Bounded Languages Meet Cellular Automata with Sparse Communication

Cellular automata are one-dimensional arrays of interconnected interacting finite automata. We investigate one of the weakest classes, the real-time one-way cellular automata, and impose an additional restriction on their inter-cell communication by bounding the number of allowed uses of the links between cells. Moreover, we consider the devices as acceptors for bounded languages in order to explore the borderline at which non-trivial decidability problems of cellular automata classes become decidable. It is shown that even devices with drastically reduced communication, that is, each two neighboring cells may communicate only constantly often, accept bounded languages that are not semilinear. If the number of communications is at least logarithmic in the length of the input, several problems are undecidable. The same result is obtained for classes where the total number of communications during a computation is linearly bounded

Conjunctive Grammars, Cellular Automata and Logic

The expressive power of the class Conj of conjunctive languages, i.e. languages generated by the conjunctive grammars of Okhotin, is largely unknown, while its restriction LinConj to linear conjunctive grammars equals the class of languages recognized by real-time one-dimensional one-way cellular automata. We prove two weakened versions of the open question Conj ?? RealTime1CA, where RealTime1CA is the class of languages recognized by real-time one-dimensional two-way cellular automata: 1) it is true for unary languages; 2) Conj ? RealTime2OCA, i.e. any conjunctive language is recognized by a real-time two-dimensional one-way cellular automaton. Interestingly, we express the rules of a conjunctive grammar in two Horn logics, which exactly characterize the complexity classes RealTime1CA and RealTime2OCA

Fast cellular automata with restricted inter-cell communication: computational capacity

A d-dimensional cellular automaton with sequential input mode is a d-dimensional grid of interconnected interacting finite automata. The distinguished automaton at the origin, the communication cell, is connected to the outside world and fetches the input sequentially. Often in the literature this model is referred to as iterative array. We investigate d-dimensional iterative arrays and one-dimensional cellular automata operating in real and linear time, whose inter-cell communication is restricted to some constant number of bits independent of the number of states. It is known that even one-dimensional one-bit iterative arrays accept rather complicated languages such as {ap│prim} or {a2n│n∈N}[16]. We show that there is an infinite strict double dimension-bit hierarchy. The computational capacity of the one-dimensional devices in question is compared with the power of communication-restricted two-way cellular automata. It turns out that the relations are quite diferent from the relations in the unrestricted case. On passing, we obtain an infinite strict bit hierarchy for real-time two-way cellular automata and, moreover, a very dense time hierarchy for every k-bit cellular automata, i.e., just one more time step leads to a proper superfamily of accepted languages.4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI

Fast cellular automata with restricted inter-cell communication: computational capacity

A d-dimensional cellular automaton with sequential input mode is a d-dimensional grid of interconnected interacting finite automata. The distinguished automaton at the origin, the communication cell, is connected to the outside world and fetches the input sequentially. Often in the literature this model is referred to as iterative array. We investigate d-dimensional iterative arrays and one-dimensional cellular automata operating in real and linear time, whose inter-cell communication is restricted to some constant number of bits independent of the number of states. It is known that even one-dimensional one-bit iterative arrays accept rather complicated languages such as {ap│prim} or {a2n│n∈N}[16]. We show that there is an infinite strict double dimension-bit hierarchy. The computational capacity of the one-dimensional devices in question is compared with the power of communication-restricted two-way cellular automata. It turns out that the relations are quite diferent from the relations in the unrestricted case. On passing, we obtain an infinite strict bit hierarchy for real-time two-way cellular automata and, moreover, a very dense time hierarchy for every k-bit cellular automata, i.e., just one more time step leads to a proper superfamily of accepted languages.4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI

Fast cellular automata with restricted inter-cell communication: computational capacity

A d-dimensional cellular automaton with sequential input mode is a d-dimensional grid of interconnected interacting finite automata. The distinguished automaton at the origin, the communication cell, is connected to the outside world and fetches the input sequentially. Often in the literature this model is referred to as iterative array. We investigate d-dimensional iterative arrays and one-dimensional cellular automata operating in real and linear time, whose inter-cell communication is restricted to some constant number of bits independent of the number of states. It is known that even one-dimensional one-bit iterative arrays accept rather complicated languages such as {ap│prim} or {a2n│n∈N}[16]. We show that there is an infinite strict double dimension-bit hierarchy. The computational capacity of the one-dimensional devices in question is compared with the power of communication-restricted two-way cellular automata. It turns out that the relations are quite diferent from the relations in the unrestricted case. On passing, we obtain an infinite strict bit hierarchy for real-time two-way cellular automata and, moreover, a very dense time hierarchy for every k-bit cellular automata, i.e., just one more time step leads to a proper superfamily of accepted languages.4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI

Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in d=1 lead to distinct classes in d >= 2. We explore the closure properties and computational complexity of these classes, including undecidability and L-, NL- and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d >= 2 has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t^{-d} unless it maps every initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy

Cellular automata with limited inter-cell bandwidth

AbstractA d-dimensional cellular automaton is a d-dimensional grid of interconnected interacting finite automata. There are models with parallel and sequential input modes. In the latter case, the distinguished automaton at the origin, the communication cell, is connected to the outside world and fetches the input sequentially. Often in the literature this model is referred to as an iterative array. In this paper, d-dimensional iterative arrays and one-dimensional cellular automata are investigated which operate in real and linear time and whose inter-cell communication bandwidth is restricted to some constant number of different messages independent of the number of states. It is known that even one-dimensional two-message iterative arrays accept rather complicated languages such as {ap∣p prime} or {a2n∣n∈N} (H. Umeo, N. Kamikawa, Real-time generation of primes by a 1-bit-communication cellular automaton, Fund. Inform. 58 (2003) 421–435). Here, the computational capacity of d-dimensional iterative arrays with restricted communication is investigated and an infinite two-dimensional hierarchy with respect to dimensions and messages is shown. Furthermore, the computational capacity of the one-dimensional devices in question is compared with the power of two-way and one-way cellular automata with restricted communication. It turns out that the relations between iterative arrays and cellular automata are quite different from the relations in the unrestricted case. Additionally, an infinite strict message hierarchy for real-time two-way cellular automata is obtained as well as a very dense time hierarchy for k-message two-way cellular automata. Finally, the closure properties of one-dimensional iterative arrays with restricted communication are investigated and differences to the unrestricted case are shown as well

Complete Symmetry in D2L Systems and Cellular Automata

We introduce completely symmetric D2L systems and cellular automata by means of an additional restriction on the corresponding symmetric devices. Then we show that completely symmetric D2L systems and cellular automata are still able to simulate Turing machine computations. As corollaries we obtain new characterizations of the recursively enumerable languages and of some space-bounded complexity classes

Upper Bound on the Products of Particle Interactions in Cellular Automata

Particle-like objects are observed to propagate and interact in many spatially extended dynamical systems. For one of the simplest classes of such systems, one-dimensional cellular automata, we establish a rigorous upper bound on the number of distinct products that these interactions can generate. The upper bound is controlled by the structural complexity of the interacting particles---a quantity which is defined here and which measures the amount of spatio-temporal information that a particle stores. Along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata; thereby elucidating why that approach is of general utility. The upper bound is tested against several relatively complex domain-particle cellular automata and found to be tight.Comment: 17 pages, 12 figures, 3 tables, http://www.santafe.edu/projects/CompMech/papers/ub.html V2: References and accompanying text modified, to comply with legal demands arising from on-going intellectual property litigation among third parties. V3: Accepted for publication in Physica D. References added and other small changes made per referee suggestion