Discrete Systems with Discrete Systems with Undecidable DynamicsUndecidable Dynamics

Marco GiuntiMarco GiuntiUniversità di CagliariUniversità di Cagliarigiunti@unica.itgiunti@unica.ithttp://www.webalice.it/marcogiunti/http://www.webalice.it/marcogiunti/

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SummarySummary

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.

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Rule 111 110 101 100 011 010 001 000

0 1 0 1 1 0 1 0

Rule number (base 2) 010110102 = 9010 (base 10)

Time 0 0 1 0 0 1 0 1 0 1 1 1 1

Time 1 0 0 1 1 0 0 0 0 1 0 0 1

12 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 212 = 4096 possible states.

A finite cellular automatonA finite cellular automaton

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64 states – 60 copies 32 states – 6 copies

16 states – 4 copies Total: 4096 = 212 states

The seventy constituent subsystems of The seventy constituent subsystems of CA rule 90CA rule 901010 – 12 cells in a circle (Wolfram 1983b) – 12 cells in a circle (Wolfram 1983b)

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Wolfram's original studies on CAsWolfram's original studies on CAs In the ’80s Wolfram (1983a) studied by extensive In the ’80s Wolfram (1983a) studied by extensive

computer simulations the simplest 256 CAs computer simulations the simplest 256 CAs (modimensional, 2 values, radius 1 neighborhood);(modimensional, 2 values, radius 1 neighborhood);

He classified them into 4 qualitative similarity types of He classified them into 4 qualitative similarity types of behavior (Wolfram 1983b, 1984b):behavior (Wolfram 1983b, 1984b):

CAs whose evolution leads toCAs whose evolution leads to

1.1. an homogeneous state, stable or periodic;an homogeneous state, stable or periodic;

2.2. a set of distinct simple structures: (i) fixed size; a set of distinct simple structures: (i) fixed size; (ii) growing size, with a repetitive or nested pattern;(ii) growing size, with a repetitive or nested pattern;

3.3. a chaotic or apparently random behavior;a chaotic or apparently random behavior;

4.4. localized complex structures, often with a quite long localized complex structures, often with a quite long life span.life span.

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Type 2 (repetitive): rule 250, 100 steps Type 2 (repetitive): rule 250, 100 steps Initial Conditions: a single black cellInitial Conditions: a single black cell

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Type 2 (nested): rule 90, 1000 steps Type 2 (nested): rule 90, 1000 steps Initial Conditions: a single black cellInitial Conditions: a single black cell

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Type 3 (random): rule 30, 1000 steps Type 3 (random): rule 30, 1000 steps Initial Conditions: a single black cellInitial Conditions: a single black cell

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Type 4 (complex): rule 110, 1000 steps Type 4 (complex): rule 110, 1000 steps Initial Conditions: a single black cellInitial Conditions: a single black cell

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Wolfram's Wolfram's Principle of Computational Principle of Computational Equivalence Equivalence ((PCEPCE)) Wolfram conjectured that, among the simplest 256 CAs, Wolfram conjectured that, among the simplest 256 CAs,

any type 4 system is computationally universal (1984b, 31);any type 4 system is computationally universal (1984b, 31); he then proved (2002, 675-689) that rule 110 (type 4) is he then proved (2002, 675-689) that rule 110 (type 4) is

universal, and that it is equivalent to the other 3 rules of universal, and that it is equivalent to the other 3 rules of type 4 (2002, 60);type 4 (2002, 60);

finally, on the basis finally, on the basis of extensive phenomenological studies of extensive phenomenological studies of the dynamics of many types of discrete systemsof the dynamics of many types of discrete systems, , Wolfram generalized this result, by putting forth the Wolfram generalized this result, by putting forth the following hypothesis:following hypothesis:

Principle of Computational Equivalence Principle of Computational Equivalence ((PCEPCE)) Almost all processes that are not obviously simple can Almost all processes that are not obviously simple can

be viewed as computations of equivalent sophistication be viewed as computations of equivalent sophistication (2002, 716-717)(2002, 716-717)

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Computational universality of type 4 Computational universality of type 4 systems – what does it mean?systems – what does it mean?

It means that It means that any any type 4 system is able to type 4 system is able to emulateemulate, , i.e.i.e. exactly reproduce, the behavior of a exactly reproduce, the behavior of a whole classwhole class of systems that is known to be of systems that is known to be computationally universal.computationally universal.

In fact, the proof of the universality of rule 110 In fact, the proof of the universality of rule 110 shows that, with appropriate initial conditions, this shows that, with appropriate initial conditions, this rule is able to emulate rule is able to emulate anyany tag system (the class of tag system (the class of tag systems is computationally universal, for any tag systems is computationally universal, for any Turing machine can in turn be emulated by an Turing machine can in turn be emulated by an appropriate tag system).appropriate tag system).

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Emulation between CAs – an exampleEmulation between CAs – an example CA 18 emulates CA 90 in two steps, with initial CA 18 emulates CA 90 in two steps, with initial

conditions 00 for 0 and 01 for 1 (Wolfram 1983b, 20)conditions 00 for 0 and 01 for 1 (Wolfram 1983b, 20)

010101 010100 010001 010000 000101 000100 000001 000000

0000 0001 0101 0100 0100 0101 0001 0000

00 01 00 01 01 00 01 00 111 110 101 100 011 010 001 000

0 1 0 1 1 0 1 0

Rule number 010110102 = 9010

111 110 101 100 011 010 001 000

0 0 0 1 0 0 1 0

Rule number 000100102 = 1810

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Consequences of Consequences of PCEPCE Ubiquity of Computational UniversalityUbiquity of Computational Universality

– ““Almost any rule whose behavior is not obviously simple Almost any rule whose behavior is not obviously simple should ultimately be capable of achieving the same level should ultimately be capable of achieving the same level of computational sophistication and should thus in effect of computational sophistication and should thus in effect be universal” (Wolfram 2002, 718)be universal” (Wolfram 2002, 718)

– Strictly speaking, this does not follow from the previous Strictly speaking, this does not follow from the previous formulation of the principle (needs premise: the behavior formulation of the principle (needs premise: the behavior of a computationally universal system is not obviously of a computationally universal system is not obviously simple). simple). It can also be viewed as a different formulation It can also be viewed as a different formulation of of PCEPCE..

Ubiquity of ComplexityUbiquity of Complexity– According to According to PCEPCE, “observers will tend to be , “observers will tend to be

computationally equivalent to the systems they observecomputationally equivalent to the systems they observe—with the inevitable consequence that they will —with the inevitable consequence that they will consider the behavior of such systems complex.” consider the behavior of such systems complex.” (Wolfram 2002, 737)(Wolfram 2002, 737)

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Further consequences of Further consequences of PCEPCE Computational IrreducibilityComputational Irreducibility

– According to According to PCEPCE, “systems one uses to make , “systems one uses to make predictions cannot be expected to make computations predictions cannot be expected to make computations that are any more sophisticated than the computations that are any more sophisticated than the computations that occur in all sorts of systems whose behavior we that occur in all sorts of systems whose behavior we might try to predict. And from this it follows that for might try to predict. And from this it follows that for many systems no systematic prediction can be done, so many systems no systematic prediction can be done, so that there is no general way to shortcut their process of that there is no general way to shortcut their process of evolution, and as a result their behavior must be evolution, and as a result their behavior must be considered computationally irreducible.” (Wolfram 2002, considered computationally irreducible.” (Wolfram 2002, 741)741)

Free Will Free Will (follows from (follows from Computational IrreducibilityComputational Irreducibility))– ““But so in the end what makes us think that there is But so in the end what makes us think that there is

freedom in what a system does? In practice the main freedom in what a system does? In practice the main criterion seems to be that we cannot readily make criterion seems to be that we cannot readily make predictions about the behavior of the systems” (Wolfram predictions about the behavior of the systems” (Wolfram 2002, 751)2002, 751)

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One more consequence of One more consequence of PCEPCE

Dynamical Undecidability ThesisDynamical Undecidability Thesis ( (DUTDUT) ) (follows from (follows from Computational IrreducibilityComputational Irreducibility))

– ““And what I suspect is that for almost any And what I suspect is that for almost any system whose behavior seems to us system whose behavior seems to us complex almost any non-trivial question complex almost any non-trivial question about what the system does after an about what the system does after an infinite number of steps will be infinite number of steps will be undecidable.” (Wolfram 2002, 755)undecidable.” (Wolfram 2002, 755)

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General definition of a General definition of a dynamical dynamical systemsystem  – – minimal time model: minimal time model: a monoida monoid

Definition 1: Definition 1: DSDSL L is a dynamical system on L is a dynamical system on L :=:= DS DSLL is a pairis a pair

((MM, (, (ggtt))ttTT) and ) and LL is a pair ( is a pair (TT, +) such that, +) such that

(i)(i) L L = ( = (TT, +) is a monoid. Any , +) is a monoid. Any tt    TT is called a is called a duration duration of the of the system, system, T T is called its is called its time settime set, and , and L L its its time model;time model;

(ii) (ii) MM is a non-empty set. Any is a non-empty set. Any xx MM is called a is called a statestate of the system, of the system, and and MM is called its is called its state space;state space;

(iii) ((iii) (ggtt))ttTT is a family indexed by is a family indexed by TT of functions from of functions from MM to to M. M. For anyFor any

t t TT, the function , the function ggtt is called the is called the state transition of duration t state transition of duration t (briefly, (briefly, t-transitiont-transition, or, or t-advance t-advance) of the system;) of the system;

(iv) for any (iv) for any vv,, t  t  T T, for any , for any xx  M, M,a)a) gg00((xx) = ) = xx;;

b)b) ggv+tv+t((xx) = ) = ggvv((ggtt((xx)).)).

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Def. 1 captures the general notion of a Def. 1 captures the general notion of a deterministic dynamical deterministic dynamical system system DeterminismDeterminism(iii)(iii) if if xx is the state at the is the state at the present instant present instant tt0 0   TT, , ggtt((xx) is ) is the state of the system after an the state of the system after an evolution of duration evolution of duration t t TT

DynamicsDynamics(iv.a)(iv.a) whatever state whatever state x x the the system is in, the evolution of system is in, the evolution of duration 0 does not modify that duration 0 does not modify that statestate(iv.b)(iv.b) any evolution of duration any evolution of duration v+t v+t can be decomposed in two can be decomposed in two successive evolutions, the first successive evolutions, the first one of duration one of duration tt, and the , and the second one of duration second one of duration vv

v+t

x

v

g0(x) = x

0

x t

t0 t+t0

gt(x)

t

t

condition (iii)

condition (iv.a)

condition (iv.b)

gv+t(x) = gv(gt(x))

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The transition graph of a The transition graph of a DSDSLL

The transition graph of a The transition graph of a DSDSLL is a directed and is a directed and labeled graph that depicts labeled graph that depicts the whole dynamics of the the whole dynamics of the system.system.

Each point of the graph Each point of the graph corresponds to exactly one corresponds to exactly one state of the state of the DSDSLL, while each , while each arrow stands for a state arrow stands for a state transition from its source to transition from its source to its target.its target.

Each arrow is labeled with Each arrow is labeled with the duration of the the duration of the corresponding state corresponding state transition.transition.

0,2,4, … 0,2,4, …

1,3,5, …

1,3,5, …

ExampleExample DSDSLL = ({= ({aa, , bb}, (}, (ggtt))ttZZ++)) LL = (Z = (Z++, +), +) if if tt is even, is even, ggtt(x(x) = ) = xxif if tt is odd, is odd, ggtt(x(x) = ) = yy,, y y ≠ ≠ xx

Any Any DSDSLL can in fact be can in fact be identified with its transition identified with its transition graph.graph.

The transition graph of any The transition graph of any DSDSLL is a category. is a category. 1818Bochum, 20/04/2012Bochum, 20/04/2012

The 1-step transition graph ofThe 1-step transition graph of a DS a DSL L

with discrete time modelwith discrete time model If the time model If the time model L L isis

discretediscrete,, i.e. i.e., , LL = (Z = (Z++, +), , +), the whole dynamics of the the whole dynamics of the system is implicitly system is implicitly depicted by the graph of its depicted by the graph of its first state transition first state transition gg11..

If If LL = (Z, +), the 1-step = (Z, +), the 1-step transition graph must also transition graph must also include include gg11..

Example (same as previous Example (same as previous slide)slide)

DSDSLL = ({= ({aa, , bb}, (}, (ggtt))ttZZ++)) LL = (Z = (Z++, +), +) if if tt is even, is even, ggtt(x(x) = ) = xxif if tt is odd, is odd, ggtt(x(x) = ) = yy,, y y ≠ ≠ xx

The 1-step transition graph of The 1-step transition graph of the the DSDSLL above above

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Emulation between dynamical Emulation between dynamical systems systems – – Intuition and examplesIntuition and examples Intuitively, a dynamical system Intuitively, a dynamical system DSDSLL1 1

emulates emulates

DSDSLL22 when the first one exactly reproduces any when the first one exactly reproduces any

state transition of the second one. However, state transition of the second one. However, state transition state transition durationsdurations are not usually are not usually preserved.preserved.

Examples –Examples – (i) a universal Turing machine (i) a universal Turing machine emulates anyemulates anyTMTM; (ii) for any ; (ii) for any TMTM, there is a , there is a CACA that emulates that emulates TMTM, and conversely; , and conversely; (iii) emulation holds between the two simple (iii) emulation holds between the two simple CAs CAs previously consideredpreviously considered ((CACA 18 emulates 18 emulates CACA 90). 90). 2020Bochum, 20/04/2012Bochum, 20/04/2012

Emulation between dynamical Emulation between dynamical systemssystems – – DefinitionDefinition

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(hv(c)) = gt(u(c)).

gt

u

hv

u

c

DSL1 emulates DSL2

:=

M Nu(N)

DSL1 = (M, (gt)tT) is a

dynamical system on L1 = (T, +)

DSL2 = (N, (hv)vV) is a

dynamical system on L2 = (V, )

N.B. The emulation function u is a quite weak structure preserving mapping of the emulated system into the emulating one.2121

Two definitionsTwo definitionsLet Let DSDSLL = ( = (MM,,  ((ggtt))ttTT) be a dynamical system on ) be a dynamical system on

monoid monoid LL = ( = (TT, +)., +).A A constituentconstituent of of DSDSLL := :=

any subsystem of any subsystem of DSDSLL whose state space whose state space

N N  MM is temporally connected in is temporally connected in MM and and includes all its past (as well as its future).includes all its past (as well as its future).DSDSLL is is indecomposableindecomposable := :=

DSDSLL has just one constituent ( has just one constituent (i.e.i.e. DSDSLL itself). itself).It is possible to show that any constituent of It is possible to show that any constituent of DSDSL L is indecomposable.is indecomposable.

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Two general resultsTwo general results Decomposition theoremDecomposition theorem [[for any dynamical system DSfor any dynamical system DSLL

with an arbitrary time model L = with an arbitrary time model L = ((T, +T, +)])] The transition graph of any dynamical system The transition graph of any dynamical system DSDSL L on a on a

monoid monoid LL exhaustively decomposes into a set of exhaustively decomposes into a set of mutually disconnected and internally connected mutually disconnected and internally connected components. Such components are in fact all the components. Such components are in fact all the constituent subsystems of constituent subsystems of DSDSLL..

Classification theorem Classification theorem [[for any indecomposable for any indecomposable dynamical system DSdynamical system DSD D with discrete time model D = with discrete time model D =

((T, +T, +), where ), where TT = = Z or Z Z or Z++]] The 1-step transition graph of any indecomposable The 1-step transition graph of any indecomposable

dynamical system dynamical system DSDSD D with discrete time model with discrete time model DD is of is of

one of the forms (i) – (vii) described in the following one of the forms (i) – (vii) described in the following slides.slides.

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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

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+)

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(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+)

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(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+)

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(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+)

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A non-trivial question about the long A non-trivial question about the long term behavior of a discrete-time term behavior of a discrete-time DSDSLL

Let Let DSDSLL = ( = (MM,,  ((ggtt))ttTT) be a dynamical system ) be a dynamical system

whose time model whose time model LL is discrete; is discrete; [[FF] given any state ] given any state xx MM, which is the form , which is the form

(i)(i)(vii) of the 1-step transition graph of the (vii) of the 1-step transition graph of the constituent to which constituent to which xx belongs? belongs?

Question Question F F is undecidable, if is undecidable, if DSDSLL is universal. is universal.

– It is easy to verify that the halting problem for any It is easy to verify that the halting problem for any system of each type (i) – (vii) is decidable; thus, if system of each type (i) – (vii) is decidable; thus, if FF were decidable, the halting problem of were decidable, the halting problem of DSDSL L itself itself

would be decidable.would be decidable.2828

Relations to Relations to DUTDUT and and PCEPCE The fact that The fact that F F is undecidable for any universal is undecidable for any universal

system can be seen as an independent, albeit system can be seen as an independent, albeit partial, confirmation of Wolfram's partial, confirmation of Wolfram's Dynamical Dynamical Undecidability Thesis Undecidability Thesis ((DUTDUT).).

As for theAs for the Principle of Computational Principle of Computational Equivalence Equivalence ((PCEPCE),), it affirms that it affirms that computational universality should be attained computational universality should be attained by any system that satisfies the quite weak by any system that satisfies the quite weak requirement of "non-obviously simple behavior" requirement of "non-obviously simple behavior" ((i.e.i.e., type 4 phenomenology)., type 4 phenomenology).

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PCE PCE vindicated?vindicated? Our previous analysis suggests as well that Our previous analysis suggests as well that

computational universality might turn out to hold under computational universality might turn out to hold under very weak conditions, forvery weak conditions, for

1.1. by the classification theorem, there is a strong by the classification theorem, there is a strong structural similarity between the 1structural similarity between the 1 step transition step transition graphs of graphs of any twoany two discrete-time systems; discrete-time systems;

2.2. therefore, reproducing the dynamics of an arbitrary therefore, reproducing the dynamics of an arbitrary discrete-time system does not seem to require a discrete-time system does not seem to require a system with especially unusual or extraordinary system with especially unusual or extraordinary features.features.

3.3. In addition, the structure preserving mapping needed In addition, the structure preserving mapping needed for universality is for universality is emulationemulation, which is itself quite weak., which is itself quite weak.

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That’s allThat’s allThank youThank you

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ReferencesReferencesGiunti 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, 3:601-644.——— (1983b), “Cellular Automata”, Los Alamos Science 9:2-21.——— (1984a), “Computer Software in Science and Mathematics”, Scientific American 56:188-203.——— (1984b), “Universality and Complexity in Cellular Automata”, in Doyne Farmer, Tommaso Toffoli, and Stephen Wolfram (eds.), Cellular Automata. Amsterdam: North Holland Publishing Company, 1-35.——— (2002), A New Kind of Science. Champaign, IL: Wolfram Media, Inc.

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Type 2 (nested): rule 90, 250 steps, Type 2 (nested): rule 90, 250 steps, Initial Conditions: a single black cellInitial Conditions: a single black cell

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Type 3 (random): rule 30, 250 steps Type 3 (random): rule 30, 250 steps Initial Conditions: a single black cellInitial Conditions: a single black cell

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Type 4 (complex): rule 110, 250 steps Type 4 (complex): rule 110, 250 steps Initial Conditions: a single black cellInitial Conditions: a single black cell

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