Discrete Systems with Undecidable Dynamics Marco Giunti Marco Giunti Università di Cagliari 1Bochum, 20/04/2012

Summary Stephen Wolfram (2002): Most discrete systems that are not obviously simple   are computationally equivalent, and thus universal (Principle of Computational Equivalence PCE);   are such that almost all non-trivial questions about the behavior of the system after an infinite number of steps are undecidable (Dynamical Undecidability Thesis DUT). I will show how it is possible to support and, perhaps, ultimately ground or explain Wolfram's principles, by looking at dynamics from a quite general and "non- classical" point of view. 2Bochum, 20/04/2012

Rule Rule number (base 2) = (base 10) Time Time cells arranged in a circle: The left neighbor of the leftmost cell is the rightmost cell. Each cell may take value 0 or 1: Thus, this CA ha 2 12 = 4096 possible states. A finite cellular automaton 3Bochum, 20/04/2012

64 states – 60 copies32 states – 6 copies 16 states – 4 copies Total: 4096 = 2 12 states The seventy constituent subsystems of CA rule – 12 cells in a circle (Wolfram 1983b) 4

Wolfram's original studies on CAs  In the ’80s Wolfram (1983a) studied by extensive computer simulations the simplest 256 CAs (modimensional, 2 values, radius 1 neighborhood);  He classified them into 4 qualitative similarity types of behavior (Wolfram 1983b, 1984b):  CAs whose evolution leads to 1.an homogeneous state, stable or periodic; 2.a set of distinct simple structures: (i) fixed size; (ii) growing size, with a repetitive or nested pattern; 3.a chaotic or apparently random behavior; 4.localized complex structures, often with a quite long life span. 5Bochum, 20/04/2012

Type 2 (repetitive): rule 250, 100 steps Initial Conditions: a single black cell 6Bochum, 20/04/2012

Type 2 (nested): rule 90, 1000 steps Initial Conditions: a single black cell 7Bochum, 20/04/2012

Type 3 (random): rule 30, 1000 steps Initial Conditions: a single black cell 8Bochum, 20/04/2012

Type 4 (complex): rule 110, 1000 steps Initial Conditions: a single black cell 9Bochum, 20/04/2012

Wolfram's Principle of Computational Equivalence (PCE)  Wolfram conjectured that, among the simplest 256 CAs, any type 4 system is computationally universal (1984b, 31);  he then proved (2002, ) that rule 110 (type 4) is universal, and that it is equivalent to the other 3 rules of type 4 (2002, 60);  finally, on the basis of extensive phenomenological studies of the dynamics of many types of discrete systems, Wolfram generalized this result, by putting forth the following hypothesis:  Principle of Computational Equivalence (PCE) Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication (2002, ) Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication (2002, ) 10Bochum, 20/04/2012

Computational universality of type 4 systems – what does it mean?  It means that any type 4 system is able to emulate, i.e. exactly reproduce, the behavior of a whole class of systems that is known to be computationally universal.  In fact, the proof of the universality of rule 110 shows that, with appropriate initial conditions, this rule is able to emulate any tag system (the class of tag systems is computationally universal, for any Turing machine can in turn be emulated by an appropriate tag system). 11Bochum, 20/04/2012

Emulation between CAs – an example  CA 18 emulates CA 90 in two steps, with initial conditions 00 for 0 and 01 for 1 (Wolfram 1983b, 20) Rule number = Rule number = Bochum, 20/04/2012

Consequences of PCE  Ubiquity of Computational Universality –“Almost any rule whose behavior is not obviously simple should ultimately be capable of achieving the same level of computational sophistication and should thus in effect be universal” (Wolfram 2002, 718) –Strictly speaking, this does not follow from the previous formulation of the principle (needs premise: the behavior of a computationally universal system is not obviously simple). It can also be viewed as a different formulation of PCE.  Ubiquity of Complexity –According to PCE, “observers will tend to be computationally equivalent to the systems they observe—with the inevitable consequence that they will consider the behavior of such systems complex.” (Wolfram 2002, 737) 13Bochum, 20/04/2012

Further consequences of PCE  Computational Irreducibility –According to PCE, “systems one uses to make predictions cannot be expected to make computations that are any more sophisticated than the computations that occur in all sorts of systems whose behavior we might try to predict. And from this it follows that for many systems no systematic prediction can be done, so that there is no general way to shortcut their process of evolution, and as a result their behavior must be considered computationally irreducible.” (Wolfram 2002, 741)  Free Will (follows from Computational Irreducibility) –“But so in the end what makes us think that there is freedom in what a system does? In practice the main criterion seems to be that we cannot readily make predictions about the behavior of the systems” (Wolfram 2002, 751) 14Bochum, 20/04/2012

One more consequence of PCE  Dynamical Undecidability Thesis (DUT) (follows from Computational Irreducibility) –“And what I suspect is that for almost any system whose behavior seems to us complex almost any non-trivial question about what the system does after an infinite number of steps will be undecidable.” (Wolfram 2002, 755) 15Bochum, 20/04/2012

General definition of a dynamical system – minimal time model: a monoid Definition 1: DS L is a dynamical system on L := DS L is a pair (M, (g t ) t  T ) and L is a pair (T, +) such that (i) L = (T, +) is a monoid. Any t  T is called a duration of the system, T is called its time set, and L its time model; (ii) M is a non-empty set. Any x  M is called a state of the system, and M is called its state space; (iii) (g t ) t  T is a family indexed by T of functions from M to M. For any t  T, the function g t is called the state transition of duration t (briefly, t-transition, or t-advance) of the system; (iv) for any v, t  T, for any x  M, a)g 0 (x) = x; b)g v+t (x) = g v (g t (x)). Bochum, 20/04/

Def. 1 captures the general notion of a deterministic dynamical system Determinism (iii) if x is the state at the present instant t 0 T, g t (x) is the state of the system after an evolution of duration t T (iii) if x is the state at the present instant t 0  T, g t (x) is the state of the system after an evolution of duration t  TDynamics (iv.a) whatever state x the system is in, the evolution of duration 0 does not modify that state (iv.b) any evolution of duration v+t can be decomposed in two successive evolutions, the first one of duration t, and the second one of duration v v+tv+t x v g 0 (x) = x 0 x t t0t0 t+t0t+t0 gt(x)gt(x) t t condition (iii) condition (iv.a) condition (iv.b) g v+t (x) = g v (g t (x)) 17Bochum, 20/04/2012

The transition graph of a DS L  The transition graph of a DS L is a directed and labeled graph that depicts the whole dynamics of the system.  Each point of the graph corresponds to exactly one state of the DS L, while each arrow stands for a state transition from its source to its target.  Each arrow is labeled with the duration of the corresponding state transition. 0,2,4, … 1,3,5, … Example  DS L = ({a, b}, (g t ) t  Z + )  L = (Z +, +)  if t is even, g t (x) = x if t is odd, g t (x) = y, y ≠ x  Any DS L can in fact be identified with its transition graph.  The transition graph of any DS L is a category. 18Bochum, 20/04/2012

The 1-step transition graph of a DS L with discrete time model  If the time model L is discrete, i.e., L = (Z +, +), the whole dynamics of the system is implicitly depicted by the graph of its first state transition g 1.  If L = (Z, +), the 1-step transition graph must also include g  1. Example (same as previous slide)  DS L = ({a, b}, (g t ) t  Z + )  L = (Z +, +)  if t is even, g t (x) = x if t is odd, g t (x) = y, y ≠ x The 1-step transition graph of the DS L above 19Bochum, 20/04/2012

Emulation between dynamical systems – Intuition and examples  Intuitively, a dynamical system DS L 1 emulates DS L 2 when the first one exactly reproduces any state transition of the second one. However, state transition durations are not usually preserved.  Examples – (i) a universal Turing machine emulates anyTM; (ii) for any TM, there is a CA that emulates TM, and conversely; (iii) emulation holds between the two simple CAs previously considered (CA 18 emulates CA 90). 20Bochum, 20/04/2012

Emulation between dynamical systems – Definition    there is an injective function u: N  M such that for any c  N, for any v  V, there is t  T such that u(h v (c)) = g t (u(c)). gtgt u hvhv u c DS L 1 emulates DS L 2 := MN u(N)u(N)   DS L 1 = (M, (g t ) tT ) is a dynamical system on L 1 = (T, +)   DS L 2 = (N, (h v ) vV ) is a dynamical system on L 2 = (V, )   N.B. The emulation function u is a quite weak structure preserving mapping of the emulated system into the emulating one. 21

Two definitions Let DS L = (M, (g t ) t  T ) be a dynamical system on monoid L = (T, +).  A constituent of DS L := any subsystem of DS L whose state space N  M is temporally connected in M and includes all its past (as well as its future).  DS L is indecomposable := DS L has just one constituent (i.e. DS L itself).  It is possible to show that any constituent of DS L is indecomposable. 22Bochum, 20/04/2012

Two general results  Decomposition theorem [for any dynamical system DS L with an arbitrary time model L = (T, +)] The transition graph of any dynamical system DS L on a monoid L exhaustively decomposes into a set of mutually disconnected and internally connected components. Such components are in fact all the constituent subsystems of DS L. The transition graph of any dynamical system DS L on a monoid L exhaustively decomposes into a set of mutually disconnected and internally connected components. Such components are in fact all the constituent subsystems of DS L.  Classification theorem [for any indecomposable dynamical system DS D with discrete time model D = (T, +), where T = Z or Z + ] The 1-step transition graph of any indecomposable dynamical system DS D with discrete time model D is of one of the forms (i) – (vii) described in the following slides. The 1-step transition graph of any indecomposable dynamical system DS D with discrete time model D is of one of the forms (i) – (vii) described in the following slides. 23Bochum, 20/04/2012

Time invertible systems (T = Z) Strictly reversible systems (T = Z + ) (i) Periodic, non-merging systems – a bidirected cycle of n nodes (n ≥ 1) (ii) Aperiodic, non-merging systems – a bidirected line, infinite in two directions Infinite Finite Infinite (iv) Aperiodic, non-merging systems – a directed line, infinite in two directions or infinite to the right only. (iii)Periodic, non-merging systems – a directed cycle of n nodes (n ≥ 1) Logically reversible systems (T = Z + ) 24

(v) Eventually periodic, non-merging systems – a directed cycle to which a possibly infinite directed line is attached Weakly irreversible systems (Finite or Infinite) (T = Z + ) 25

(vi) Eventually periodic, merging systems – a directed cycle to which the roots of a finite number of possibly infinite trees (either with respect to the number of levels – infinite height, or to the number of nodes in some level – infinite thickness) are attached; either at least two trees attach to the cycle, or the unique tree attached to it has different branches (i.e. it is not just a directed line) Strongly irreversible systems (Finite or Infinite) (T = Z + ) 26

(vii) Aperiodic, merging systems – a directed line infinite in two directions or to the right only, to which the roots of a possibly infinite number of possibly infinite trees (in either height or thickness) are attached Strongly irreversible systems (Infinite) (T = Z + ) 27

A non-trivial question about the long term behavior of a discrete-time DS L  Let DS L = (M, (g t ) t  T ) be a dynamical system whose time model L is discrete;  [F] given any state x  M, which is the form (i)  (vii) of the 1-step transition graph of the constituent to which x belongs?  Question F is undecidable, if DS L is universal. –It is easy to verify that the halting problem for any system of each type (i) – (vii) is decidable; thus, if F were decidable, the halting problem of DS L itself would be decidable. 28

Relations to DUT and PCE  The fact that F is undecidable for any universal system can be seen as an independent, albeit partial, confirmation of Wolfram's Dynamical Undecidability Thesis (DUT).  As for the Principle of Computational Equivalence (PCE), it affirms that computational universality should be attained by any system that satisfies the quite weak requirement of "non-obviously simple behavior" (i.e., type 4 phenomenology). 29Bochum, 20/04/2012

PCE vindicated?  Our previous analysis suggests as well that computational universality might turn out to hold under very weak conditions, for 1.by the classification theorem, there is a strong structural similarity between the 1­step transition graphs of any two discrete-time systems; 2.therefore, reproducing the dynamics of an arbitrary discrete-time system does not seem to require a system with especially unusual or extraordinary features. 3.In addition, the structure preserving mapping needed for universality is emulation, which is itself quite weak. 30Bochum, 20/04/2012

That’s all Thank you 31Bochum, 20/04/2012

References Giunti M. (1997), Computation, Dynamics, and Cognition. New York: Oxford University Press. Giunti M., Mazzola C. (2012), “Dynamical systems on monoids: Toward a general theory of deterministic systems and motion”, in: Minati G., Abram M., Pessa E. (eds.), Methods, models, simulations and approaches towards a general theory of change. Singapore: World Scientific. Mazzola C., Giunti M. (2012), “Reversible dynamics and the directionality of time”, in: Minati G., Abram M., Pessa E. (eds.), Methods, models, simulations and approaches towards a general theory of change. Singapore: World Scientific. Wolfram S. (1983a), “Statistical Mechanics of Cellular Automata”, Reviews of Modern Physics 55, 3: ——— (1983b), “Cellular Automata”, Los Alamos Science 9:2-21. ——— (1984a), “Computer Software in Science and Mathematics”, Scientific American 56: ——— (1984b), “Universality and Complexity in Cellular Automata”, in Doyne Farmer, Tommaso Toffoli, and Stephen Wolfram (eds.), Cellular Automata. Amsterdam: North Holland Publishing Company, ——— (2002), A New Kind of Science. Champaign, IL: Wolfram Media, Inc. 32Bochum, 20/04/2012

Type 2 (nested): rule 90, 250 steps, Initial Conditions: a single black cell 33Bochum, 20/04/2012

Type 3 (random): rule 30, 250 steps Initial Conditions: a single black cell 34Bochum, 20/04/2012

Type 4 (complex): rule 110, 250 steps Initial Conditions: a single black cell 35Bochum, 20/04/2012