Extracting Unsatisfiable Cores for LTL via Temporal Resolution

TL;DR

This paper introduces a novel method for extracting more precise unsatisfiable cores in Linear Temporal Logic (LTL) using optimized temporal resolution graphs, improving debugging and analysis of LTL formulas.

Contribution

The authors develop and implement a new approach for extracting fine-grained unsatisfiable cores in LTL by leveraging optimized resolution graphs, surpassing existing tools in precision.

Findings

Extracted UCs are often significantly smaller than original formulas.

The approach produces more fine-grained UCs than existing tools.

The method maintains acceptable runtime and memory usage.

Abstract

Unsatisfiable cores (UCs) are a well established means for debugging in a declarative setting. Still, there are few tools that perform automated extraction of UCs for LTL. Existing tools compute a UC as an unsatisfiable subset of the set of top-level conjuncts of an LTL formula. Using resolution graphs to extract UCs is common in other domains such as SAT. In this article we construct and optimize resolution graphs for temporal resolution as implemented in the temporal resolution-based solver TRP++, and we use them to extract UCs for propositional LTL. The resulting UCs are more fine-grained than the UCs obtained from existing tools because UC extraction also simplifies top-level conjuncts instead of treating them as atomic entities. For example, given an unsatisfiable LTL formula of the form $\phi \equiv ({\bf G} \psi) \wedge {\bf F} \psi'$ existing tools return $\phi$ as a UC irrespective of the complexity of $\psi$ and $\psi'$, whereas the approach presented in this article continues to remove parts not required for unsatisfiability inside $\psi$ and $\psi'$. Our approach also identifies groups of occurrences of a proposition that do not interact in a proof of unsatisfiability. We implement our approach in TRP++. Our experimental evaluation demonstrates that our approach (i) extracts UCs that are often significantly smaller than the input formula with an acceptable overhead and (ii) produces more fine-grained UCs than competing tools while remaining at least competitive in terms of run time and memory usage. The source code of our tool is publicly available.