Minimal strings in a regular language with respect to a partial order on the alphabet

Abstract

Let L be a regular language and “⩽” a partial order on its alphabet Σ; the partial order “⩽” may be extended to Σ ∗ by defining a 1 a 2⋯ a k ⩽ b 1 b 2⋯ b m if k⩽ m and a j ⩽ b j for 1⩽ j⩽ k. We show that L={x: x is a minimal string in L} is a regular language. A similar property does not hold, however, for the context-free languages. We construct a finite-state machine for the language L by showing that a certain type of “marked” finite-state machines are equivalent to the standard finite-state machines. Interestingly, the number of states N in the minimum deterministic finite-state machine for L may be as large as O( N 2), where N is the number of states in the minimum deterministic finite-state machine for L. The motivation for considering the minimal string problem is that it has important applications in “common sense reasoning” of temporal events in artificial intelligence.