Mortality�and�periodicity�of reversible�dynamical�systems Jarkko Kari, Nicolas Ollinger
Periodicity of Cellular Automata One-dimensional cellular automaton (CA) is a shift commuting, continuous F : S Z − → S Z . In algorithmic considerations, F is effectively specified by its local rule f : S 2 r +1 − → S which determines F by F ( x ) i = f ( x i − r , . . . , x i + r ) .
Point x ∈ S Z is periodic if F n ( x ) = x for some n ≥ 1, and the CA is periodic if all x ∈ S Z 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 w n that differs of its n ’th image in the middle: a F n ≠ a
Lemma. A periodic CA is uniformly periodic. Proof. Suppose F is not uniformly periodic. Then, for every n ≥ 1 there is a finite word w n that differs of its n ’th image in the middle: a F n ≠ a Points x ∈ S Z that contain a copy of w n for every n ≥ 1 are not periodic: a a a a 4 2 3 1 ≠ a 1 ≠ a 2 ≠ a 3 ≠ a 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: a b c d e f g q
Periodicity of Turing Machines A Turing machine with moving head: d’ a b c d e f g q’ 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 ∈ Σ Z is the tape content, and • i ∈ Z is the position on the tape.
A Turing machine with moving tape: a b c d e f g q
A Turing machine with moving tape: d’ a b c d e f g q’ q δ ( q, d ) = ( q ′ , d ′ , R )
A Turing machine with moving tape: a b c e f g d’ q’ Configurations are pairs c = ( q, x ) where • q ∈ Q is the state, and • x ∈ Σ Z is the tape content. Turing machine is always positioned at cell zero.
A Turing machine with moving tape: a b c e f g d’ q’ 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. a b c d e f g q + 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. a b c d’ e f g q’ + 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.
a b c d e f g q _ If the state is coupled with symbol ”-” then the inverse TM is simulated instead.
a b c x e f g r _ If the state is coupled with symbol ”-” then the inverse TM is simulated instead.
a b c d e f g q + 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 + ← → − .
a b c d e f g q _ 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. a b c d e f g q +
• 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. Q�x � Z Z c H�x � Z Z
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. Q�x � Z Z H�x � 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 + ← → − . Q�x � Z Z q + H�x � Z Z
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 + ← → − . Q�x � Z Z q - H�x � Z Z
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 + ← → − . Q�x � Z Z q - q + H�x � Z Z
If M only contains periodic and mortal configurations then M ′ is periodic. Q�x � Z Z q - q + H�x � Z Z
But if M contains some non-periodic, non-mortal configuration then M ′ is not periodic Q�x � Z Z H�x � 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. Q�x � Z Z q - q + H�x � Z Z 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 ∈ S Z is mortal if F n ( x ) 0 ∈ H for some n ≥ 0.
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