Abstract
A binary grammar is a relational grammar with two nonterminal alphabets, two terminal alphabets, a set of pairs of productions and the pair of the initial nonterminals that generates the binary relation, i.e., the set of pairs of strings over the terminal alphabets. This paper investigates the binary context-free grammars as mutually controlled grammars: two context-free grammars generate strings imposing restrictions on selecting production rules to be applied in derivations. The paper shows that binary context-free grammars can generate matrix languages whereas binary regular and linear grammars have the same power as Chomskyan regular and linear grammars.
Highlights
A “traditional” phrase-structure grammar is a generative computational mechanism that produces strings over some alphabet starting from the initial symbol and sequentially applying production rules that rewrite sequences of symbols [1,2,3].According to the forms of production rules, phrase-structure grammars and their languages are divided into four families: regular, context-free, context-sensitive, and recursively enumerable [4,5].Regular and context-free grammars, which have good computational and algorithmic properties, are widely used in modeling and studying of phenomena appearing in linguistics, computer science, artificial intelligence, biology, etc. [6,7]
Though binary grammars are asynchronous systems by their definitions, we showed that they can work in synchronized mode (Lemmas 1–3), i.e., the both grammars in a binary relation generate strings with derivations where the grammars apply some productions in each step, and stop at the same time
We have studied the generative capacity of binary context-free grammars, and showed that binary regular and linear grammars have the same power as their Chomskyan alternatives, i.e., traditional regular and linear grammars, respectively (Lemmas 6 and 7 and Corollary 1)
Summary
A “traditional” phrase-structure grammar ( known as a Chomskyan grammar) is a generative computational mechanism that produces strings (words) over some alphabet starting from the initial symbol and sequentially applying production rules that rewrite sequences of symbols [1,2,3]. The second is partial parallelism where all or some nonterminal symbols (not terminal symbols) are written in each step of the derivations: absolutely parallel grammars [26]—all nonterminals of the sentential form are rewritten in one derivation step; Indian parallel grammars [27]—all occurrences of one letter are replaced (according to one rule); Russian parallel grammars [28]—which combines the context-free and Indian parallel feature; scattered context grammars [29]—in which only a fixed number of symbols can be replaced in a step but the symbols can be different; concurrently controlled grammars [30]—the control over a parallel application of the productions is realized by a Petri net with different parallel firing strategies Another perspective in using the notion of parallelism with grammars is a grammar system, which is a system of several phrase-structure grammars with own axioms, symbols and rewriting productions that can work simultaneously and generate own strings. Several other papers [35,36,37,38,39,40] investigated the properties of relational grammars and applied in solving problems appeared in natural and visual language processing
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