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Tiling Problem and Undecidability in Cellular Automata

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

We consider the following algorithmic questions concerning cellular automata. All problems are decision problems, that is, the answer for each input instance is either yes or no. All problems considered are undecidable, i.e., no algorithm can solve them. We only consider problems whose undecidability is proved using a reduction from the tiling problem or its variant.

Injectivity

Input::

Cellular automaton A.

Question::

Is A injective (i.e., reversible)?

There is an algorithm that solves INJECTIVITY for one-dimensional CA (Amoroso and Patt 1972), but the problem is undecidable among two-dimensional CA (Kari 1990, Kari 1994a). The problem is semi-decidable in any dimension.

Surjectivity

Input::

Cellular automaton A.

Question::

Is A surjective?

Also, SURJECTIVITY is decidable among one-dimensional CA. The two-dimensional question is, however, undecidable (Kari 1994a). The complement of the problem (i.e., non-surjectivity) is semi-decidable in any dimension.

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Abbreviations

Aperiodic Tile Set:

A two-dimensional tile set that admits a valid tiling of the plane, but does not admit any valid periodic tilings. The smallest known aperiodic set of Wang tiles contains 11 tiles (Jeandel and Rao 2015). See also (Culik 1996; Kari 1996).

Cellular Automaton (CA):

A d-dimensional cellular automaton consists of an infinite d-dimensional grid of cells, indexed by ℤd. Each cell stores an element of a finite state set S. Configuration c: ℤd → S specifies the states of all cells. The set of all configurations is S ℤd. The neighborhood vector \( N=\left({\overrightarrow{n}}_1,{\overrightarrow{n}}_2,\dots, {\overrightarrow{n}}_m\right) \) is a sequence of m distinct elements of ℤd specifying the relative locations of the neighbors of the cells: A cell located at \( \overrightarrow{x}\in {\boldsymbol{\mathbb{Z}}}^d \) has m neighbors, in positions \( \overrightarrow{x}+{\overrightarrow{n}}_1,\overrightarrow{x}+{\overrightarrow{n}}_2,\dots, \overrightarrow{x}+{\overrightarrow{n}}_m \). Finally, the local update rule f:S m → S specifies the new state of a cell, based on the old states of its neighbors. In one step, configuration c is transformed into configuration e where, for all \( \overrightarrow{x}\in {\boldsymbol{\mathbb{Z}}}^d, \)

$$ e\left(\overrightarrow{x}\right)=f\left[c\left(\overrightarrow{x}+{\overrightarrow{n}}_1\right), c\left(\overrightarrow{x}+{\overrightarrow{n}}_2\right),\dots, c\left(\overrightarrow{x}+{\overrightarrow{n}}_m\right)\right]. $$

The mapping c → e is the global transition function, or the CA function, \( G:{S^{\boldsymbol{\mathbb{Z}}}}^{{}^d}\to {S^{\boldsymbol{\mathbb{Z}}}}^{{}^d} \) specified by the CA \( \mathcal{A} \) = (d,S,N,f).

Decision Problem:

A decision problem is an algorithmic question with a yes/ no answer. The problem has an input (called the instance of the problem) and a well-defined answer “yes” or “no” associated to each instance.

Equicontinuous Cellular Automaton:

Cellular automaton G is called equicontinuous if, for every finite \( A\subseteq {\boldsymbol{\mathbb{Z}}}^d \), there exists a finite \( B\subseteq {\boldsymbol{\mathbb{Z}}}^d \) such that any two initial configurations that agree inside B will agree inside A for all subsequent steps. In other words

$$ \begin{array}{c}\hfill \forall \overrightarrow{x}\in B:c\left(\overrightarrow{x}\right)=e\left(\overrightarrow{x}\right)\hfill \\ {}\hfill \Rightarrow \forall t\in \boldsymbol{\mathbb{N}} \mathrm{and}\forall \overrightarrow{x}\in A:{G}^t(c)\left(\overrightarrow{x}\right)={G}^t(e)\left(\overrightarrow{x}\right)\hfill \end{array}. $$

This means that equicontinuous CA can be reliably simulated in finite windows. It is known that a CA is equicontinuous if and only if it is ultimately periodic \( \left[17\right]:\exists n,p\in \boldsymbol{\mathbb{N}}:{G}^n={G}^{n+p} \). In this sense, the dynamics of equicontinuous CA is trivial.

Injective (Reversible) Cellular Automaton:

A CA is injective if every configuration has at most one pre-image, that is, the global transition function is one-to-one. It is well known that a CA is injective if and only if it is bijective (every configuration has a unique pre-image), which, in turn, is equivalent to reversibility. (There exists an inverse CA that traces the CA back in time).

Limit Set:

The limit set of a CA is its maximal attractor. In other words, it is the compact and translation invariant set

$$ {\displaystyle \underset{i=0}{\overset{\infty }{\cap }}G\left({s}^{{\boldsymbol{\mathbb{Z}}}^d}\right)} $$

where G is the global transition function and S is the state set.

Nilpotent Cellular Automaton:

Nilpotent CA has trivial dynamics. A CA is called nilpotent if its limit set contains only one configuration. The unique element of the limit set is the quiescent configuration. This is equivalent to every initial configuration eventually becoming the quiescent configuration.

Periodic Tiling:

A tiling that is invariant under some nonzero translation. A two-dimensional? tiling is called totally periodic if it is invariant under two linearly independent translations. A totally periodic tiling is automatically periodic in horizontal and vertical directions, which means that it consists of a rectangular pattern that is repeated horizontally and vertically to fill the plane. A two-dimensional tile set that admits a valid periodic tiling automatically admits also totally periodic tiling.

More generally, a d-dimensional tiling t: ℤd → T is periodic with period \( \overrightarrow{p}\in {\boldsymbol{\mathbb{Z}}}^d,\overrightarrow{p}\ne \overrightarrow{0} \), if \( t\left(\overrightarrow{x}\right)=t\left(\overrightarrow{x}+\overrightarrow{p}\right) \) for all \( \overrightarrow{x}\in {\boldsymbol{\mathbb{Z}}}^d \). It is totally periodic if it is periodic with d linearly independent periods \( {\overrightarrow{p}}_1,{\overrightarrow{p}}_2,\dots {\overrightarrow{p}}_d \). Note that when d > 2, it is possible that a tile set admits a periodic tiling but does not admit any totally periodic tiling.

Recursive and Recursively Enumerable (RE):

A formal language L is called recursive if it is decidable whether a given word belongs to L. The language is called recursively enumerable (RE for short) if this membership problem is semi-decidable.

Semi-algorithm:

An algorithm-like procedure for a decision problem that correctly returns a positive answer on positive input instances, but on negative instances runs forever without ever returning an answer.

Semi-decidability:

A decision problem is called semi-decidable if there is a semi-algorithm for it. For example, the decision problem Turing machine halting on blank tape is semi-decidable since one can simulate any given Turing machine step-by-step until (if ever) it halts.

Sensitive Cellular Automata:

Cellular automaton G is called sensitive to initial conditions if there exists a finite set B ⊆ ℤd of cells such that for every configuration c and every finite set A ⊆ ℤ of cells there exists a configuration e and time t ≥0 such that \( e\left(\overrightarrow{x}\right)=d\left(\overrightarrow{x}\right)\;\mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\ \overrightarrow{x}\in A \mathrm{but}\ne {G}^t(e)\left(\overrightarrow{x}\right)\ne {G}^t(c)\left(\overrightarrow{x}\right) \) for some \( \overrightarrow{x}\in B \). This means that arbitrarily distant modifications to any configuration c may propagate to a fixed observation window B.

Surjective Cellular Automaton:

A cellular automaton (CA) is called surjective if every configuration has a pre-image, that is, if its global transition function \( G:{S}^{{\boldsymbol{\mathbb{Z}}}^d}\to {S}^{{\boldsymbol{\mathbb{Z}}}^d} \) is surjective.

The Finite Tiling Problem:

A decision problem where we are given a set T of Wang tiles and a specific blank tile B∈T whose all four edges are colored with the same color. A tiling t: ℤ → T is called finite if

$$ \left\{\overrightarrow{x}\in {\boldsymbol{\mathbb{Z}}}^2\Big|t\left(\overrightarrow{x}\right)\ne B\right\} $$

is a finite set. If \( t\left(\overrightarrow{x}\right)=B \) for all \( \overrightarrow{x}\in {\boldsymbol{\mathbb{Z}}}^2, \) then t is called trivial. The finite tiling problem asks whether there exist nontrivial valid finite tilings. The problem is undecidable but semi-decidable.

The Halting Problem:

Turing machine halting on blank tape is the decision problem whose input is a Turing machine \( M \) = (Q, Γ, \( \delta \), q 0, q h , b) and the answer is positive if and only if the Turing machine eventually enters its halting state q h when started in the initial state q 0 on a totally blank tape, i. e., initially every tape location has the blank symbol b.

The Periodic Tiling Problem:

The decision problem to determine if a given set of Wang tiles admits a periodic tiling. The problem is undecidable, but it is semi-decidable (Gurevich and Koryakov 1972).

The Tiling Problem (Also Known as the Domino Problem):

The decision problem that gets as input a finite tile set \( T \) and asks whether there exists a valid tiling by \( T \). The tiling problem was proved undecidable for Wang tiles by R.Berger (1966). Its complement (i.e., “Does there not exist a valid tiling”) is semi-decidable.

The Tiling Problem with a Seed Tile:

The decision problem that gets as input a tile set \( T \) and one tile five and asks whether \( T \) admits a tiling that contains tiles at least once. The problem is undecidable for Wang tiles, but its complement is semi-decidable (Wang 1961).

Tiling:

Tiling is a covering of the plane using tiles. A valid Wang tiling by a Wang tile set T is an assignment t: ℤ2 → T of tiles to cells such that the local matching rule is satisfied between all adjacent tiles. We say that T admits tilingt.

A more general definition: a d-dimensional tiling using tile set \( T \) = (d,T,N,R) is a mapping t: ℤ2 → T. Tiling t is valid at cell \( \overrightarrow{x}\in {\boldsymbol{\mathbb{Z}}}^d \) if

$$ t\left(\overrightarrow{x}+{\overrightarrow{n}}_1,\overrightarrow{x}+{\overrightarrow{n}}_2,\dots, \overrightarrow{x}+{\overrightarrow{n}}_m\right)\in R. $$

Tiling t is called valid if it is valid at every cell \( \overrightarrow{x}\in {\boldsymbol{\mathbb{Z}}}^d \).

Topological Entropy:

The topological entropy h(G) of a CA G measures the complexity of its dynamics. For any finite A ⊆ ℤd and positive integer n, we define the equivalence relation ≡ A,n among initial configurations as follows: For all c,e ∈S ℤd

$$ \begin{array}{c}\hfill c \equiv A{,}_ne\hfill \\ {}\hfill \iff \forall \overrightarrow{x}\in A,0\le t<n:\hfill \\ {}\hfill {G}^t(c)\left(\overrightarrow{x}\right)={G}^t(e)\left(\overrightarrow{x}\right).\hfill \end{array} $$

In other words, two configurations are equivalent if we cannot observe any difference in their orbits in region A within the first n time instances. Let us denote by N G (A,n) the number of equivalence classes of ≡ A,n . Then the topological entropy is

$$ h(G)=\underset{A}{ \sup}\;\underset{n\to \infty }{ \lim}\frac{ \log {N}_G\left(A,n\right)}{n} $$

where the supremum is over all finite A ⊆ ℤd. The entropy always exists. In the one-dimensional case, the entropy is always a finite, nonnegative number. If d≥2, then the entropy can also be infinite.

Turing Machine (TM):

Turing machines are computation devices commonly used to formally define the concept of an algorithm. They also provide us with the most basic undecidable decision problems. A Turing machine consists of a finite state control unit that moves along an infinite tape. The tape has symbols written in cells that are indexed by ℤ. Depending on the state of the control unit and the symbol currently scanned on the tape, the machine may overwrite the tape symbol, change the internal state, and move along the tape one cell to the left or right. We formally define a Turing machine as a six-tuple \( M=\left(Q,\Gamma, \delta, {q}_0,{q}_h,b\right) \) where Q and Γ are finite sets (the state alphabet and the tape alphabet, respectively), q 0,q h ∈Q are the initial and the halting states, respectively, b∈Γ is the blank symbol, and \( \updelta :Q\times \Gamma \to Q\times \Gamma \times \left\{-1,1\right\} \) is the transition function that specifies the moves of the machine.

A configuration (or instantaneous description) of the machine is a triplet (q, i, t) where \( q\in Q \) is the current state, i∈ℤ is the position of the machine on the tape, and t: ℤ → Γ describes the content of the tape. In one time step, configuration (q, i, t) becomes \( \left(q^{\prime },i+d,t^{\prime}\right) \) if \( \updelta \left(q,t(i)\right)=\left(q^{\prime },y,d\right) \) and \( t^{\prime }(i)=y \) and \( {t}^{\prime }(j)=t(j) \) for all \( j\ne i \). We denote this move by

$$ \left(q,i,t\right)\vdash \left({q}^{\prime },i+d,{t}^{\prime}\right) $$

The reflexive, transitive closure of is denoted by ⊢∗, that is

$$ \left(q,i,t\right)\left({q}^{\prime },{i}^{\prime },{t}^{\prime}\right) $$

if and only if (q′, i′, t′) can be reached from (q, i, t) by executing zero or more Turing machine moves.

(Un)Decidability:

Some decision problems cannot be solved by any algorithm. Such problems are called undecidable. In contrast, decidable decision problems are solved by some algorithm. An example of an undecidable problem is the Turing machine halting on blank tape.

Wang Tile:

A Wang tile is a unit square tile with colored edges. Tiles have an orientation, i.e., they may not be rotated or reflected. The colors give a local matching rule that specifies which tiles may be placed next to each other: Two adjacent tiles must have identical colors on the abutting edges. A Wang tile set consists of a finite number of Wang tiles.

A more general definition: a d-dimensional tile set is a quadruple \( T \) = (d, T,N,R) where T is a finite set whose elements are called tiles, \( N=\left({\overrightarrow{n}}_1,{\overrightarrow{n}}_2,\dots, {\overrightarrow{n}}_m\right) \) is a neighborhood vector of m distinct elements of ℤd, and R ⊆ T m is a relation of allowed patterns. The neighborhood vector has the same interpretation as in the definition of cellular automata: it gives the relative locations of the neighbors of cells.

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Acknowledgments

This research is supported by the Academy of Finland grants 211967 and 131558.

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  1. Department of Mathematics, University of Turku, Turku, Finland

    Jarkko Kari

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Correspondence to Jarkko Kari .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Kari, J. (2015). Tiling Problem and Undecidability in Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_552-5

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