Closure properties of hyper-minimized automata
RAIRO - Theoretical Informatics and Applications
Two deterministic finite automata are almost equivalent if they disagree in acceptance only for finitely many inputs. An automaton A is hyper-minimized if no automaton with fewer states is almost equivalent to A. A regular language L is canonical if the minimal automaton accepting L is hyper-minimized. The asymptotic state complexity s * (L) of a regular language L is the number of states of a hyper-minimized automaton for a language finitely different from L. In this paper we show that: (1)
... e show that: (1) the class of canonical regular languages is not closed under: intersection, union, concatenation, Kleene closure, difference, symmetric difference, reversal, homomorphism, and inverse homomorphism; (2) for any regular languages L1 and L2 the asymptotic state complexity of their sum L1 ∪ L2, intersection L1 ∩ L2, difference L1 − L2, and symmetric difference L1 ⊕ L2 can be bounded by s * (L1) · s * (L2). This bound is tight in binary case and in unary case can be met in infinitely many cases. (3) For any regular language L the asymptotic state complexity of its reversal L R can be bounded by 2 s * (L) . This bound is tight in binary case. (4) The asymptotic state complexity of Kleene closure and concatenation cannot be bounded. Namely, for every k ≥ 3, there exist languages K, L, and M such that s * (K) = s * (L) = s * (M ) = 1 and s * (K * ) = s * (L · M ) = k. These are answers to open problems formulated by Badr et al. [RAIRO-Theor.