Nondeterministic Syntactic Complexity

We introduce a new measure on regular languages: their nondeterministic syntactic complexity. It is the least degree of any extension of the ‘canonical boolean representation’ of the syntactic monoid. Equivalently, it is the least number of states of any subatomic nondeterministic acceptor. It turns out that essentially all previous structural work on nondeterministic state-minimality computes this measure. Our approach rests on an algebraic interpretation of nondeterministic finite automata as deterministic finite automata endowed with semilattice structure. Crucially, the latter form a self-dual category.

One-Time Nondeterministic Computations

We introduce the concept of one-time nondeterminism as a new kind of limited nondeterminism for finite state machines and pushdown automata. Roughly speaking, one-time nondeterminism means that at the outset the computation is nondeterministic, but whenever it performs a guess, this guess is fixed for the rest of the computation. We characterize the computational power of one-time nondeterministic finite automata (OTNFAs) and one-time nondeterministic pushdown devices. Moreover, we study the descriptional complexity of these machines. For instance, we show that for an [Formula: see text]-state OTNFA with a sole nondeterministic state, that is nondeterministic for only one input symbol, [Formula: see text] states are sufficient and necessary in the worst case for an equivalent deterministic finite automaton. In case of pushdown automata, the conversion of a nondeterministic to a one-time nondeterministic as well as the conversion of a one-time nondeterministic to a deterministic one turn out to be non-recursive, that is, the trade-offs in size cannot be bounded by any recursive function.

Descriptional Complexity of the Forever Operator

We examine the descriptional complexity of the forever operator, which assigns the language [Formula: see text] to a regular language [Formula: see text], and we investigate the trade-offs between various models of finite automata. We consider complete and partial deterministic finite automata, nondeterministic finite automata with single or multiple initial states, alternating, and Boolean finite automata. We assume that the argument and the result of this operation are accepted by automata belonging to one of these six models. We investigate all possible trade-offs and provide a tight upper bound for 32 of 36 of them. The most interesting result is the trade-off from nondeterministic to deterministic automata given by the Dedekind number [Formula: see text]. We also prove that the nondeterministic state complexity of [Formula: see text] is [Formula: see text] which solves an open problem stated by Birget [The state complexity of [Formula: see text] and its connection with temporal logic, Inform. Process. Lett. 58 (1996) 185–188].

State Complexity of Insertion

It is well known that the resulting language obtained by inserting a regular language to a regular language is regular. We study the nondeterministic and deterministic state complexity of the insertion operation. Given two incomplete DFAs of sizes m and n, we give an upper bound (m+2)·2mn−m−1·3m and find a lower bound for an asymp-totically tight bound. We also present the tight nondeterministic state complexity by a fooling set technique. The deterministic state complexity of insertion is 2Θ(mn) and the nondeterministic state complexity of insertion is precisely mn+2m, where m and n are the size of input finite automata. We also consider the state complexity of insertion in the case where the inserted language is bifix-free or non-returning.

NONDETERMINISTIC STATE COMPLEXITY OF PROPORTIONAL REMOVALS

The language [Formula: see text] consists of first halfs of strings in L. Many other variants of a proportional removal operation have been considered in the literature and a characterization of removal operations that preserve regularity is known. We consider the nondeterministic state complexity of the operation [Formula: see text] and, more generally, of polynomial removals as defined by Domaratzki (J. Automata, Languages and Combinatorics 7(4), 2002). We give an O(n2) upper bound for the nondeterministic state complexity of polynomial removals and a matching lower bound in cases where the polynomial is a sum of a monomial and a constant, or when the polynomial has rational roots.

THE RANGES OF STATE COMPLEXITIES FOR COMPLEMENT, STAR, AND REVERSAL OF REGULAR LANGUAGES

We examine the deterministic and nondeterministic state complexity of complements, stars, and reversals of regular languages. Our results are as follows: (1) The nondeterministic state complexity of the complement of an n-state nfa language over a five-letter alphabet may reach each value from log n to 2n. (2) The state complexity of the star (reversal) of an n-state dfa language over a growing alphabet may reach each value from 1 to [Formula: see text] (from log n to 2n, respectively). (3) The nondeterministic state complexity of the star (reversal) of an n-state nfa binary language may reach each value from 1 to n + 1 (from n - 1 to n + 1, respectively). We also obtain some partial results on the nondeterministic state complexity of complements of binary regular languages. As a bonus, we get an exponential number of values that are non-magic in the binary case.