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\(n\)-ary grammars and the description of mapping of languages. (English) Zbl 0194.31602
An \(n\)-ary grammar is a generalization of a phrase-structure grammar. It consists of a system of \(n\) terminal alphabets, \(n\) nonterminal alphabets, a set of productions and an initial \(n\)-tuple of nonterminals. Each production is an \(n\)-tuple of common productions or empty places. Similar systems have been used elsewhere for definition of languages. In the paper three different types of derivation \((\alpha,\beta\) and \(\gamma)\) are considered whose common feature is that components of a production are applied simultaneously to corresponding components of intermediate \(n\)-tuples of words. An \(\alpha\)-generation is a straightforward generalization of the usual “top-down” definition of generation in phrase structure grammars (N. Chomsky). The \(\beta\) generation can be regarded as a generalization of “bottom-up” generation used in definition of Backus normal forms. For common context-free grammars these definitions are known to be equivalent, this is not true for context-free \(n\)-ary grammars. A condition under which an \(\alpha\)-generation is also a \(\beta\)-generation is derived. The Chomsky classification of phrase-structure grammars and languages is generalized for \(n\)-ary grammars and relations. The standard closure properties and other properties of classes of relations generated by different types of \(n\)-ary grammars are studied. The class of relation defined by right-linear grammars are identical for \(\alpha\) and \(\beta\)-generations and is precisely equal to the class of relation defined by nondeterministic \(n\)-tape finite one-way nonwriting automata.
In the last section binary grammars are used to describe translations of languages. \(\beta\) and \(\gamma\) generations seem to be particularly convenient for this application. Finally, an attempt is made to classify the complexity of alphabetical mappings by the type of binary grammar which is required for their description.
Reviewer: Karel Čulik II

MSC:
68Q42 Grammars and rewriting systems
68Q45 Formal languages and automata
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References:
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