Total Outcome Logic: Proving Termination and Nontermination in Programs with Branching

2024-10-31

arXiv.org

AI Summary

Termination analysis of programs has long relied on disparate, specialized logics, making it difficult to uniformly characterize termination, non-termination, and partial correctness—especially for nondeterministic and probabilistic programs. To address this, we propose the first unified program logic framework that simultaneously models and reasons about total correctness, partial correctness, and non-termination within a single formalism. Our logic extends Hoare logic by integrating incorrectness logic and semantics for nondeterminism and probability, and establishes a rigorous metatheory—including soundness and relative completeness. We validate its expressiveness and practicality through multiple case studies, demonstrating support for compositional verification across programming paradigms. The framework significantly enhances the generality, unification, and engineering applicability of termination analysis.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Probabilistic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance

Abstract

While there is a long tradition of reasoning about termination (and nontermination) in the context of program analysis, specialized logics are typically needed to give different termination guarantees. This includes partial correctness, where termination is not guaranteed, and total correctness, where it is guaranteed. We present Total Outcome Logic, a single logic which can express the full spectrum of termination conditions and program properties offered by the aforementioned logics. Total Outcome Logic extends termination and incorrectness reasoning across different kinds of branching effects, so that a single metatheory powers this reasoning in different kinds of programs, including nondeterministic and probabilistic. We demonstrate the utility of this approach through a variety of case studies.

Problem

• Unify reasoning for diverse program termination criteria
• Extend logic to handle nondeterministic and probabilistic programs
• Subsume multiple taxonomies of correctness logics

Innovation

• Unified logic for diverse termination conditions
• Extends reasoning across branching effects
• Subsumes multiple correctness logics