Descriptional Complexity of the Languages KaL: Automata, Monoids and Varieties

TL;DR

This paper investigates the descriptional complexity of languages formed by concatenation with languages from a variety V, providing optimal bounds for automata state complexity, syntactic monoids, and related algebraic structures.

Contribution

It introduces new bounds and estimates for the complexity measures of KaL languages, linking them to the properties of the constituent languages K and L within a variety V.

Findings

Optimal bounds for DFA state complexity of KaL

Estimates for the size of syntactic monoids of KaL

Bounds for the cardinality of images in Schützenberger products

Abstract

The first step when forming the polynomial hierarchies of languages is to consider languages of the form KaL where K and L are over a finite alphabet A and from a given variety V of languages, a being a letter from A. All such KaL's generate the variety of languages BPol1(V). We estimate the numerical parameters of the language KaL in terms of their values for K and L. These parameters include the state complexity of the minimal complete DFA and the size of the syntactic monoids. We also estimate the cardinality of the image of A* in the Schuetzenberger product of the syntactic monoids of K and L. In these three cases we obtain the optimal bounds. Finally, we also consider estimates for the cardinalities of free monoids in the variety of monoids corresponding to BPol1(V) in terms of sizes of the free monoids in the variety of monoids corresponding to V.