Prime Normal Form and Equivalence of Simple Grammars

• Published in: Implementation and Application of Automata pp. 78 - 89
• Main Authors: Bastien, Cédric, Czyzowicz, Jurek, Fraczak, Wojciech, Rytter, Wojciech
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2006; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; Automata. Abstract machines. Turing machines; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Exact sciences and technology; Software; Theoretical computing; Randomization; Concatenation; Automaton; Character string; Grammar form; Normal form; Context free grammar; Parallelism; Deterministic approach; Polynomial time
• ISBN: 3540310231; 9783540310235
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11605157_7

A prefix-free language is a 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 a 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 log2n) and O(n5 polylog v) deterministic time, and O(n4 polylog n) randomized 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. Our improvement is based on a version of Caucal’s algorithm from [1].