Winning Sets of Regular Languages: Descriptional and Computational Complexity

TL;DR

This paper explores the complexity of winning sets in regular language games, establishing their regularity, bounds on their state complexity, and analyzing the computational difficulty of related decision problems.

Contribution

It introduces a framework for analyzing word-construction games with variable turn orders, proving regularity of winning sets and providing bounds and complexity results.

Findings

Winning sets are regular languages when the target language is regular

Bounds on state complexity of winning sets are established

Membership and intersection problems are analyzed for computational complexity

Abstract

We investigate certain word-construction games with variable turn orders. In these games, Alice and Bob take turns on choosing consecutive letters of a word of fixed length, with Alice winning if the result lies in a predetermined target language. The turn orders that result in a win for Alice form a binary language that is regular whenever the target language is, and we prove some upper and lower bounds for its state complexity based on that of the target language. We also consider the computational complexity of membership and intersection problems of winning sets.