Complexity in Prefix-Free Regular Languages

Abstract

We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next provide the tight bounds on state complexity of symmetric difference, and deterministic and nondeterministic state complexity of difference and cyclic shift of prefix-free languages

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

“…The fourth claim is from [22]. The results for union, symmetric difference and intersection were proved in [22], and that for difference in [23].…”

“…Thus, the mn−2 remaining states are pairwise distinguishable. Using some results from [22,23] we define another prefix-free witness stream that meets all the bounds except those for syntactic complexity and atom complexity. Moreover, all the bounds are met by dialects over minimal alphabets.…”

“…(A ternary witness was also used in [22] for the reverse, but it had more complicated transitions than our witness.) The fourth claim is from [22]. The results for union, symmetric difference and intersection were proved in [22], and that for difference in [23].…”

“…The second and third claims were proved in Theorem 4. (A ternary witness was also used in [22] for the reverse, but it had more complicated transitions than our witness.) The fourth claim is from [22].…”

“…The complexity of operations on prefix-free languages was studied in [19,22,23], but most complex prefix-free languages were not considered. As every prefix-free language has an empty quotient, the restricted and unrestricted complexities are the same for binary operations.…”