We investigate computational resources used by alternating Turing machines (ATMs) to accept Szilard languages (SZLs) of context-free matrix grammars (MGs). The main goal is to relate these languages to parallel complexity classes such as NC1 and NC2. We prove that unrestricted and leftmost-1 SZLs of context-free MGs, without appearance checking, can be accepted by ATMs in logarithmic time and space. Hence, these classes of languages belong to NC1 (under ALOGTIME reduction). Unrestricted SZLs of context-free MGs with appearance checking can be accepted by ATMs in logarithmic space and square logarithmic time. Consequently, this class is contained in NC2. We conclude with some results on SZLs of MGs with phrase-structure rules.
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 N C 1 and AC 1 , 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 U E * -uniform N C 1 . 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 AC 1 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.