Operations on Unambiguous Finite Automata

Abstract

A nondeterministic finite automaton is unambiguous if it has at most one accepting computation on every input string. We investigate the state complexity of basic regular operations on languages represented by unambiguous finite automata. We get tight upper bounds for reversal ([Formula: see text]), intersection ([Formula: see text]), left and right quotients ([Formula: see text]), positive closure ([Formula: see text]), star ([Formula: see text]), shuffle ([Formula: see text]), and concatenation ([Formula: see text]). To prove tightness, we use a binary alphabet for intersection and left and right quotients, a ternary alphabet for star and positive closure, a five-letter alphabet for shuffle, and a seven-letter alphabet for concatenation. For complementation, we reduce the trivial upper bound [Formula: see text] to [Formula: see text]. We also get some partial results for union and square.

Image-Binary Automata

“…This super-polynomial blowup (even for unary alphabet and even if the output automaton is allowed to be ambiguous) refuted a conjecture that it may be possible to complement UFAs with a polynomial blowup [7]. An upper bound (for general alphabets and requiring the output to be a UFA) of O(2 0.79n ) was shown in [16]; see also [15] for an (unpublished) improvement.…”

Lower Bounds for Unambiguous Automata via Communication Complexity

“…This super-polynomial blowup (even for unary alphabet and even if the output automaton is allowed to be ambiguous) refuted a conjecture that it may be possible to complement UFAs with a polynomial blowup [3]. A non-trivial upper bound (for general alphabets and outputting a UFA) was shown by Jirásek et al [9]: 9]). Let A be a UFA with n ≥ 7 states that recognizes a language L ⊆ Σ * .…”

“…The state complexity has been well studied for various types of automata and language operations, see, e.g., [9] and the references therein for some known results. For example, it was shown in [7] that complementing an NFA with n states may require Θ(2 n ) states.…”

Ambiguity, Nondeterminism and State Complexity of Finite Automata

“…To conclude this section, we mention that Okhotin [35] has studied the state complexity of determinization of unary UFAs and Jirasek et al [22] recently studied the state complexity of operations on UFAs.…”