Tiling Problem and Undecidability in Cellular Automata

Kari, Jarkko

doi:10.1007/978-0-387-30440-3_552

Tiling Problem and Undecidability in Cellular Automata

Jarkko Kari²

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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.

Input::: Cellular Automaton A
Question::: Is A injective (i. e. reversible )?

There is an algorithm that solves Injectivity for one‐dimensional CA [1]. But the problem is undecidable among two‐dimensional CA [9,11]. The problem is semi‐decidable in any dimension.

Input::: Cellular Automaton A
Question::: Is A surjective ?

Also Surjectivity is decidable among one‐dimensional CA. The two‐dimensional question is, however, undecidable [11]. The complement of the problem (i. e. non‐surjectivity) is semi‐decidable in any dimension.

Input::: Cellular Automaton A
Question::: Is A...

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Abbreviations

Cellular automata (CA):

A d‑dimensional cellular automaton consists of an infinite d‑dimensional grid of cells, indexed by $ { \mathbb{Z}^d } $. Each cell stores an element of a finite state set S. Configuration $ { c \colon \mathbb{Z}^d\to S } $ specifies the states of all cells. The set of all configurations is $ { S^{\mathbb{Z}^d} } $. The neighborhood vector $ { N=(\vec{n}_1, \vec{n}_2, \dots ,\vec{n}_m) } $ is a sequence of m distinct elements of $ { \mathbb{Z}^d } $ specifying the relative locations of the neighbors of the cells: A cell located at $ { \vec{x}\in\mathbb{Z}^d } $ has m neighbors, in positions $ { \vec{x}+\vec{n}_1, \vec{x}+\vec{n}_2, \dots ,\vec{x}+\vec{n}_m } $. Finally, the local update rule $ { f \colon S^m\to 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 $ { \vec{x}\in\mathbb{Z}^d } $,

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

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

Tiles:

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 $ { {\mathcal T}=(d,T,N,R) } $ where T is a finite set whose elements are called tiles, $ { N=(\vec{n}_1, \vec{n}_2, \dots ,\vec{n}_m) } $ is a neighborhood vector of m distinct elements of $ { \mathbb{Z}^d } $ and $ { R\subseteq 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.

Tiling :

A covering of the plane using tiles. A valid Wang tiling by a Wang tile set T is an assignment $ { t \colon \mathbb{Z}^2\to T } $ of tiles to cells such that the local matching rule is satisfied between all adjacent tiles. We say that T admits tiling t.

More general definition: A d‑dimensional tiling using tile set $ { {\mathcal T}=(d, T, N, R) } $ is a mapping $ { t \colon \mathbb{Z}^d\to T } $. Tiling t is valid at cell $ { \vec{x}\in\mathbb{Z}^d } $ if

$$ t(\vec{x}+\vec{n}_1, \vec{x}+\vec{n}_2, \dots ,\vec{x}+\vec{n}_m) \in R\:. $$

Tiling t is called valid if it is valid at every cell $ { \vec{x}\in\mathbb{Z}^d } $.

Periodic tiling :

A tiling that is invariant under some non-zero 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 \colon \mathbb{Z}^d\to T } $ is periodic with period $ { \vec{p}\in\mathbb{Z}^d } $, $ { \vec{p}\neq\vec{0} } $, if $ { t(\vec{x}) = t(\vec{x}+\vec{p}) } $ for all $ { \vec{x}\in\mathbb{Z}^d } $. It is totally periodic if it is periodic with d linearly independent periods $ { \vec{p}_1, \vec{p}_2, \dots ,\vec{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.

Aperiodic tile set:

A two‐dimensional tile set that admits a valid tiling of the plane, but does not admit any valid periodic tilings. Smallest known aperiodic set of Wang tiles contain 13 tiles [3,13]

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 $ { \mathbb{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 6-tuple $ { {\mathcal M} = (Q,\Gamma,\delta, q_0, q_h, b) } $ where Q and Γ are finite sets (the state alphabet and the tape alphabet, respectively), $ { q_0,q_h\in Q } $ are the initial and the halting states, respectively, $ { b\in\Gamma } $ is the blank symbol and $ { \delta \colon Q\times\Gamma \to Q\times\Gamma\times\{-1,1\} } $ 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\in \mathbb{Z} } $ is the position of the machine on the tape and $ { t \colon \mathbb{Z}\to \Gamma } $ describes the content of the tape. In one time step configuration $ { (q,i,t) } $ becomes $ { (q^{\prime},i+d,t^{\prime}) } $ if $ { \delta(q,t(i))=(q^{\prime},\gamma, d) } $ and $ { t^{\prime}(i)=\gamma } $ and $ { t^{\prime}(j)=t(j) } $ for all $ { j\neq i } $. We denote this move by

$$ (q,i,t)\vdash (q^{\prime},i+d,t^{\prime})\:. $$

The reflexive, transitive closure of $ { \vdash } $ is denoted by $ { \vdash^* } $, that is,

$$ (q,i,t)\vdash^* (q^{\prime},i^{\prime},t^{\prime}) $$

if and only if $ { (q^{\prime},i^{\prime},t^{\prime}) } $ can be reached from $ { (q,i,t) } $ by executing zero or more Turing machine moves.

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.

The halting problem :

Turing machine halting on blank tape is the decision problem whose input is a Turing machine $ { {\mathcal M} = (Q,\Gamma,\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.

(Un)decidability:

Some decision problems can not 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.

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 for ever 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.

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.

The tiling problem :

The decision problem that gets as input a tile set $ { {\mathcal T} } $, and asks whether there exists a valid tiling by $ { {\mathcal T} } $. The tiling problem was proved undecidable for Wang tiles by R. Berger [2]. 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 $ { {\mathcal T} } $ and one tile s, and asks whether $ { {\mathcal T} } $ admits a tiling that contains tile s at least once. The problem is undecidable for Wang tiles, but its complement is semi‐decidable [24].

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 [7].

The finite tiling problem:

A decision problem where we are given a set T of Wang tiles and a specific blank tile $ { B\in T } $ whose all four edges are colored with the same color. A tiling $ { t \colon \mathbb{Z}\to T } $ is called finite if

$$ \{\vec{x}\in\mathbb{Z}^2 | t(\vec{x})\neq B\} $$

is a finite set. If $ { t(\vec{x})=B } $ for all $ { \vec{x}\in\mathbb{Z}^2 } $ then t is called trivial. The finite tiling problem asks whether there exist non‐trivial valid finite tilings. The problem is undecidable but semi‐decidable.

Surjective cellular automata:

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

Injective (reversible ) CA:

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

$$ \bigcap_{i=0}^{\infty} G(S^{\mathbb{Z}^d}) $$

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

Nilpotent cellular automata:

Nilpotent CA have 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.

Equicontinuous CA:

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

$$ \forall \vec{x}\in B \colon c(\vec{x})=e(\vec{x})\\ \Longrightarrow \forall t\in \mathbb{N}\enskip\text{and}\enskip\forall \vec{x}\in A \colon G^t(c)(\vec{x})=G^t(e)(\vec{x})\:. $$

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 [17]: $ { \exists n,p\in \mathbb{N} \colon G^n=G^{n+p} } $. In this sense the dynamics of equicontinuous CA is trivial.

Sensitive CA:

Cellular automaton G is called sensitive to initial conditions if there exists a finite set $ { B\subseteq\mathbb{Z}^d } $ of cells such that for every configuration c and every finite set $ { A \subseteq \mathbb{Z} } $ of cells there exists a configuration e and time $ { t \geq 0 } $ such that $ { e(\vec{x})=d(\vec{x}) } $ for all $ { \vec{x}\in A } $ but $ { G^t(e)(\vec{x}) \neq G^t(c)(\vec{x}) } $ for some $ { \vec{x}\in B } $. This means that arbitrarily distant modifications to any configuration c may propagate to a fixed observation window B.

Topological entropy :

The topological entropy h(G) of a CA G measures the complexity of its dynamics. For any finite $ { A\subseteq \mathbb{Z}^d } $ and positive integer n we define the equivalence relation $ { \equiv_{A,n} } $ among initial configurations as follows: For all $ { c,e\in S^{\mathbb{Z}^d} } $

$$ c \equiv_{A,n} e \\ \Leftrightarrow \forall\vec{x}\in A, 0\leq t < n \colon G^t(c)(\vec{x}) = G^t(e)(\vec{x})\:. $$

In other words, two configurations are equivalent if we can not 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 $ { \equiv_{A,n} } $. Then the topological entropy is

$$ h(G) = \sup_{A} \lim_{n\rightarrow\infty} \frac{\log N_G(A,n)}{n} $$

where the supremum is over all finite $ { A\subseteq\mathbb{Z}^d } $. The entropy always exists. In the one‐dimensional case the entropy is always a finite, non‐negative number. If $ { d\geq 2 } $ then the entropy can also be infinite.

Bibliography

Primary Literature

Amoroso S, Patt Y (1972) Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures. J Comput Syst Sci 6:448–464
MathSciNet MATH Google Scholar
Berger R (1966) The Undecidability of the Domino Problem. Mem Am Math Soc 66:1–72
Google Scholar
Culik K II (1996) An aperiodic set of 13 Wang tiles. Discret Math 160: 245–251
MathSciNet MATH Google Scholar
Culik K II, Yu S (1988) Undecidability of CA Classification Schemes. Complex Syst 2:177–190
MathSciNet MATH Google Scholar
Durand B, Formenti E, Varouchas G (2003) On undecidability of equicontinuity classification for cellular automata. In: Proceedings of Discrete Models for Complex Systems, Lyon, France, 16–19 June 2003, pp. 117–128
Google Scholar
Finelli M, Manzini G, Margara L (1998) Lyapunov Exponents versus Expansivity and Sensitivity in Cellular Automata. J Complex 14:210–233
MathSciNet MATH Google Scholar
Gurevich YS, Koryakov IO (1972) Remarks on Berger's paper on the domino problem. J Sib Math J 13:319–321
MATH Google Scholar
Hurd LP, Kari J, Culik K (1992) The topological Entropy of Cellular Automata is Uncomputable. Ercodic Theor Dyn Syst 12:255–265
MathSciNet MATH Google Scholar
Kari J (1990) Reversibility of 2D cellular automata is undecidable. Physica D 45:379–385
MathSciNet ADS MATH Google Scholar
Kari J (1992) The nilpotency problem of one‐dimensional cellular automata. SIAM J Comput 21:571–586
MathSciNet MATH Google Scholar
Kari J (1994) Reversibility and surjectivity problems of cellular automata. J Comput Syst Sci 48:149–182
MathSciNet MATH Google Scholar
Kari J (1994) Rice's theorem for the Limit Sets of Cellular Automata. Theoret Comput Sci 127:229–254
MathSciNet MATH Google Scholar
Kari J (1996) A small aperiodic set of Wang tiles. Discret Math 160:259–264
MathSciNet MATH Google Scholar
Kari J (2008) Undecidable properties on the dynamics of reversible one‐dimensional cellular automata. In: Proceedings of Journées Automates Cellulaires, Uzés, France, 21–25 April 2008
Google Scholar
Kari J, Ollinger N Periodicity and immortality in reversible computing. (in press)
Google Scholar
Kari J, Papasoglu P (1999) Deterministic aperiodic tile sets. J Geom Funct Anal 9:353–369
MathSciNet MATH Google Scholar
Kurka P (1997) Languages, equicontinuity and attractors in cellular automata. Ergod Theory Dyn Syst 17:417–433
MathSciNet MATH Google Scholar
Lukkarila V (2007) The 4‐way deterministic tiling problem is undecidable. (in press)
Google Scholar
Moore EF (1962) Machine models of self‐reproduction. In: Proceedings of the Symposia in Applied Mathematics 14:17–33
Google Scholar
Myhill J (1963) The Converse to Moore's Garden-of-Eden Theorem. In: Proceedings of the American Mathematical Society 14:685–686
MathSciNet MATH Google Scholar
Robinson RM (1971) Undecidability and Nonperiodicity for Tilings of the plane. Invent Math 12:177–209
MathSciNet ADS Google Scholar
Shereshevsky MA (1993) Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indag Math 4:203–210
MathSciNet MATH Google Scholar
Sutner K (1989) A note on the Culik-Yu classes. Complex Syst 3:107–115
MathSciNet MATH Google Scholar
Wang H (1961) Proving theorems by pattern recognition – II. Bell Syst Techn J 40:1–42
Google Scholar

Books and Reviews

Codd EF (1968) Cellular Automata. Academic Press, New York
MATH Google Scholar
Garzon M (1995) Models of massive parallelism: analysis of cellular automata and neural networks. Springer, New York
MATH Google Scholar
Hedlund G (1969) Endomorphisms and automorphisms of shift dynamical systems. Math Syst Theory 3:320–375
MathSciNet MATH Google Scholar
Kari J (2005) Theory of cellular automata: a survey. Theoret Comput Sci 334:3–33
MathSciNet MATH Google Scholar
Toffoli T, Margolus N (1987) Cellular Automata Machines. MIT Press, Cambridge
Google Scholar
Wolfram S (ed) (1986) Theory and Applications of Cellular Automata. World Scientific Press, Singapore
MATH Google Scholar
Wolfram S (2002) A New Kind of Science. Wolfram Media, Canada
MATH Google Scholar

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Acknowledgments

Research supported by the Academy of Finland grant 211967.

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Department of Mathematics, University of Turku, Turku, Finland
Jarkko Kari

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RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA
Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Kari, J. (2009). Tiling Problem and Undecidability in Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_552

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DOI: https://doi.org/10.1007/978-0-387-30440-3_552
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