Complexity of Suffix-Free Regular Languages

A sequence \((L_k,L_{k+1} \dots )\) of regular languages in some class \({\mathcal C}\), where n is the state complexity of \(L_n\), is called a stream. A stream is most complex in class \({\mathcal C}\) if its languages together with their dialects (that is, languages that differ only very slightly from the languages in the stream) meet the state complexity bounds for boolean operations, product (concatenation), star, and reversal, have the largest syntactic semigroups, and have the maximal numbers of atoms, each of which has maximal state complexity. It is known that there exist such most complex streams in the class of regular languages, and also in the classes of right, left, and two-sided ideals. In contrast to this, we prove that there does not exist a most complex stream in the class of suffix-free regular languages. However, we do exhibit one ternary suffix-free stream that meets the bound for product and whose restrictions to binary alphabets meet the bounds for star and boolean operations. We also exhibit a quinary stream that meets the bounds for boolean operations, reversal, size of syntactic semigroup, and atom complexities. Moreover, we solve an open problem about the bound for the product of two languages of state complexities m and n in the binary case by showing that it can be met for infinitely many m and n.

Two transition semigroups play an important role for suffix-free languages: semigroup \(\mathbf {T}^{\leqslant 5}(n)\) is the largest suffix-free semigroup for \(n\leqslant 5\), while semigroup \(\mathbf {T}^{\geqslant 6}(n)\) is largest for \(n=2,3\) and \(n\geqslant 6\). We prove that all witnesses meeting the bounds for the star and the second witness in a product must have transition semigroups in \(\mathbf {T}^{\leqslant 5}(n)\). On the other hand, witnesses meeting the bounds for reversal, size of syntactic semigroup and the complexity of atoms must have semigroups in \(\mathbf {T}^{\geqslant 6}(n)\).