> |
| DATA. Differential Analytic Turing Automata | | DATA. Differential Analytic Turing Automata |
| | | |
Line 2,841: |
Line 2,741: |
| 32. http://stderr.org/pipermail/inquiry/2004-June/001641.html | | 32. http://stderr.org/pipermail/inquiry/2004-June/001641.html |
| 33. http://stderr.org/pipermail/inquiry/2004-June/001642.html | | 33. http://stderr.org/pipermail/inquiry/2004-June/001642.html |
− | 34.
| |
| | | |
| NKS Forum | | NKS Forum |
Line 2,879: |
Line 2,778: |
| 32. http://forum.wolframscience.com/showthread.php?postid=1489#post1489 | | 32. http://forum.wolframscience.com/showthread.php?postid=1489#post1489 |
| 33. http://forum.wolframscience.com/showthread.php?postid=1490#post1490 | | 33. http://forum.wolframscience.com/showthread.php?postid=1490#post1490 |
− | 34.
| |
− |
| |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| </pre> | | </pre> |