On the Complexity of Szilard Languages of Matrix Grammars

Abstract

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 Complexity of Szilard Languages of Matrix Grammars Revisited

“…Szilard languages of context-free MGs with appearance checking, without any restriction on derivations, can be accepted by a deterministic log-space Turing machine in O(n log n) time. Consequently, this class is contained in AC 1 , which is an improvement over the result in [6], where we proved that Szilard languages of contextfree MGs with appearance checking are contained in N C 2 . Section 4 is dedicated to the complexity of leftmost Szilard languages of MGs.…”

“…Hence, A has to consider all possibilities of computing the net effect of a matrix m j , 1 ≤ j ≤ k, occurring in the Szilard word, with respect to the policies of m j . By using a divide and conquer algorithm in [6] we proved that each language in SZM ac (CF ) can be recognized by an indexing ATM in logarithmic space and square logarithmic time. Consequently [30], the class of Szilard languages of MGs with appearance checking is in N C 2 .…”

“…for each 1 ≤ l ≤ m. Otherwise, ℘ n returns 0 6. Here s(i) α η i ,r+1 is actually the sum s (i) A l where αη i ,r+1 is the l th nonterminal in N .…”

On Szilard Languages of Labelled Insertion Grammars

“…The main contribution of the paper is the association of the well-known concept of Szilard languages with insertion grammars and compare the Szilard languages obtained by these grammars with the family of languages in Chomsky hierarchy. The idea of Szilard languages is well investigated in formal language theory and their closure properties, decidability aspects, complexity aspects for Chomsky grammars, matrix grammars, parallel communicating grammar systems, communicating distributive grammar systems have been investigated in [22,16,14,17,15]. Also, the idea of derivation languages (as Szilard and Control languages) has been introduced for DNA and membrane computing models in [13,21].…”