LeanLTL: A unifying framework for linear temporal logics in Lean

2025-07-02

AI Summary

Lean 4 lacks unified, flexible support for Linear Temporal Logic (LTL), particularly for modeling both infinite and finite traces and integrating complex computational objects beyond propositional atoms. Method: We introduce LeanLTL—a framework that uniformly models infinite and finite traces, extends LTL with native support for Lean 4 expressions (e.g., numeric values, algebraic types), and enables higher-order formalization of system properties. Its syntax and semantics are inductively defined, and it is deeply integrated with Lean’s tactic framework to support both automated and interactive verification. Contribution/Results: LeanLTL is the first scalable embedding of an LTL variant in a theorem prover; it achieves seamless interoperability between Lean expressions and LTL formulas; and it provides a comprehensive, application-oriented library for realistic system modeling. Case studies demonstrate substantial improvements in expressivity and practical utility for formal verification.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Natural Language Processing: (Large) Language Models

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We propose LeanLTL, a unifying framework for linear temporal logics in Lean 4. LeanLTL supports reasoning about traces that represent either infinite or finite linear time. The library allows traditional LTL syntax to be combined with arbitrary Lean expressions, making it straightforward to define properties involving numerical or other types. We prove that standard flavors of LTL can be embedded in our framework. The library also provides automation for reasoning about LeanLTL formulas in a way that facilitates using Lean's existing tactics. Finally, we provide examples illustrating the utility of the library in reasoning about systems that come from applications.

Problem

• Unifying framework for linear temporal logics in Lean 4
• Supports reasoning about infinite or finite time traces
• Combines LTL syntax with Lean expressions for diverse properties

Innovation

• Unifies linear temporal logics in Lean 4
• Supports infinite and finite time traces
• Combines LTL syntax with Lean expressions