Enhancing Unsatisfiable Cores for LTL with Information on Temporal Relevance

TL;DR

This paper introduces a method to enhance unsatisfiable cores of LTL formulas with temporal relevance information, aiding debugging and understanding of unsatisfiability in formal specifications.

Contribution

It proposes a novel approach to extract temporal relevance data from resolution proofs to improve unsatisfiable core explanations in LTL.

Findings

Enhanced unsatisfiable cores with temporal relevance aid debugging.

Method successfully implemented in TRP++ and evaluated experimentally.

Source code is publicly available for further use.

Abstract

LTL is frequently used to express specifications in many domains such as embedded systems or business processes. Witnesses can help to understand why an LTL specification is satisfiable, and a number of approaches exist to make understanding a witness easier. In the case of unsatisfiable specifications unsatisfiable cores (UCs), i.e., parts of an unsatisfiable formula that are themselves unsatisfiable, are a well established means for debugging. However, little work has been done to help understanding a UC of an unsatisfiable LTL formula. In this paper we suggest to enhance a UC of an unsatisfiable LTL formula with additional information about the time points at which the subformulas of the UC are relevant for unsatisfiability. For example, in "(G p) and (X not p)" the first occurrence of "p" is really only "relevant" for unsatisfiability at time point 1 (time starts at time point 0). We present a method to extract such information from the resolution graph of a temporal resolution proof of unsatisfiability of an LTL formula. We implement our method in TRP++, and we experimentally evaluate it. Source code of our tool is available.