Prime normal form and equivalence of simple grammars

Authors:
Cédric Bastien;Jurek Czyzowicz;Wojciech Fraczak;Wojciech Rytter
Affiliations:
Dépt d'informatique, Université du Québec en Outaouais, Gatineau PQ, Canada;Dépt d'informatique, Université du Québec en Outaouais, Gatineau PQ, Canada;Dépt d'informatique, Université du Québec en Outaouais, Gatineau PQ, Canada and IDT Canada Inc., Ottawa, Ont., Canada and Institute of Informatics, Warsaw University, Warsaw, Poland;Institute of Informatics, Warsaw University, Warsaw, Poland
Venue:
Theoretical Computer Science - Implementation and application of automata
Year:
2006

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Abstract

A prefix-free language is prime if it cannot be decomposed into a concatenation of two prefix-free languages. We show that we can check in polynomial time if a language generated by a simple context-free grammar is prime. Our algorithm computes a canonical representation of a simple language, converting its arbitrary simple grammar into prime normal form (PNF); a simple grammar is in PNF if all its nonterminals define primes. We also improve the complexity of testing the equivalence of simple grammars. The best previously known algorithm for this problem worked in O(n13) time. We improve it to O(n7 log2 n) and O(n5 polylog v) time, where n is the total size of the grammars involved, and v is the length of a shortest string derivable from a nonterminal, maximized over all nonterminals.