State Complexity of Basic Operations on Finite Languages

Incomplete Transition Complexity of Basic Operations on Finite Languages

“…Câmpeanu et al [3] studied the state complexity of the concatenation of a mstate complete DFA with a n-state complete DFA over an alphabet of size k and proposed the upper bound…”

“…In this paper we give tight upper bounds for the transition complexity of all the above operations. We correct the upper bound for the state complexity of concatenation [3], and show that if the right automaton is larger than the left one, the upper bound is only reached using an alphabet of variable size. Note that, the difference between the state complexity for non necessarily complete DFAs and for complete DFAs is at most one.…”

“…For finite languages, Salomaa and Yu [11] showed that the state complexity of the determinization of a nondeterministic automaton (NFA) with m states and k symbols is Θ(k m 1+log k ) (lower than 2 m as it is the case for general regular languages). Câmpeanu et al [3] studied the operational state complexity of concatenation, Kleene star, and reversal. Finally, Han and Salomaa [6] gave tight upper bounds for the state complexity of union and intersection on finite languages.…”

The Average State Complexity of Rational Operations on Finite Languages

“…More precisely, for every integer h ≥ 3, they gave a language X h of length Θ(h 2 ), containing Θ(h) words, whose state complexity is Θ(h2 h ). Using another measure on finite sets of words, Campeanu, Culik, Salomaa and Yu proved in [3,4] that if the set X is a finite language over an alphabet of at least three letters having state complexity n ≥ 4, the state complexity of X * is 2 n−3 + 2 n−4 in the worst case. In addition when X is not necessarily finite, the state complexity of X * is 2 n−1 + 2 n−2 in the worst case [16,17].…”

Complexity of Operations on Cofinite Languages

“…This topic is intensively studied since almost the beginning of automata theory (see [1][2][3][4] for recent results). Researchers are mainly interested in automata that represent natural subclasses of regular languages such as finite languages [5] or prefix-free regular languages [6]. Most articles focus on the worst case state complexity of basic regular operations such as set constructions, concatenation, Kleene star and reversal.…”