Equational and Inductive Reasoning for Maude in Athena

2026-04-21

AI Summary

This work addresses the limited native support for inductive and deductive reasoning in Maude’s equational reduction, which hinders its applicability in formal verification. To bridge this gap, the paper introduces the maude2athena framework, which provides the first parameterized translation of many-sorted membership equational logic—augmented with structural axioms such as associativity and commutativity—into the Athena theorem-proving language. This translation preserves semantic fidelity, avoids exponential blowup, and yields compact proof obligations amenable to natural-deduction–based verification. The resulting framework supports induction, equational chaining, case analysis, and proof by contradiction, thereby effectively narrowing the divide between model checking and interactive theorem proving.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Data transparency and provenance

Abstract

In the rewriting logic framework, equational-based specifications are used to define deterministic functional behavior, abstract data types, and canonical representations of data. These specifications include a (possibly order-sorted) signature and equations interpreted modulo structural axioms, such as associativity, commutativity, and identity. While equational rewriting provides a powerful basis for execution and symbolic reasoning, it does not by itself offer native support for inductive or deductive reasoning. This paper presents maude2athena, a framework that systematically translates Maude's equational theories into Athena, a theorem proving language designed to support natural deduction proofs over many-sorted first-order logic specifications, including inductive reasoning, equational chaining, case-based reasoning, and proofs by contradiction. The translation supports induction-based reasoning modulo structural axioms with parametric induction rules; it faithfully encodes membership equational logic in a many-sorted setting without exponential blowup under reasonable conditions. This approach preserves the semantics of the original specification, while ensuring that the translation remains compact and amenable to deductive reasoning. This work helps bridge the gap between model checking and theorem proving, enabling formal verification efforts that can benefit from both of these approaches.

Problem

• equational reasoning
• inductive reasoning
• theorem proving
• formal verification
• rewriting logic

Innovation

• equational reasoning
• inductive reasoning
• Maude
• Athena
• membership equational logic