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

Approximating Language Edit Distance Beyond Fast Matrix Multiplication: Ultralinear Grammars Are Where Parsing Becomes Hard!

In 1975, a breakthrough result of L. Valiant showed that parsing context free grammars can be reduced to Boolean matrix multiplication, resulting in a running time of O(n^omega) for parsing where omega <= 2.373 is the exponent of fast matrix multiplication, and n is the string length. Recently, Abboud, Backurs and V. Williams (FOCS 2015) demonstrated that this is likely optimal; moreover, a combinatorial o(n^3) algorithm is unlikely to exist for the general parsing problem. The language edit distance problem is a significant generalization of the parsing problem, which computes the minimum edit distance of a given string (using insertions, deletions, and substitutions) to any valid string in the language, and has received significant attention both in theory and practice since the seminal work of Aho and Peterson in 1972. Clearly, the lower bound for parsing rules out any algorithm running in o(n^omega) time that can return a nontrivial multiplicative approximation of the language edit distance problem. Furthermore, combinatorial algorithms with cubic running time or algorithms that use fast matrix multiplication are often not desirable in practice. To break this n^omega hardness barrier, in this paper we study additive approximation algorithms for language edit distance. We provide two explicit combinatorial algorithms to obtain a string with minimum edit distance with performance dependencies on either the number of non-linear productions, k^*, or the number of nested non-linear production, k, used in the optimal derivation. Explicitly, we give an additive O(k^*gamma) approximation in time O(|G|(n^2 + (n/gamma)^3)) and an additive O(k gamma) approximation in time O(|G|(n^2 + (n^3/gamma^2))), where |G| is the grammar size and n is the string length. In particular, we obtain tight approximations for an important subclass of context free grammars known as ultralinear grammars, for which k and k^* are naturally bounded. Interestingly, we show that the same conditional lower bound for parsing context free grammars holds for the class of ultralinear grammars as well, clearly marking the boundary where parsing becomes hard

Synthesis, structure and power of systolic computations

AbstractA variety of problems related to systolic architectures, systems, models and computations are discussed. The emphases are on theoretical problems of a broader interest. Main motivations and interesting/important applications are also presented. The first part is devoted to problems related to synthesis, transformations and simulations of systolic systems and architectures. In the second part, the power and structure of tree and linear array computations are studied in detail. The goal is to survey main research directions, problems, methods and techniques in not too formal a way