Syntactic Complexity of Regular Ideals

Abstract

The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that n n−1 , n n−1 + n − 1, and n n−2 + (n − 2)2 n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced.

Complexity of proper prefix-convex regular languages

“…Hence the syntactic semigroup of L n (a, b, c, d) has size n n−1 as well. The fact that at least four letters are needed was proved in [14]. 2.…”

Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

“…Hence the syntactic semigroup of L n (a, b, c, d) has size n n−1 as well. The fact that at least four letters are needed was proved in [15].…”

Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

“…The complexity of suffix-closed languages was studied in [10] in the restricted case, and the syntactic semigroup of these languages, in [14,17,20]; however, most complex suffix-closed languages have not been examined. c, d, e), the complexity κ(A S ) satisfies: 1.…”