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Trade-Offs Between Time, Space, Cooperation, and Communication Complexity for CD Grammar Systems

100%

Cojocaru L.

Fundamenta Informaticae

2011

tom Vol. 107, nr 4

345-378

In this paper we investigate the communication and cooperation phenomenon in Cooperating Distributed Grammar Systems (henceforth CDGSs). In this respect, we define several complexity structures and two complexity measures, the cooperation and communication complexity measures. These measures are studied with respect to the derivation modes and fairness conditions under which CDGSs may work. We deal with trade-offs between time, space, cooperation, and communication complexity of languages generated by CDGSs with regular, linear, and context-free components. Cooperation and communication processes in CDGSs with regular and linear components are of equal complexity. The two (equal) cooperation and communication complexity measures are either constant or linear, as functions of lengths of words in the generated language. The same result holds for the cooperation and communication complexity of q-fair languages generated by CDGSs with regular and linear components. For the case of non-constant communication (cooperation) the time and space used by a nondeterministicmultitape Turingmachine to recognizeweakly q-fair languages are linear, as is the communication (cooperation) complexity. For CDGSs with context-free components the cooperation and communication complexity may be different. These measures are either linear or logarithmic functions, in terms of lengths of words in the generated language. In order to find trade-offs between time, space, cooperation, and communication complexity of languages generated by CDGSs with context-free components we study asymptotic behavior of growth functions characteristic to these measures. We prove that languages generated by CDGSs with context-free components are accepted by nondeterministicmultitape Turing machines either in quadratic time, linear space, and with cooperation complexity that varies from logarithmic to linear, or in polynomial or exponential time and space, and linear cooperation complexity.

The Complexity of Szilard Languages of Matrix Grammars Revisited

63%

Cojocaru L. , Mäkinen E.

Fundamenta Informaticae

2013

tom Vol. 123, nr 4

381--399

The regulated rewriting mechanism is one of the most efficient methods to augment the Chomsky hierarchy with a large variety of language classes. In this paper we investigate the derivation mechanism in regulated rewriting grammars such as matrix grammars, by studying their Szilard languages. We focus on the complexity of Szilard languages associated with unrestricted and leftmost-like derivations in matrix grammars, with or without appearance checking. The reason is twofold. First, to relate these classes of languages to parallel complexity classes such as NC1 and AC1, and, second, to improve some previous results. We prove that unrestricted Szilard languages and certain leftmost Szilard languages of context-free matrix grammars, without appearance checking, can be accepted by indexing alternating Turing machines in logarithmic time and space. Consequently, these classes are included in UE-uniform NC1. Unrestricted Szilard languages of matrix grammars with appearance checking can be accepted by deterministic Turing machines in O(n log n) time and O(log n) space. Leftmost-like Szilard languages of context-free matrix grammars, with appearance checking, can be recognized by nondeterministic Turing machines by using the same time and space resources. Hence, all these classes are included in AC1.