Galina Jirásková

State complexity of some operations on binary regular languages

We investigate the state complexity of some operations on binary regular languages. In particular, we consider the concatenation of languages represented by deterministic finite automata, and the reversal and complementation of languages represented by nondeterministic finite automata. We prove that the upper bounds on the state complexity of these operations, which were known to be tight for larger alphabets, are tight also for binary alphabets.

STATE COMPLEXITY OF CONCATENATION AND COMPLEMENTATION

We investigate the state complexity of concatenation and the nondeterministic state complexity of complementation of regular languages. We show that the upper bounds on the state complexity of concatenation are also tight in the case that the first automaton has more than one accepting state. In the case of nondeterministic state complexity of complementation, we show that the entire range of complexities, up to the known upper bound can be produced.

State complexity of concatenation and complementation of regular languages

We investigate the state complexity of concatenation and the nondeterministic state complexity of complementation of regular languages. We show that the upper bounds on the state complexity of concatenation are also tight in the case that the first automaton has more than one accepting state. In the case of nondeterministic state complexity of complementation, we show that the entire range of complexities, up to the known upper bound can be produced.

State complexity of cyclic shift

The cyclic shift of a language L, defined as SHIFT(L) = {vu | uv ∈ L}, is an operation known to preserve both regularity and context-freeness. Its descriptional complexity has been addressed in Maslovs pioneering paper on the state complexity of regular language operations [Soviet Math. Dokl. 11 (1970) 1373-1375], where a high lower bound for partial DFAs using a growing alphabet was given. We improve this result by using a fixed 4-letter alphabet, obtaining a lower bound (n - 1)! 2 (n-1)(n-2) , which shows that the state complexity of cyclic shift is 2 n2+n log n-o(n) for alphabets with at least 4 letters. For 2- and 3-letter alphabets, we prove 2 ⊖(n2) state complexity. We also establish a tight 2n 2 + 1 lower bound for the nondeterministic state complexity of this operation using a binary alphabet.

On the State Complexity of Complements, Stars, and Reversals 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 in the range from lognto 2n. 1 The state complexity of the star (reversal) of an n-state DFA language over a growing alphabet may reach each value in the range from 1 to

COMPLEXITY IN UNION-FREE REGULAR LANGUAGES

\frac{3}{4}2^n

On the State Complexity of Star of Union and Star of Intersection

(from lognto 2n, respectively). 1 The nondeterministic state complexity of the star (reversal) of an n-state NFA binary language may reach each value in the range from 1 to n+ 1 (from ni¾? 1 to n+ 1, respectively). We also obtain some partial results on the nondeterministic state complexity of the complements of binary regular languages. As a bonus, we get an exponential number of values that are non-magic, which improves a similar result of Geffert (Proc. 7th DCFS, Como, Italy, 23---37).

Note on Minimal Finite Automata

We continue the investigation of union-free regular languages that are described by regular expressions without the union operation. We also define deterministic union-free languages as languages accepted by one-cycle-free-path deterministic finite automata, and show that they are properly included in the class of union-free languages. We prove that (deterministic) union-freeness of languages does not accelerate regular operations, except for the reversal in the nondeterministic case.

Reversal of binary regular languages

The state complexity of the star of union of an m-state DFA language and an n-state DFA language is proved to be 2 m+n−1−2 m−1−2 n−1+1 for every alphabet of at least two letters. The state complexity of the star of intersection is established as 3/4 2 mn for every alphabet of six or more letters. This improves the recent results of A. Salomaa, K. Salomaa and Yu (“State complexity of combined operations”, Theoret. Comput. Sci., 383 (2007) 140-152).

MAGIC NUMBERS AND TERNARY ALPHABET

We show that for all n and α such that 1 ≤ n ≤ α ≤ 2n there is a minimal n-state nondeterministic finite automaton whose equivalent minimal deterministic automaton has exactly α states.