Syntactic Complexity of Bifix-Free Languages

Abstract

Abstract. We study the properties of syntactic monoids of bifix-free regular languages. In particular, we solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a bifix-free language with state complexity n is at most (n−1) n−3 +(n−2) n−3 +(n−3)2 n−3 for n 6. The main proof uses a large construction with the method of injective function. Since this bound is known to be reachable, and the values for n 5 are known, this completely settles the problem. We also prove that (n − 2) n−3 + (n − 3)2 n−3 − 1 is the minimal size of the alphabet required to meet the bound for n 6. Finally, we show that the largest transition semigroups of minimal DFAs which recognize bifix-free languages are unique up to renaming the states.

Complexity of bifix-free regular languages

“…The syntactic complexity of bifix-free languages was shown to be (n − 1) n−3 + (n − 2) n−3 + (n − 3)2 n−3 for n ≥ 6 [24]. Moreover, the transition semigroup of a minimal DFA D n of a witness language meeting the bound must be W ≥6 bf (n), which is a transition semigroup containing three types of transformations and can be defined as follows:…”

“…Following [24], we say that an unordered pair {p, q} of distinct states from Q M is colliding in T (n) if there is a transformation t ∈ T (n) such that 0t = p and rt = q for some r ∈ Q M . A pair of states is focused by a transformation u ∈ T (n) if u maps both states of the pair to a single state r ∈ Q M ∪ {n − 2}.…”

“…To meet the bound for syntactic complexity an alphabet of size at least (n − 3) + ((n − 2) n−3 − 1) + (n − 3)(2 n−3 − 1) = (n − 2) n−3 + (n − 3)2 n−3 − 1 is required, and so a witness stream cannot have a smaller number of letters. Our stream contains the DFAs from [24,Definition 4], which have the transition semigroup W ≥6 bf (n). Definition 14 (Most complex stream, [24,Definition 4]).…”

“…In [5] the tight bound on the state complexity of basic operations on bifix-free languages were established; however, the witnesses were different for particular cases. The syntactic complexity complexity of bifix-free languages was first studied in [6], where a lower bound was established, and then the formula was shown to be an upper bound in [24].…”

Syntactic Complexity of Regular Ideals

“…In the proof for the upper bounds for left and two-sided ideals we use the method of injective function, which is generally applicable for other subclasses of regular languages (see [12] for suffix-free and [31] for bifix-free languages). The proofs presented here are the first that apply this method to syntactic complexity.…”