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(n1/3) variables, and this bound is tight. We show that any unary NFA with n states can be simulated by a CNF grammar with O(n2/3) variables. Finally, for larger alphabets we show that there exist languages which can be accepted by an n-state DFA, but which require Ω(n/log n) variables in any equivalent CNF grammar.

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