Verifying Termination and Reduction Properties about Higher-Order Logic Programs

Abstract

Abstract. We describe two checkers for verifying termination and reduction properties about higher-order logic programs. The reduction checker verifies that the result of a program execution is structurally smaller (or equal) than the inputs to the program. The termination checker guarantees that the inputs of the recursive calls are structurally smaller than the inputs of the original call taking into account reduction properties. At the heart of both checkers lies an inference system to reason about structural properties which are described by higher-order subterm relations. This approach provides a logical foundation for proving properties such as termination and reduction and factors the effort required for each one of them. Moreover, it allows the study of proof-theoretical properties, soundness and completeness and different optimizations. The termination and reduction checker are implemented as part of the Twelf system and have been used on a wide variety of examples, including proofs about typed assembly language and those in the area of proof-carrying code.

Beluga: A Framework for Programming and Reasoning with Deductive Systems (System Description)

“…We formulated an algorithm [1] to ensure that all cases are covered, and plan to implement it over the next few months. Verifying termination will follow ideas in Twelf [13,9] for checking that arguments in recursive calls are indeed smaller.…”

Reasoning with Higher-Order Abstract Syntax and Contexts: A Comparison

“…Intuitively, pattern matching on a contextual object of type A[Ψ ] is exhaustive if we cover all constructors of type A plus the cases described by parameter variables, which cover the possibility that we have used an assumption from the context Ψ . For termination checking, we believe the ideas from [Pie05] can be easily adapted.…”

Mechanized metatheory model-checking

“…Determining whether a logic program defines a recursive relation is an orthogonal issue, but see, e.g. the termination analysis approach taken in the Twelf system [38]. Nevertheless, we do not believe completeness is necessary for our approach to be useful, even for systems with undecidable predicates such as first-order sequent calculi or undecidable typing or subtyping relations.…”

“…Analyses for checking modes, coverage, termination, and other program properties can be used to verify program properties; this technique plays an important role in the Twelf system [37,38]. This approach is also possible (and seems likely to be helpful) in αProlog, but such analyses have not yet been adapted to the setting of nominal logic programming.…”