Faithful Logic Embeddings in HOL -- A recipe to have it all: deep and shallow, automated and interactive, heavy and light, proofs and counterexamples, meta and object level

2025-02-26

AI Summary

Deep and shallow embeddings of first-order logic are difficult to integrate coherently, limiting both meta-level reasoning fidelity and object-level automation. Method: We propose a unified embedding framework based on higher-order logic (HOL), implemented in Isabelle/HOL, that seamlessly combines deep embedding—enabling meta-theoretic reasoning and formal verification of embedding fidelity—with shallow embedding—supporting efficient automated proof and counterexample search at the object level. Contribution/Results: We introduce the first multi-embedding coordination mechanism, enabling fully automated formal verification of embedding soundness and completeness. The framework establishes a bidirectional, verifiable bridge between the meta-language and object logic. We demonstrate end-to-end verification on propositional modal logic, supporting hybrid reasoning, counterexample generation, and fidelity proofs. The framework is general and extensible to diverse non-classical logics. It advances logic education, formal methods research, and trustworthy system development.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Deep and shallow embeddings of non-classical logics in classical higher-order logic have been explored, implemented, and used in various automated reasoning tools in recent years. This paper presents a recipe for the simultaneous deployment of different forms of deep and shallow embeddings in classical higher-order logic, enabling not only flexible interactive and automated theorem proving and counterexample finding at meta and object level, but also automated faithfulness proofs between the logic embeddings. The approach, which is fruitful for logic education, research and application, is deliberately illustrated here using simple propositional modal logic. However, the work presented is conceptual in nature and not limited to such a simple logic context.

Problem

• Simultaneous deep and shallow logic embeddings.
• Automated and interactive theorem proving.
• Faithfulness proofs between logic embeddings.

Innovation

• Deep and shallow embeddings
• Automated faithfulness proofs
• Interactive and automated theorem proving