Mortalityandperiodicityof reversibledynamicalsystems Jarkko Kari, - - PowerPoint PPT Presentation

Mortalityandperiodicityof reversibledynamicalsystems

Jarkko Kari, Nicolas Ollinger

Periodicity of Cellular Automata

One-dimensional cellular automaton (CA) is a shift commuting, continuous F : SZ − → SZ. In algorithmic considerations, F is effectively specified by its local rule f : S2r+1 − → S which determines F by F(x)i = f(xi−r, . . . , xi+r).

Point x ∈ SZ is periodic if F n(x) = x for some n ≥ 1, and the CA is periodic if all x ∈ SZ are periodic, i.e., all orbits are periodic. CA F is uniformly periodic if F n is the identity map, for some n ≥ 1. Note: Periodic CA are necessarily reversible: Function F is a

• homeomorphism. The inverse F −1 is also a CA, the inverse

CA.

• Lemma. A periodic CA is uniformly periodic.

• Lemma. A periodic CA is uniformly periodic.
• Proof. Suppose F is not uniformly periodic. Then, for every

n ≥ 1 there is a finite word wn that differs of its n’th image in the middle:

a ≠a Fn

• Lemma. A periodic CA is uniformly periodic.
• Proof. Suppose F is not uniformly periodic. Then, for every

n ≥ 1 there is a finite word wn that differs of its n’th image in the middle:

a ≠a Fn

Points x ∈ SZ that contain a copy of wn for every n ≥ 1 are not periodic:

a ≠a

1 1

a ≠a

2 2

a ≠a

3 3

a ≠a

4 4

Periodicity problem: Characterize local rules of periodic CA.

• Theorem. It is undecidable if a given 1D CA is periodic.

In order to prove this result we need to consider (reversible) Turing machines as dynamical systems.

Periodicity of Turing Machines

A Turing machine with moving head:

q a b c d e f g

Periodicity of Turing Machines

A Turing machine with moving head:

q a b c d e f g d’ q’

δ(q, d) = (q′, d′, R)

Periodicity of Turing Machines

A Turing machine with moving head:

a b c e f g d’ q’

Configurations are triplets c = (q, x, i) where

• q ∈ Q is the state,
• x ∈ ΣZ is the tape content, and
• i ∈ Z is the position on the tape.

A Turing machine with moving tape:

q a b c d e f g

A Turing machine with moving tape:

q a b c e f g q’ d d’

δ(q, d) = (q′, d′, R)

A Turing machine with moving tape:

a b c e f g q’ d’

Configurations are pairs c = (q, x) where

• q ∈ Q is the state, and
• x ∈ ΣZ is the tape content.

Turing machine is always positioned at cell zero.

A Turing machine with moving tape:

a b c e f g q’ d’

TM with moving tape have a compact configuration space (under the product topology), and the TM transitions are continuous.

Configuration c is called periodic if c ⊢n c for some n ≥ 1. Note that periodicity under moving head and moving tape are not equivalent concepts. A TM is periodic if all configurations are periodic, and it is uniformly periodic if c ⊢n c for all c and some fixed n. Note: Periodic TM are necessarily reversible: the transition function is bijective. In this case the inverse function is computed by another Turing machine, the inverse TM.

• Lemma. The following are equivalent for a Turing machine

M with at least two tape symbols: (1) M is periodic under the moving tape mode, (2) M is uniformly periodic under the moving tape mode, (3) M is periodic under the moving head mode, (4) M is uniformly periodic under the moving head mode.

For any given reversible Turing machine M one can effectively construct a reversible cellular automaton F that is periodic if and only if M is periodic.

For any given reversible Turing machine M one can effectively construct a reversible cellular automaton F that is periodic if and only if M is periodic.

q a b c d e f g

+

CA configurations consist of two ”tracks”: (1) The first track stores tape symbols of the Turing machine, (2) The second track stores in some position(s) a Turing machine state together with a symbol + or - indicating whether the machine is running forwards or backwards in

• time. Other positions contain an arrow ← or →.

For any given reversible Turing machine M one can effectively construct a reversible cellular automaton F that is periodic if and only if M is periodic.

q’ a b c d’ e f g

+

The CA local rule simulates one transition according to the TM, provided the TM is surrounded locally (before and after the transition) by arrows pointing towards it.

q a b c d e f g

_

If the state is coupled with symbol ”-” then the inverse TM is simulated instead.

r a b c x e f g

_

If the state is coupled with symbol ”-” then the inverse TM is simulated instead.

q a b c d e f g

+

But if the immediate neighborhood before or after the move contains an ”error” (another Turing machine state, or an arrow pointing to the wrong direction) then instead of TM transition we simply swap the direction + ← → −.

q a b c d e f g

_

But if the immediate neighborhood before or after the move contains an ”error” (another Turing machine state, or an arrow pointing to the wrong direction) then instead of TM transition we simply swap the direction + ← → −.

• The cellular automaton we constructed is reversible (due

to the reversibility of the Turing machine).

• The cellular automaton we constructed is reversible (due

to the reversibility of the Turing machine).

• If the TM is not periodic then the CA is not periodic: The

CA configuration simulating the non-periodic TM configuration is not periodic.

q a b c d e f g

+

• The cellular automaton we constructed is reversible (due

to the reversibility of the Turing machine).

• If the TM is not periodic then the CA is not periodic: The

CA configuration simulating the non-periodic TM configuration is not periodic.

• If the TM is periodic then the CA is periodic: Each TM

state in a CA configuration either (a) never sees an ”error” (in which case it cycles), or (b) eventually sees an ”error” and swaps the direction. In this case the TM bounces back-and-forth in time, hence also cycles periodically.

We have established the undecidability of the CA periodicity problem, once we prove the following:

• Theorem. It is undecidable whether a given reversible Turing

machine is periodic.

We have established the undecidability of the CA periodicity problem, once we prove the following:

• Theorem. It is undecidable whether a given reversible Turing

machine is periodic. To prove this result we resort to the undecidable mortality problem of Turing machines.

Mortality of Turing Machines

Let H ⊆ Q be a subset of TM states. Configuration c is mortal if there is n ≥ 0 such that c ⊢n (h, x) for some h ∈ H. Otherwise c is immortal. Note that it does not matter whether a moving head or moving tape mode is used.

Qx Z

Z

Hx Z

Z

c

The TM is mortal if all configurations are mortal. It easily follows from compactness that a mortal TM is uniformly mortal : there exists N such that all configurations reach a halting state within N steps.

Qx Z

Z

Hx Z

Z

Theorem (Hooper 1966). It is undecidable whether a given Turing machine is mortal. Based on Hooper’s proof idea we established the following

• Theorem. It is undecidable whether a given reversible Turing

machine is mortal.

From TM mortality to TM periodicity

Given a reversible Turing machine M with state set Q, one effectively constructs a reversible TM M ′ with the state set Q × {+, −} such that in states q+ and q− the TM M or its inverse M −1 are simulated, respectively, unless the current or the next state is in H, in which case one simply swaps + ← → −.

Qx Z

Z

Hx Z

Z

q+

From TM mortality to TM periodicity

Given a reversible Turing machine M with state set Q, one effectively constructs a reversible TM M ′ with the state set Q × {+, −} such that in states q+ and q− the TM M or its inverse M −1 are simulated, respectively, unless the current or the next state is in H, in which case one simply swaps + ← → −.

Qx Z

Z

Hx Z

Z

q-

From TM mortality to TM periodicity

Given a reversible Turing machine M with state set Q, one effectively constructs a reversible TM M ′ with the state set Q × {+, −} such that in states q+ and q− the TM M or its inverse M −1 are simulated, respectively, unless the current or the next state is in H, in which case one simply swaps + ← → −.

Qx Z

Z

Hx Z

Z

q+ q-

If M only contains periodic and mortal configurations then M ′ is periodic.

Qx Z

Z

Hx Z

Z

q+ q-

But if M contains some non-periodic, non-mortal configuration then M ′ is not periodic

Qx Z

Z

Hx Z

Z

If we can decide periodicity of M ′ then we determine whether all configurations in M are periodic or mortal. Then (due to uniform periodicity/mortality) we can also determine whether all configurations in M are mortal.

Qx Z

Z

Hx Z

Z

q+ q-

As mortality of reversible TM is undecidable, so is periodicity.

So far, Mortality Periodicity RTM Undecidable Undecidable RCA Undecidable What about mortality of RCA ?

Mortality of reversible CA

Let H ⊆ S be a subset of the state set S of a RCA F. Point x ∈ SZ is mortal if F n(x)0 ∈ H for some n ≥ 0.

The CA F is called mortal if all x ∈ SZ are mortal. By compactness, mortality implies uniform mortality : There is a bound N such that (∀x ∈ SZ) (∃n ≤ N) F n(x)0 ∈ H. We have the following: Theorem (Kari, Lukkarila). It is undecidable if a given reversible 1D CA is mortal (w.r.t a given halting set H.)

The CA F is called mortal if all x ∈ SZ are mortal. By compactness, mortality implies uniform mortality : There is a bound N such that (∀x ∈ SZ) (∃n ≤ N) F n(x)0 ∈ H. We have the following: Theorem (Kari, Lukkarila). It is undecidable if a given reversible 1D CA is mortal (w.r.t a given halting set H.) The proof is based on the undecidability of the domino problem among Wang tile sets that are NW- and SE-deterministic.

A B C D

In any valid tiling, NE-to-SW diagonals uniquely determine the diagonals above and below them: The tiles of the next diagonal are determined locally.

C A C D B C A C D B

In any valid tiling, NE-to-SW diagonals uniquely determine the diagonals above and below them: The tiles of the next diagonal are determined locally.

C A C D B C A C D B D D B B C C C C A A A

In any valid tiling, NE-to-SW diagonals uniquely determine the diagonals above and below them: The tiles of the next diagonal are determined locally.

C A C D B C A C D B D D B B C C C C A A A D D B C A C B C A C

In any valid tiling, NE-to-SW diagonals uniquely determine the diagonals above and below them: The tiles of the next diagonal are determined locally.

C A C D B C A C D B D D B B C C C C A A A D D B C A C B C A C B C A C D B C A C D B

To make a reversible 1D CA out of a NW- and SE- deterministic tile set we add new tiles for the missing color

• combinations. These ”error” -tiles form the halting set H.

A B C D

B A

?

C

To make a reversible 1D CA out of a NW- and SE- deterministic tile set we add new tiles for the missing color

• combinations. These ”error” -tiles form the halting set H.

A B C D

B A

?

C

Then: If the original tile set does not admit valid tilings, presence of H is guaranteed in all orbits. By a suitable signalling technique the information can be propagated to all cells, including position 0.

The undecidability of the mortality problem then follows from the undecidability of the domino problem among deterministic tile sets. Mortality Periodicity RTM Undecidable Undecidable RCA Undecidable Undecidable

Final remarks

• All problems considered are r.e. because of the uniform

bounds.

• These are the first undecidability results that concern

reversible 1D CA.

• We also considered a third reversible computation model:

Reversible counter machines (RCM). Due to the lack of compactness, the periodicity and mortality are not equivalent to their uniform variants. In fact: uniform variants are decidable while the non-uniform variants are undecidable.

Open problems

• Does there exist an ”aperiodic” reversible Turing machine:

A RTM that does not have any periodic configurations, under the moving tape mode ? (A non-reversible example is known, due to Blondel, Cassaigne, Nichitiu.)

• Is it decidable whether a given RTM has a periodic orbit ?

(In the non-reversible case we have proved the problem undecidable, as also for reversible counter machines.)

Open problems

• Does there exist an ”aperiodic” reversible Turing machine:

A RTM that does not have any periodic configurations, under the moving tape mode ? (A non-reversible example is known, due to Blondel, Cassaigne, Nichitiu.)

• Is it decidable whether a given RTM has a periodic orbit ?

(In the non-reversible case we have proved the problem undecidable, as also for reversible counter machines.)

Thank You