On the Structure and Complexity of Rational Sets of Regular Languages

TL;DR

This paper provides a theoretical analysis of rational sets of regular languages (RSRLs), focusing on their closure properties and the complexity of membership checking, thereby underpinning the semantics of the FQL test specification language.

Contribution

It offers a systematic semantic foundation for FQL by analyzing RSRLs' closure properties and membership complexity, both in general and finite cases.

Findings

RSRLs have specific closure properties under set operations.

Membership checking for RSRLs varies in complexity.

Some open problems remain in the theoretical analysis.

Abstract

In a recent thread of papers, we have introduced FQL, a precise specification language for test coverage, and developed the test case generation engine FShell for ANSI C. In essence, an FQL test specification amounts to a set of regular languages, each of which has to be matched by at least one test execution. To describe such sets of regular languages, the FQL semantics uses an automata-theoretic concept known as rational sets of regular languages (RSRLs). RSRLs are automata whose alphabet consists of regular expressions. Thus, the language accepted by the automaton is a set of regular expressions. In this paper, we study RSRLs from a theoretic point of view. More specifically, we analyze RSRL closure properties under common set theoretic operations, and the complexity of membership checking, i.e., whether a regular language is an element of a RSRL. For all questions we investigate both the general case and the case of finite sets of regular languages. Although a few properties are left as open problems, the paper provides a systematic semantic foundation for the test specification language FQL.