1. Discrete dynamical systems and intrinsic computability Marco Giunti University of Cagliari, Italy giunti@unica.it http://edu.supereva.it/giuntihome.dadacasa

2. Outline General thesis – Computation theory is a special branch of dynamical systems theory, and its objects (the computational systems) are a special kind of discrete dynamical systems. The specific difference of these objects, i.e. the property of being computational, can be thought as an intrinsic property of their dynamics. 1.Dynamical systems: definition and examples. 2.Computational systems: definition 1 and why, by this definition, being computational is not intrinsic. 3.Dynamically effective representations of discrete systems and a refined, intrinsic, definition 2 of a computational system. 4.Possible consequences for computability theory: (some) non-recursive functions may turn out to be computable by a particular class of intrinsic computational systems.

3. A Dynamical System (DS ) is a mathematical model that expresses the idea of a deterministic system (discrete/continuous, revers./irrevers.)  A Dynamical System (DS) is a set theoretical structure (M, (g t ) tT ) such that: 1.the set M is not empty; M is called the state- space of the system; 2.the set T, is either Z, Z + (integers) or R, R + (reals); T is called the time set; 3.(g t ) tT is a family of functions from M to M; each function g t is called a state transition or a t-advance of the system; 4.for any t and w   T, for any x M, a.g 0 (x) = x; b.g t+w (x) = g w (g t (x)).

4. Intuitive meaning of the definition of dynamical system gt+wgt+w x gwgw x g0g0 x gtgt t0t0 t0+tt0+t gt(x)gt(x) t gtgt

5. Example of a continuous DS (Galilean model of free fall)  Explicit specification Let F = (M, (g t ) tT ) such that M = SV and S = V = T = real numbers g t (s, v) = (s + vt + at 2 /2, v + at)  Implicit specification Let F = (M, (g t ) tT ) such that M = SV and S = V = T = real numbers ds(t)/dt = v(t), dv(t)/dt = a

6. A standard functional scheme of a Turing machine  A physical realization of a Turing machine is any concrete system which satisfies (implements, works according to) the abstract functional scheme below. Control unit Internal memory External memory Read/write head Read/write/move head ajaj qiqi ajaj qiqi.. :... q i a j :a k Lq m... :...... :... akLakL qmqm

7. Example of a discrete DS ( Functional scheme of a Turing machine)  The abstract functional scheme of a Turing machine can be identified with the discrete dynamical system T = (M, (g t ) tT ) such that: M = PCS, where P = Z (integers) is the set of the possible relative positions of the read/write/move head, C is the set of the possible contents of the whole external memory, and Q is the set of the possible contents of the internal memory; T = Z + (non-negative integers); let g be the function from M to M determined by the machine table of the functional scheme; then, g 0 is the identity function on M and, for any t 0, g t is the t-th iteration of g.

8. Computational systems: intuitive concept  Extensional characterization: by the term computational system I refer to any device of the kind studied by standard computation theory; e.g. Turing machines, register machines, cellular automata, finite state automata, etc. Discreteness and determinism are two properties shared by all such devices; thus, so called analog computers are not computational systems is this sense.  Intensional characterization: the computational systems can be identified with those discrete, deterministic dynamical systems that can be represented effectively.

9. The crucial question: What is an effective representation of a discrete dynamical system?  A natural definition (perhaps the most natural definition?) of effective representation is as follows:  an effective representation of a discrete dynamical system DS = (M, (g t ) tT ) is a pair (u, DS # ) such that: 1. DS # = (N, (h t ) tT ) is a discrete dynamical system, where N is either Z + or a finite initial segment of Z + ; 2. u: N  M is an isomorphism of DS # in DS; 3. for any t  T, h t is a recursive function.

10. The first definition of a computational system. Is it intrinsic?  If we buy the previous definition of an effective representation of a discrete dynamical system, we can then define:  DS is a computational 1 system iff DS is a discrete dynamical system, and there is an effective representation of DS.  Question: is the property of being computational 1 intrinsic to the dynamics of the discrete system DS? In fact, DS might admit two isomorphic numeric representations, such that one is recursive and the other is not. In this case, the property of being computational 1 could not be said to be intrinsic to the dynamics of DS, for it would depend on the numeric representation of the dynamics we choose.

11. Being computational 1 is not intrinsic  There is a discrete DS such that: it is obviously computational 1 (i.e., it has an effective representation = it has a recursive numeric representation); but, it also has a numeric representation that is not recursive (i.e., the first two conditions of the definition of effective representation are satisfied, but not the third).  Surprisingly enough, this system is DS 1 = (Z +, (s n ) nZ + ), i.e., the discrete dynamical system generated by iterating the successor function s.

12. DS 1 = (Z +, (s n ) n  Z + ) is computational 1, but not intrinsic. Sketch of proof (1/2)  Obviously, a recursive numeric representation of DS 1 = (Z +, (s n ) nZ + ), is (i, DS 1 ), where i: Z +  Z + is the identity function.  Consider an arbitrary bijection p: Z +  Z + and the “new successor function” s p on Z + corresponding to the order induced by p:

13. DS 1 = (Z +, (s n ) n  Z + ) is computational 1, but not intrinsic. Sketch of proof (2/2)  Thus, (p -1, DS p ), where DS p is the discrete dynamical system generated by s p, is a numeric representation of DS 1.  How many representations (p -1, DS p ) are there?  As many as the number of bijections p of the non-negative integers.  But the number of such bijections is uncountable.  Therefore, there is p* such that (p* -1, DS p* ) is a non-recursive numeric representation of DS 1. Q.E.D.

14. The previous proof is surprising It is odd to realize that a dynamical system like DS p*, which has exactly the same structure as the sequence of the natural numbers, is generated by a non-recursive pseudo-successor function s p*, and that (p* ‑ 1, DS p* ) thus constitutes a bona fide non-recursive numeric representation of DS 1, which, in contrast, is generated by the authentic successor function that is obviously recursive.

15. Might ( p* ‑ 1, DS p* ) be not a bona fide numeric representation of the dynamics of DS 1 ?  Compare the “good” representation (i, DS 1 ) with the “odd” one (p* ‑ 1, DS p* ):  if we are given the whole structure of DS 1 (i.e., the successor function s: Z +  Z + ), we can mechanically produce the identity function i by simply starting from state 0 and counting 0, then moving to state s(0) = 1 and counting 1, and so forth;  but it seems that, for any starting state, moving back and forth along the structure of DS 1 and counting whenever we reach a new state won’t allow us to produce such a complex p* ‑ 1.

16. The odd representation ( p* ‑ 1, DS p* ) is not dynamically effective  Thus, it seems that the “good” representation (i, DS 1 ) can be constructed effectively by means of a mechanical procedure that takes as given the whole structure of the state space M of DS 1 ;  while the “odd” one (p* ‑ 1, DS p* ) cannot be constructed effectively in this way.  To distinguish the two kinds of representations, let us then introduce the concept of a dynamically effective representation.

17. Dynamically effective representation (condition 3 is not formal)  A dynamically effective representation of a discrete dynamical system DS = (M, (g t ) tT ) is a pair (u, DS # ) such that: 1. DS # = (N, (h t ) tT ) is a discrete dynamical system, where N is either Z + or a finite initial segment of Z + ; 2. u: N  M is an isomorphism of DS # in DS; 3. the enumeration u: N  M can be constructed effectively by means of a mechanical procedure that takes as given the whole structure of the state space M of DS (and nothing more).

18. Lines for a formal analysis of condition 3  Condition 3 of the previous definition can be analyzed once we make clear what we mean by: whole structure of the state space; mechanical procedure that takes such a structure as given.  In extreme synthesis: the state-space structure can be identified with a special kind of connected (infinite) graph, which can assume nine types of general forms;  the mechanical procedure is the one executed by a special kind of ideal machine, which can move back and forth along the edges of such graphs and “count” 0, 1, 2,..., n,... whenever it reaches a new node.

19. The second definition of a computational system. Is it intrinsic?  Thus, we now have two possible formal explications of the intuitive idea of an effective representation of a discrete DS;  the first definition is the basis for the concept of a computational 1 system. But this concept is not intrinsic to the dynamics of DS, for it depends on the way we numerically represent such dynamics;  on the basis of the second definition, we can now define: DS is a computational 2 system iff DS is a discrete dynamical system, and there is a dynamically effective representation of DS.  Question: is the property of being computational 2 intrinsic to the dynamics of the discrete system DS?

20. Being computational 2 is intrinsic  First, being computational 2 is intrinsic to the dynamics of a discrete DS in an obvious, but not trivial, sense: for DS has a numeric representation (u, DS # ) whose enumeration u: N  M is constructed effectively by means of a mechanical procedure that takes as given the whole structure of the state space M of DS, i.e., the dynamics of DS.  Second, there is a strong informal argument in favor of the conjecture that any two dynamically effective representations of the same DS are either both recursive or both non-recursive.

21. Two scenarios for computability theory  If (i) we buy the second definition of a computational system and (ii) the previous conjecture is true, there are two possible scenarios: 1. any computational 2 system DS is intrinsically recursive, i.e., for any dynamically effective representation (u, DS # = (N, (h t ) tT )) of DS, the dynamics (h t ) tT turns out to be recursive; 2. some computational 2 system DS is intrinsically non-recursive, i.e., for any dynamically effective representation (u, DS # = (N, (h t ) tT )) of DS, the dynamics (h t ) tT turns out to be non-recursive.

22. Consequences for Turing-Church’s thesis as a mathematical thesis  Turing-Church’s thesis (TC-thesis) can be interpreted in many different ways. The Mathematical TC-thesis (MTC-thesis) can be expressed as follows:  any numeric function that can be computed by a computational system (in the intuitive sense) is recursive.  But then, provided that computational 2 is a good explication for the intuitive concept of a computational system, it is clear that the truth of either scenario (1) or scenario (2) entails, respectively, the truth or falsity of MTC-thesis.

23. That’s all Thank you