HUMAN COMPUTATION FROM A STRICTLY DYNAMICAL POINT OF VIEW

Transcription

1 Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, 2013 HUMAN COMPUTATION FROM A STRICTLY DYNAMICAL POINT OF VIEW Marco Giunti - ALOPHIS, Università di Cagliari

2 SUMMARY 1/2 SHOW HOW TURING'S THESIS CAN BECOME THE CENTRAL EMPIRICAL CLAIM OF A STRICTLY DYNAMICAL THEORY OF HUMAN COMPUTATION Reconstruct the main steps of Turing's justification of Church's thesis, and enucleate Turing's thesis (TT). Explicitly define the type of machine (HTM) on which such justification is based. Following Gödel, point out a serious flaw in Turing's argument and in particular in his justification of (TT). Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

3 SUMMARY 2/2 On the basis of Gödel's criticism, define a more general type of machine (ETM) that fully retains the dynamical content of HTMs, but lacks their algorithmic nature. Retrieve the algorithmic nature of HTMs by defining a third type of machine (n dtm). Use this type of machine to reinstate a more general version of Turing's thesis (TT*). Finally, explain why (TT*) can naturally be interpreted as the central claim (MTT) of a new empirical theory of human computation. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

4 TURING 1936, beginning of sec. 9: How to justify Church's Thesis (CT)? No attempt has yet been made to show that the "computable" numbers include all numbers which would naturally be regarded as computable. The real question at issue is "What are the possible processes which can be carried out in computing a number?" That is to say: IN ORDER TO JUSTIFY CT, WE NEED AN ANALYSIS OF HUMAN COMPUTING IN GENERAL Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

5 HUMAN COMPUTING JOB (PHC) Phenomenon of Human Computation (PHC). Any activity of a human being that consists in executing a purely mechanical or effective procedure (an algorithm) A mechanical procedure (or an algorithm) is a finite set of clear-cut formal instructions for symbol manipulation; given a finite collection of initial data, a human being must be able to carry out such instructions in a definite sequence of steps, with the exclusive aid of paper and pencil (or equivalent external devices), and without resorting to any special insight or ingenuity. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

6 TURING 1936, sec. 9.I: Analysis of an arbitrary 1-d Human Computing Job Hyp1. [1-dimensional external support] I assume that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares. Hyp2. [Finite number of symbols] The number of symbols which may be printed is finite. Hyp3. [Instantaneous condition 1/3] The behaviour of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. instantaneous condition = (state_of_mind, observed_symbols) Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

7 TURING 1936, sec. 9.I: Analysis of an arbitrary 1-d Human Computing Job Hyp4. [Finite reading neighborhood] There is a bound B to the number of symbols or squares which the computer can observe at one moment. Hyp5. [Finite number of possible states of mind] The number of states of mind which need be taken into account is finite. Hyp6. [Simple operations] Every such operation consists of some change of the physical system consisting of the computer and his tape. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

8 TURING 1936, sec. 9.I: Analysis of an arbitrary 1-d Human Computing Job Hyp7. [State of the system] We know the state of the system if we know the sequence of symbols on the tape, which of these are observed by the computer (possibly with a special order), and the state of mind of the computer. state = (tape_content, posit_of_observ_symbs, state_of_mind) Hyp8. [Write one symbol at a time] In a simple operation not more than one symbol is altered. Hyp9. [Write within observed neighborhood] The squares whose symbols are changed are always "observed" squares. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

9 TURING 1936, sec. 9.I: Analysis of an arbitrary 1-d Human Computing Job Hyp10. [Finite moving neighborhood] Each of the new observed squares is within L squares of an immediately previously observed square. Hyp11. [Instantaneous condition 2/3] The operation actually performed is determined, as has been suggested on p. 250, by the state of mind of the computer and the observed symbols. Hyp12. [Instantaneous condition 3/3] In particular, they determine the state of mind of the computer after the operation is carried out. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

10 TURING 1936, sec. 9.1:Turing's Thesis (TT) and Human like Turing Machines (HTMs) Turing's Thesis (TT) We may now construct a machine to do the work of this computer. Functional Description of a HUMAN LIKE TURING MACHINE (HTM) To each state of mind of the computer corresponds an "mconfiguration" of the machine. The machine scans B squares corresponding to the B squares observed by the computer. In any move the machine can change a symbol on a scanned square or can change any one of the scanned squares to another square distant not more than L squares from one of the other scanned squares. The move which is done, and the succeeding configuration, are determined by the scanned symbol and the m-configuration. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

11 TURING 1936, sec. 9.1: UPSHOT JUSTIFICATION OF CHURCH'S THESIS The machines just described [HTMs] do not differ very essentially from computing machines as defined in 2, and corresponding to any machine of this type [for any HTM] a computing machine [an ordinary TM] can be constructed to compute the same sequence, that is to say [by TT] the sequence computed by the computer. [Church's Thesis is thus justified] Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

12 Human like Turing Machine (HTM) DS HTM = (Q C P, (g t ) t Z +) 1/3 VARIABLE DOMAIN STATE VARIABLE a A := {b, a 1,..., a m } NO q Q := {q 1,..., q n } YES p P := the set of all integers Z YES c r C := the set of all functions c: P A such that c(p) b for at most finitely many p P R := a finite set of integers such that 0 R (read/write neighborhood) YES s S := the set of all functions s: R A NO NO m M := a finite set of integers such that 0 M (move neighborhood) NO Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

13 Human like Turing Machine (HTM) DS HTM = (Q C P, (g t ) t Z +) 2/3 FUNCTION CODOMAIN DEFINITION READ(c, p, R) S READ(c, p, R) := s S such that, for any r R, s(r) = c(p + r) WRITE(a, c, p, r) C WRITE(a, c, p, r) := c C such that, for any x P: if x = p + r, c (x) = a; else, c (x) = c(x) MOVE(p, m) P MOVE(p, m) := p + m A(q, s) A A(q, s) := for any q and s, A(q, s) is listed in a given finite table T1 R(q, s) R R(q, s) := for any q and s, R(q, s) is listed in a given finite table T2 M(q, s) M M(q, s) := for any q and s, M(q, s) is listed in a given finite table T3 Q(q, s) Q Q(q, s) := for any q and s, Q(q, s) is listed in a given finite table T4 Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

14 Human like Turing Machine (HTM) DS HTM = (Q C P, (g t ) t Z +) 3/3 a Human like Turing Machine (HTM) is the dynamical system DS HTM = (Q C P, (g t ) t Z +) univocally specified by the 3-component difference equation below. q = Q(q, READ(c, p, R)) c = WRITE(A(q, READ(c, p, R)), c, p, R(q, READ(c, p, R))) p = MOVE(p, M(q, READ(c, p, R))) Note that any ordinary Turing Machine (TM) is in fact an HTM such that R = {0} and M = {-1, 0, 1}. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

15 THE CRUCIAL POINT OF TURING'S ARGUMENT TURING'S THESIS (TT) All the force of the argument rests on the adequacy of Turing's analysis of an arbitrary 1-d human computing job (1-d PHC). If any1-d PHC does indeed satisfy Turing's 12 hypotheses, then there is no doubt that: (TT) For any 1-d PHC, there is a human like Turing machine HTM that does exactly the work of the human computer. And, if (TT) is OK, then the whole argument is OK. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

16 ARE ALL 12 HYPOTHESES REALLY TENABLE? In particular, Hyp5? Turing, in Proc. Lond. Math. Soc. 42 (1936), p. 250, gives an argument which is supposed to show that mental procedures cannot carry any farther than mechanical procedures. However, this argument is inconclusive, because it depends on the supposition that a finite mind is capable of only a finite number of distinguishable states. What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing. Therefore, although at each stage of the mind s development the number of its possible states is finite, there is no reason why this number should not converge to infinity in the course of its development. (Kurt Gödel, reported in Wang, Hao, From Mathematics to Philosophy, Routledge & Kegan Paul, 1974, p. 325) Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

17 NUMBER OF STATES OF MIND MAY GROW WITHOUT BOUND Gödel's criticism points out that there is no obvious limit to the human capacity to attain new mental states. Potential infinity of mental states number seems to be a direct consequence of the most basic fact that mental states are not simple units, but have a compositional character. In HTMs, not only are possible internal states a finite number, but they lack any compositional structure. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

18 ENHANCED TURING MACHINE (ETM) Among the 12 hypotheses, reject those which are not tenable features of any PHC: Hyp1. [1-dimensional external support] Hyp5. [Finite number of possible states of mind] Perhaps, Hyp10. [Finite moving neighborhood] If we do this, we arrive at the concept of a more general system an Enhanced Turing Machine (ETM). Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

19 ENHANCED TURING MACHINE (ETM) DS ETM = (Q C P, (g t ) t Z +) 1/3 VARIABLE DOMAIN STATE VARIABLE a A := {b, a 1,..., a m } NO q Q := a possibly infinite set YES p P := the set of all integer n-tuples Z n YES c r C := the set of all functions c: P A such that c(p) b for at most finitely many p P R := a finite set of integer n-tuples such that 0 R (read/write neighborhood) YES s S := the set of all functions s: R A NO NO m M := a (finite) set of integer n-tuples such that 0 M (move neighborhood) NO Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

20 ENHANCED TURING MACHINE (ETM) DS ETM = (Q C P, (g t ) t Z +) 2/3 FUNCTION CODOMAIN DEFINITION READ(c, p, R) S READ(c, p, R) := s S such that, for any r R, s(r) = c(p + r) WRITE(a, c, p, r) C WRITE(a, c, p, r) := c C such that, for any x P: if x = p + r, c (x) = a; else, c (x) = c(x) MOVE(p, m) P MOVE(p, m) := p + m A(q, s) A A(q, s) := for any q and s, A(q, s) exists and is unique R(q, s) R R(q, s) := for any q and s, R(q, s) exists and is unique M(q, s) M M(q, s) := for any q and s, M(q, s) exists and is unique Q(q, s) Q Q(q, s) := for any q and s, Q(q, s) exists and is unique Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

21 ENHANCED TURING MACHINE (ETM) DS ETM = (Q C P, (g t ) t Z +) 3/3 an Enhanced Turing Machine (ETM) is the dynamical system DS ETM = (Q C P, (g t ) t Z +) univocally specified by the 3-component difference equation below. q = Q(q, READ(c, p, R)) c = WRITE(A(q, READ(c, p, R)), c, p, R(q, READ(c, p, R))) p = MOVE(p, M(q, READ(c, p, R))) Note that any human like Turing machine HTM is in fact an ETM such that P = Z 1 = Z, both R and M are finite subsets of Z, and Q is finite. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

22 In general, ETMs are not algorithmic Enhanced Turing Machines retain all the purely dynamical aspects of HTMs and TMs, but ETMs do not preserve their algorithmic or procedural character. For, as Q may be infinite, the four action functions A, R, M, Q cannot in general be expressed as finite tables, and so they cannot be interpreted as a finite set of conditional instructions for symbol manipulation. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

23 HOW TO PRESERVE IN ETMs THE ALGORITHMIC NATURE of HTMs? If we just allow the number of internal states to be infinite, we lose the algorithmic nature of Turing Machines (HTMs or TMs). Thus, we need some constraint on the structure of internal states, so that an algorithmic interpretation of the four action functions A, R, M, Q be always possible. The first step in this direction is to recognize the compositional character of internal states. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

24 INTERNAL STATE of an n-dimensional TM An internal state of an n-dtm is the finite sequence formed by the contents of a finite sequence of registers R[B], R[X 1 ],..., R[X v ], where the content of R[B] is one of the symbols in B = {q 1,..., q n }; each q i is called a simple internal state; the content of R[X i ] is a finite string member of X i, where X i is a subset of the set of all strings built out of a given finite alphabet A i = {a 1i,..., a mi }; an equivalence relation i is given on the set X i, which fixes the identity conditions for the strings in X i ; (X i, i ) is called a data-type (or an object-type). Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

25 AUXILIARY RELATIONS AND FUNCTIONS of an n-dimensional TM Besides compositional internal states, we also admit auxiliary relations and functions whose domains and codomains are specified data-types. Thus, auxiliary relations and functions all operate on strings of symbols. More precisely: an auxiliary function f 1,..., f u,... is a k-arguments function whose domains and codomain are specified data-types; an auxiliary relation C 1,..., C d,... is a k-ary relation whose domains are specified data-types. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

26 SIMPLE n-dimensional TM [n-dhtm] A simple n-dtm is an ETM where Q = B. As Q = B, in a simple n-dtm there is only a finite number of internal states (the simple ones), so that the four functions A, R, M, Q are always definable by cases by means of a conditional of the following form: q i1 ; s j1 : a k1 ; r p1 ; m w1 ; q s1 where each subsequent condition is q i2 ; s j2 : a k2 ; r p2 ; m w2 ; q s2 evaluated only if the previous one fails;... ;... :... ;... ;... ;... a is the symbol in the 0 position of the [a finite number of sixtuples] presently scanned neighborhood, and else : a; 0; 0; q q is the present simple internal state. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

27 n-dimensional Turing Machine 1/2 M is an n-dtm : = (i) M is a simple n-dtm or (ii) M is an ETM such that its functions A, R, M, Q are definable by cases by a conditional of the following form: q i1, x 1,...,x v ; C d1 (s,x 1,...,x v ) : f a1 (s,x 1,...,x v ); f r1 (s,x 1,...,x v ); f p1 (s,x 1,...,x v ); q s1, f u1 1 (s,x 1,...,x v ),..., f u v 1 (s,x 1,...,x v ) q i2, x 1,...,x v ; C d2 (s,x 1,...,x v ) : f a2 (s,x 1,...,x v ); f r2 (s,x 1,...,x v ); f p2 (s,x 1,...,x v ); q s2, f u1 2 (s,x 1,...,x v ),..., f u v 2 (s,x 1,...,x v )... ;... :... ;... ;... ;... where x i is a variable whose [a finite number of sixtuples] domain is X i, and s is the else : a; 0; 0; q scanned pattern Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

28 n-dimensional Turing Machine 2/2 for any data type (X i, i ), both X i and i are decidable by an n-dtm; any auxiliary relation in the antecedent of the above conditional is decidable by an n-dtm, and any auxiliary function in the consequent is computable by an n-dtm. Note that the previous recursive definition makes it quite obvious that whatever is computable by an n dtm is computable by an ordinary TM as well. (A rigorous proof of this fact would be by induction on the complexity of an n-dtm.) Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

29 TURING'S THESIS REINSTATED (TT*) We can now recast (TT) in terms of nd TMs: (TT*) For any PHC, there is an n-dtm that does exactly the work of the human computer. This more general version of Turing's thesis is not subject to Gödel's criticism. Furthermore, as any PHC does seem to satisfy all Turing's hypotheses except hyp1 and hyp5, and perhaps hyp10, (TT*) has indeed a quite high degree of a-priori plausibility. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

30 TOWARD AN EMPIRICAL THEORY OF HUMAN COMPUTATION As Turing himself pointed out, his argument in favor of (TT), as well as mine in favor of (TT*), is in fact an appeal to intuition, and it thus has a speculative or a priori character. However, the high generality and flexibility of n dtm architecture opens up the possibility of actually constructing n dtms which are empirically correct dynamical models (i.e. Galilean models) of real phenomena of human computation. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

31 TURING'S THESIS AS AN EMPIRICAL CLAIM (MTT) In this perspective, Turing's thesis thus becomes the basic methodological claim of a new empirical theory of human computation: (MTT) For any specific phenomenon PHC of human computation, there is an appropriate n-dimensional Turing machine n dtm, and an empirical interpretation I n dtm,phc, such that (n dtm, I n dtm,phc ) turns out to be a Galilean model of PHC. Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31

32 THANK YOU Cagliari Colloquium on the Extended Mind, Dynamicism, and Computation. Cagliari, June 10, /31