Nondeterministic Cellular Automata Research Articles

Topological dynamics of Nondeterministic Cellular Automata

Cellular Automata (CA) are discrete dynamical systems and an abstract model of parallel computation. Nondeterministic Cellular Automata (NCA) are the class of multi-valued functions obtained by allowing nondeterminism in CA. In this study we extend to multi-valued functions the definition of some important topological properties and investigate the differences between the dynamical behaviour of one-dimensional NCA and one-dimensional CA in such classes.

Uniform continuity of relations and nondeterministic cellular automata

This paper provides a definition of uniformly continuous relations between uniform spaces and studies relationships with continuous relations between topological spaces associated with the uniform spaces. Using the notion of uniform continuity, we give a global characterisation of nondeterministic cellular automata on a group and a possibly infinite alphabet.

A logical approach to locality in pictures languages

This paper deals with descriptive complexity of picture languages of any dimension by fragments of existential second-order logic:1) We generalize to any dimension the characterization by Giammarresi et al. (1996) of the class of recognizable picture languages in existential monadic second-order logic.2) We state natural logical characterizations of the class of picture languages of any dimension d≥1 recognized in linear time on nondeterministic cellular automata, a robust complexity class that contains, for d=1, all the natural NP-complete problems.Our characterizations are essentially deduced from normalization results for first-order and existential second-order logics over pictures.

Stochastic Cellular Automata: Correlations, Decidability and Simulations

This paper introduces a simple formalism for dealing with deterministic, non- deterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this feature allows for strictly more behaviors (for instance, number conserving stochastic cellular automata require these local probabilistic correlations). We also show that several problems which are deceptively simple in the usual definitions, become undecidable when we allow for local probabilistic correlations, even in dimension one. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochas- tic settings. Although the intrinsic simulation relation is shown to become undecidable in dimension two and higher, we provide explicit tools to prove or disprove the existence of such a simulation between any two given stochastic cellular automata. Those tools rely upon a characterization of equality of stochastic global maps, shown to be equivalent to the existence of a stochastic coupling between the random sources. We apply them to prove that there is no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality, as well as a universal non-deterministic cellular automaton.

Stochastic Cellular Automata: Correlations, Decidability and Simulations

This paper introduces a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this feature allows for strictly more behaviors for instance, number conserving stochastic cellular automata require these local probabilistic correlations. We also show that several problems which are deceptively simple in the usual definitions, become undecidable when we allow for local probabilistic correlations, even in dimension one. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. Although the intrinsic simulation relation is shown to become undecidable in dimension two and higher, we provide explicit tools to prove or disprove the existence of such a simulation between any two given stochastic cellular automata. Those tools rely upon a characterization of equality of stochastic global maps, shown to be equivalent to the existence of a stochastic coupling between the random sources. We apply them to prove that there is no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality, as well as a universal non-deterministic cellular automaton.

Intrinsic Simulations between Stochastic Cellular Automata

The paper proposes a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in a unifying and composable manner. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. We then provide explicit tools to prove or disprove the existence of such a simulation between two stochastic cellular automata, even though the intrinsic simulation relation is shown to be undecidable in dimension two and higher. The key result behind this is the caracterization of equality of stochastic global maps by the existence of a coupling between the random sources. We then prove that there is a universal non-deterministic cellular automaton, but no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality.

Non-deterministic cellular automata and languages

We summarize results on non-deterministic cellular language acceptors. The non-determinism is regarded as limited resource. For parallel devices, it is natural to bound the non-determinism in time and/or space. Depending on the length of the input, the number of allowed non-deterministic state transitions as well as the number of non-deterministic cells at all is limited. We centre our attention to real-time, linear-time, and unrestricted-time computations and discuss the computational power of these machines. Speed-up results and the possibility to reduce the non-determinism as well as closure properties of languages acceptable with a constant number of non-deterministic transitions are presented. By considering the relations with context-free languages, several relations between the devices in question are implied. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved, and open problems for further research.

A Recursive Padding Technique on Nondeterministic Cellular Automata

We present a tight time-hierarchy theorem for nondeterministic cellular automata by using a recursive padding argument. It is shown that, if t2(n) is a time-constructible function and t2(n) grows faster than t1(n + 1), then there exists a language which can be accepted by a t2(n)-time nondeterministic cellular automaton but not by any t1(n)-time nondeterministic cellular automaton.

A SAT-based parser and completer for pictures specified by tiling

Pictures or patterns have been formally specified by different methods such as grammars. An alternative approach is based on tiling systems (TS) (Wang tiles are an analogous and equivalent formalism), whereby the picture is obtained by first covering it with a specified set of 2 × 2 tiles, then by performing a pixel by pixel mapping. TS are a powerful technique: the corresponding pictures can be recognized by non-deterministic cellular automata, which are more powerful than the four-ways automata. The difficulty to write such specifications for non-elementary pictures, and the NP-complete computational complexity of TS picture recognition have so far blocked any attempt to application. We have implemented a recognizer and generator for TS pictures in a very attractive, unconventional way, by transforming the tiling problem into a SAT (Boolean satisfiability) one, then using an efficient off-the-shelf SAT-solver. The prototype is fast enough to experiment on reasonably sized samples, and has the bonus of being able to complete or extrapolate a partial or noisy picture. The tool is invaluable to assist in writing picture specification. A series of examples shows how to specify patterns using TS.

Computations on Nondeterministic Cellular Automata

This work concerns the trade-offs between the dimension and the time and space complexity of computations on nondeterministic cellular automata. We assume that the space complexity is the diameter of area in space involved in computation. It is proved that (1) every nondeterministic cellular automata (NCA) A of dimension r , computing a predicate P with time complexity T ( n ) and space complexity S ( n ) can be simulated by r -dimensional NCA with time and space complexity O ( T 1/( r +1) S r /( r +1) ) and by r +1 dimensional NCA with time and space complexity O ( T 1/2 + S ), where T and S are functions constructible in time, (2) for any predicate P and integer r >1 if A is a fastest r -dimensional NCA computing P with time complexity T ( n ) and space complexity S ( n ), then T = O ( S ), and (3) if T r , P is the time complexity of a fastest r -dimensional NCA computing predicate P then T r +1, P = O (( T r , P ) 1− r /( r +1) 2 ), T r +1, P = O (( T r , P ) 1+2/ r ).Similar problems for deterministic cellular automata (CA) are discussed.

Some results concerning 2-D on-line tessellation acceptors and 2-D alternating finite automata

A two-dimensional nondeterministic on-line tessellation acceptor (2-NOTA) is a special type of real-time two-dimensional nondeterministic cellular automaton in which data flows from the upper-left corner to the lower-right corner. A two-dimensional alternating finite automaton (2-AFA) is an alternating finite automaton with a two-dimensional rectangular input whose input head can move in all four directions on the input. In this paper, we show that 2-NOTAs and 2-AFAs are incomparable. This answers in the negative an open question posed by Ito (this journal, 1989). Closure properties of the classes of languages (i.e., sets of two-dimensional patterns) accepted by two-way, three-way, and four-way two-dimensional alternating finite automata and two-dimensional alternating finite automata with only universal states (2-UFAs) are also obtained which answer several open questions posed by Inoue and Takanami (1988).