Alternation on cellular automata

AbstractIn this paper we consider several notions of alternation in cellular automata: non-uniform, uniform and weak alternation. We study relations among these notions and with alternating Turing machines. It is proved that the languages accepted in polynomial time by alternating Turing machines are those accepted by alternating cellular automata in polynomial time for all the proposed alternating cellular automata. In particular, this is true for the weak model where the difference between existential and universal states is omitted for all the cells except the first one. It is proved that real time alternation in cellular automata is strictly more powerful than real time alternation in Turing machines, with only one read-write tape. Moreover, it is shown that in linear time uniform and weak models agree

A Simple Cellular Automation that Solves the Density and Ordering Problems

Cellular automata (CA) are discrete, dynamical systems that perform computations in a distributed fashion on a spatially extended grid. The dynamical behavior of a CA may give rise to emergent computation, referring to the appearance of global information processing capabilities that are not explicitly represented in the system's elementary components nor in their local interconnections.1 As such, CAs o?er an austere yet versatile model for studying natural phenomena, as well as a powerful paradigm for attaining ?ne-grained, massively parallel computation. An example of such emergent computation is to use a CA to determine the global density of bits in an initial state con?guration. This problem, known as density classi?cation, has been studied quite intensively over the past few years. In this short communication we describe two previous versions of the problem along with their CA solutions, and then go on to show that there exists yet a third version | which admits a simple solution

On the Expressive Power of 2-Stack Visibly Pushdown Automata

Visibly pushdown automata are input-driven pushdown automata that recognize some non-regular context-free languages while preserving the nice closure and decidability properties of finite automata. Visibly pushdown automata with multiple stacks have been considered recently by La Torre, Madhusudan, and Parlato, who exploit the concept of visibility further to obtain a rich automata class that can even express properties beyond the class of context-free languages. At the same time, their automata are closed under boolean operations, have a decidable emptiness and inclusion problem, and enjoy a logical characterization in terms of a monadic second-order logic over words with an additional nesting structure. These results require a restricted version of visibly pushdown automata with multiple stacks whose behavior can be split up into a fixed number of phases. In this paper, we consider 2-stack visibly pushdown automata (i.e., visibly pushdown automata with two stacks) in their unrestricted form. We show that they are expressively equivalent to the existential fragment of monadic second-order logic. Furthermore, it turns out that monadic second-order quantifier alternation forms an infinite hierarchy wrt words with multiple nestings. Combining these results, we conclude that 2-stack visibly pushdown automata are not closed under complementation. Finally, we discuss the expressive power of B\"{u}chi 2-stack visibly pushdown automata running on infinite (nested) words. Extending the logic by an infinity quantifier, we can likewise establish equivalence to existential monadic second-order logic