WHAT IS A PHYSICAL REALIZATION OF A COMPUTATIONAL SYSTEM?

Transcription

1 II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, 2013 WHAT IS A PHYSICAL REALIZATION OF A COMPUTATIONAL SYSTEM? Marco Giunti - ALOPHIS, Università di Cagliari

2 THE PROBLEM PHYSICAL REALIZATION OF A COMPUTATIONAL SYSTEM KEY NOTION FOR FUNCTIONALISM SYMBOLIC APPROACH TO COGNITIVE SCIENCE AI NEVERTHELESS, NO SHARED ANALYSIS OR DEFINITION!!! II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

3 PREVIOUS ATTEMPTS Emulation Approach A computational system and its multiple physical realizations are thought as dynamical systems The realization relation is analyzed as a structure preserving mapping from the state space of the computational system into the state space of the physical one; same kind of mapping (an emulation function) between one computational system and many physical systems MULT. REALIZ. EXPLAINED? II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

4 WHY EMULATION APPROACH DOES NOT WORK 1/2 Emulation is a structure preserving mapping between two mathematical systems (i.e., dynamical systems) the realizing system (the physical one) is not a real or concrete system, but another mathematical or abstract system INFINITE REGRESS, or PLATONISM II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

5 WHY EMULATION APPROACH DOES NOT WORK 2/2 Emulation based strategy provides us with an in principle solution (it could be done, but no detail) In principle solution is not the kind of answer we have in mind when we ask what it means for a computational system to be realized by a physical one This question does not ask for abstract in principle answers, but for detailed and concrete ones NO REFERENCE TO HIGH LEVEL / LOW LEVEL RATHER: ABSTRACT / CONCRETE II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

6 A NEW STRATEGY COMPUTATIONAL SYSTEMS ARE COMPLEX OBJECTS MORE SIMILAR TO EMPIRICALLY CORRECT DYNAMICAL MODELS, THEN TO DYNAMICAL SYSTEMS TOUT COURT. SEARCH SOLUTION AMONG MODELING RELATIONS DS/PHENOMENA NOT: EMULATION RELATIONS DS/DS II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

7 THREE PARTS OF COMPUTAT. SYSTEMS MATHEMATICAL PART: DS = (M, (g t ) t T ) DISCRETE, N-COMPONENT DYNAMICAL SYSTEM COMPUTATIONAL SETUP: H = (F, B F ) theoretical part F real part B F INTERPRETATION: I DS,H LINKS the dynamical system DS with the setup H ANALYSIS SOLUTION REALIZATION PROBLEM II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

8 The setup H = (F, B F ) 1/2 The theoretical part F is a functional description of: organization and internal functioning of a real system type AS F a causal scheme CS F of AS F s external interactions. Must include the specification of: initial conditions that an arbitrary evolution of any real system of type AS F must satisfy; boundary conditions during the whole subsequent evolution; and, possibly, the final conditions under which the evolution terminates. II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

9 The setup H = (F, B F ) 2/2 The real part B F is the (fuzzy) set of all real or concrete systems which satisfy the functional description F or, that is the same: B F is the (fuzzy) set of all real systems of type AS F whose temporal evolutions are all constrained by the causal scheme CS F B F is called the realization domain (or application domain) of H. Any b F B F is called an F-realizer. II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

10 The Turing machine setup H TM = (F TM, B F TM) external memory read/write/move head q 1 read/write internal memory q 1 0 1Rq 2 q 1 1 : 1Lq 1 q 2 0 : 1Rq 1 q 2 1 : 1Hq 2 : control unit II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

11 The Turing machine setup H TM = (F TM, B F TM) external memory read/write/move head q 1 read/write internal memory q 1 0 : 1Rq 2 q 1 1 : 1Lq 1 q 2 0 : 1Rq 1 q 2 1 : 1Hq 2 control unit II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

12 The Turing machine setup H TM = (F TM, B F TM) external memory read/write/move head q 2 read/write internal memory q 1 0 : 1Rq 2 q 1 1 : 1Lq 1 q 2 0 : 1Rq 1 q 2 1 : 1Hq 2 control unit II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

13 The Turing machine dynamical system DS TM = (Q C Z, (g t ) t Z +) 1/3 VARIABLE DOMAIN STATE VARIABLE q Q = {q 1,..., q n } YES c C YES p Z YES a A = {b, a 1,..., a m } NO m {-1, 0, +1} NO II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

14 The Turing machine dynamical system DS TM = (Q C Z, (g t ) t Z +) 2/3 FUNCTION CODOMAIN DEFINITION READ(c, p) A = {b, a 1,..., a m } READ(c, p) := c(p) WRITE(a, c, p) C WRITE(a, c, p) := c C such that, for any x Z: if x = p, c (x) = a; else, c (x) = c(x) MOVE(p, m) Z MOVE(p, m) := p + m A(q, a) A = {b, a 1,..., a m } A(q, a) := for any q and a, A(q, a) is listed in a given finite table T1 M(q, a) {-1, 0, +1} M(q, a) := for any q and a, M(q, a) is listed in a given finite table T2 Q(q, a) Q = {q 1,..., q n } Q(q, a) := for any q and a, Q(q, a) is listed in a given finite table T3 II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

15 The Turing machine dynamical system DS TM = (Q C Z, (g t ) t Z +) 3/3 The Turing machine dynamical system DS TM = (Q C Z, (g t ) t Z +) is the dynamical system univocally specified by the 3-component difference equation below. q = Q(q, READ(c, p)) c = WRITE(A(q, READ(c, p)), c, p) p = MOVE(p, M(q, READ(c, p))) II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

16 The interpretation I DS,H An interpretation I DS,H of DS on H consists in identifying each component C i of the state space M with (a subset of) all the possible values V(M i ) of a magnitude M i of setup H, and the time set T of DS with all the possible values V(T) of the time magnitude T of setup H. I DS,H is thus an m-tuple of m = n+1 statements I DS,H = <C 1 V(M 1 ),..., C n V(M n ), T = V(T)> II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

17 The intended interpretation of the TM dynamical system on the TM setup the Turing machine dynamical system DS TM = (Q C Z, (g t ) t Z +) the Turing machine setup H TM = (F TM, B FTM ) the intended interpretation of DS TM on H TM I(DS TM, H TM ) = <Q = V(Q), C = V(C), Z = V(P), Z + = V(T)> Q is the content of the internal memory C is the whole content of the external memory P is the position of the head II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

18 I DS,H : possible states of setup H given an interpretation I DS,H = <C 1 V(M 1 ),..., C n V(M n ), T = V(T)> it is possible to define: x is a state of H relative to I DS,H := x V(M 1 )... V(M n ) V(M 1 )... V(M n ) is called the state space of H relative to I DS,H, and is indicated by M. II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

19 I DS,H : the instantaneous state of an F-realizer of setup H = (F, B F ) Let b F B F be an arbitrary F-realizer of setup H. x is the state of b F at instant i relative to I DS,H := x = (x 1,..., x n ), where x j is the value (if any) at instant i T of magnitude M j of b F Note that, depending on the instant i, the value at instant i of magnitude M j of b F may not exist. If this is the case, the state of b F at instant i does not exist either. II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

20 I DS,H : the initial states of b F Relative to the interpretation I DS,H, we may define the set of all those states in M (if any) which are states of b F at the initial instant of some of its evolutions. C bf := {x: x M and x is the state of H at instant i, for some i T such that i is the initial instant of some evolution of b F } C bf is called the set of the initial states of b F II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

21 I DS,H : possibly correct interpretation of DS on H Note that, depending on the interpretation I DS,H and the realizer b F, C bf may be empty, or C bf may not be a subset of M. Since we are not interested in this kind of somewhat pathological interpretations, we define: I DS,H is a possibly correct interpretation of DS on H := for some b F B F, C bf and for any b F B F, C bf M. II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

22 I DS,H : correct interpretation of DS on H For any x C bf, let I bf,x := {i x : i x is the initial instant of some evolution of b F, and x is the state of b F at instant i x }. In other words, i x I bf,x is any of the initial instants of b F at which x is the state of b F. I DS,H is a correct interpretation of DS on H := I DS,H is a possibly correct interpretation of DS on H and, for any b F B F, for any x C bf, for any i x I bf,x, for any t T, if the state of b F at instant t + i x exists, then g t (x) = the state of b F at instant t + i x. II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

23 I DS,H : realization of DS on H Finally, we can now define: I DS,H is a realization of DS on H = (F, B F ) := the theoretical part F of the setup H, in conjunction with the specification of both DS and I DS,H, entail that I DS,H is a correct interpretation of DS on H. II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

24 The intended interpretation of DS TM on H TM is a realization of DS TM on H TM It is easy to verify that, in the case of a Turing machine, the intended interpretation I(DS TM, H TM ) of the Turing machine dynamical system DS TM on the Turing machine setup H TM is a realization of DS TM on H TM. I maintain that this fact is not peculiar to Turing machines, but it is one of the essential features of all computational systems. II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

25 A-priori vs. a-posteriori correct interpretation Computational systems are characterized by a form of a-priori (or purely theoretical) interpretation of the mathematical part on the setup part. The a-priori character of the interpretation of computational systems distinguishes these systems from ordinary dynamical models of phenomena. For this second kind of dynamical systems, the correctness of the interpretation of the mathematical part on the setup part (i.e., the phenomenon) is not a- priori, but a-posteriori or empirical. II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25

26 THANK YOU II Cagliari-Urbino Meeting. Metaphor, analogy, reasoning. Urbino, May 27-28, /25