A New Constructive Method for the One-Letter Context-Free Grammars

Constructive methods for obtaining the regular grammar counterparts for some sub-classes of the context free grammars (cfg) have been investigated by many researchers. An important class of grammars for which this is always possible is the one-letter cfg. We show in this paper a new constructive method for transforming arbitrary one-letter cfg to an equivalent regular expression of star-height 0 or 1. Our new result is considerably simpler than a previous construction by Leiss, and we also propose a new normal form for a regular expression with single-star occurrence. Through an alphabet factorization theorem, we show how to go beyond the one-letter cfg in a straight-forward way.Singapore-MIT Alliance (SMA

Chrobak Normal Form Revisited, with Applications

Abstract. It is well known that any nondeterministic finite automata over a unary alphabet can be represented in a certain normal form called the Chrobak normal form [1]. We present a very simple conversion pro-cedure working in O(n3) time. Then we extend the algorithm to improve two trade-offs concerning conversions between different representations of unary regular languages. Given an n-state NFA, we are able to find a regular expression of size O ( n2 logn) describing the same language (which improves the previously known O(n2) size bound [8]) and a context-free grammar in Chomsky normal form with O(√n logn) nonterminals (which improves the previously known O(n2/3) bound [3]). As a byproduct of our conversion procedure, we get an alternative proof of the Chrobak normal form theorem. We believe that its efficiency and simplicity make the effort of reproving an already known result worth-while. Key-words: unary automata, descriptional complexity

Simulating finite automata with context-free grammars

We consider simulating finite automata (both deterministic and nondeterministic) with context-free grammars in Chomsky normal form (CNF). We show that any unary DFA with n states can be simulated by a CNF grammar with O(n^{1/3}) variables, and this bound is tight. We show that any unary NFA with n states can be simulated by a CNF grammar with O(n^{2/3}) variables. Finally, for larger alphabets we show that there exist languages which can be accepted by an n-state DFA, but which require \u3a9(n/logn) variables in any equivalent CNF grammar