Nondeterministic Descriptional Complexity of Regular Languages

Abstract

We investigate the descriptional complexity of operations on finite and infinite regular languages over unary and arbitrary alphabets. The languages are represented by nondeterministic finite automata (NFA). In particular, we consider Boolean operations, catenation operations – concatenation, iteration, λ-free iteration – and the reversal. Most of the shown bounds are tight in the exact number of states, i.e. the number is sufficient and necessary in the worst case. Otherwise tight bounds in the order of magnitude are shown.

Complement on Prefix-Free, Suffix-Free, and Non-Returning NFA Languages

“…This means that O(F (n)) states are sufficient for an NFA to accept the complement of a unary NFA language. Moreover, in [12] a unary n-state NFA language is described such that every NFA accepting its complement needs at least F (n − 1) states. In [14], using a fooling set method, the lower bound F (n−1)+1 is proved for a non-returning language.…”

“…Ellul [7] gave binary O(n)-state witness languages. Holzer and Kutrib [12] proved the lower bound 2 n−2 for a binary n-state NFA language. Finally, a binary n-state NFA language meeting the upper bound 2 n was described by Jirásková in [15].…”

“…Finally, a binary n-state NFA language meeting the upper bound 2 n was described by Jirásková in [15]. In the unary case, the complexity of complementation is known to be in e Θ( √ n ln n) [12,14]. In this paper, we investigate the complementation operation on prefix-free, suffix-free, and non-returning languages.…”

Nondeterministic State Complexity of Positional Addition

“…In particular, Birget [2] has shown that the complement of a language recognized by an n-state NFA may require an NFA with as many as 2 n states, and this result was later improved by Jirásková [9] who reduced the alphabet for the witness language from {a, b, c, d} to {a, b}. The systematic study of nondeterministic state complexity, that is, state complexity with respect to NFAs, of different operations was started by Holzer and Kutrib [6], who obtained, in particular, the precise results for union, intersection and concatenation. More recently Jirásková and Okhotin [10] determined the nondeterministic state complexity of cyclic shift, Gruber and Holzer [4] established precise results for scattered substrings and scattered superstrings, Domaratzki and Okhotin [3] studied k-th power of a language, L k , while Han, K. Salomaa and Wood [5] considered the standard operations on NFAs in the context of prefix-free languages.…”

Lower bounds for the transition complexity of NFAs

“…The results are surveyed and more references can be found, for example, in [6,16,[27][28][29]. The state complexity of nondeterministic finite automaton (NFA) operations has been investigated by Holzer and Kutrib [13,14]. The minimal DFA equivalent to an arbitrary DFA can be constructed efficiently, whereas Jiang and Ravikumar [19] have shown that minimization of NFAs is PSPACE-complete, even if the input is given as a DFA, and Malcher [24] has shown that minimization remains NP-complete for classes of NFAs that are nearly deterministic.…”