On the Structure and Complexity of Rational Sets of Regular Languages

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.

💡 Research Summary

This paper presents a foundational theoretical study on Rational Sets of Regular Languages (RSRLs), a concept central to the semantics of the FQL test specification language for ANSI C. An RSRL is defined as a regular language K over an alphabet Δ, coupled with a regular language substitution φ that maps each symbol in Δ to a regular language over a base alphabet Σ. The set represented, R = (K, φ), is {φ(w) | w ∈ K}, which is a set of regular languages. The research is motivated by the need for a formal and manipulable foundation for test coverage specifications in FQL, where test goals are expressed as regular languages over program control-flow edges.

The core investigation is twofold. First, the paper meticulously analyzes the closure properties of RSRLs under standard set-theoretic operations. The operations considered include product, Kleene star, complement, union, intersection, set difference, and symmetric difference, along with their point-wise and Cartesian variants. The analysis distinguishes between general RSRLs, finite RSRLs (which correspond to FQL’s practical use case), and finite RSRLs with a fixed substitution φ. Key results show that RSRLs are closed under union, product, and Kleene star in the general case, with union being expressible without altering φ—a practical advantage. A significant negative result is that the complement of an RSRL is never an RSRL, stemming from the fact that the set of all languages (2^Σ*) is uncountable while any RSRL is always countable. For intersection, difference, and symmetric difference, the closure property for general RSRLs is left as an open problem. However, the paper proves that all operations yield an RSRL when applied to finite RSRLs, which is highly relevant as FQL generates finite sets of test goals.

Second, the paper investigates the computational complexity of decision problems for Kleene-star-free RSRLs, with a primary focus on the membership problem: given a regular language L and an RSRL R, is L ∈ R? The authors analyze the complexity of this problem and provide an algorithm for checking membership, discussing its associated costs. This analysis is crucial for understanding the performance characteristics of the test case generation engine built upon FQL.

In conclusion, the paper establishes a systematic semantic foundation for FQL by exploring the algebraic structure and computational aspects of RSRLs. The findings on closure properties inform query optimization and manipulation within the FQL framework, while the complexity analysis sheds light on the inherent costs of core operations. By bridging formal language theory and practical software testing needs, this work provides essential insights for both the theoretical understanding of RSRLs and the efficient implementation of specification-based testing tools. Several properties are noted as open problems, inviting further research in this area.