[Paper Review] Embedding and Automating Conditional Logics in Classical Higher-Order Logic

Christoph Benzmueller, Dov M. Gabbay+2 more|arXiv (Cornell University)|Jun 18, 2011

Logic, programming, and type systems21 references4 citations

TL;DR

This paper presents a sound and complete embedding of conditional logics into classical higher-order logic (HOL), enabling the use of off-the-shelf higher-order theorem provers and model finders for automated reasoning within and about conditional logics. The approach leverages selection function semantics and demonstrates practical effectiveness through experiments with HOL-based reasoning systems, achieving automated verification of correspondence results between axioms and semantic conditions.

ABSTRACT

A sound and complete embedding of conditional logics into classical higher-order logic is presented. This embedding enables the application of off-the-shelf higher-order automated theorem provers and model finders for reasoning within and about conditional logics.

Motivation & Objective

• To provide a uniform, formal framework for specifying and reasoning about conditional logics, which previously lacked a coherent proof-theoretic foundation.
• To overcome the limitations of existing proof methods that are restricted to specific, limited axiomatizations of conditional logics.
• To enable automated reasoning within and about conditional logics using general-purpose higher-order reasoning systems.
• To establish a practical and scalable methodology for theorem proving and model generation in conditional logics, including extensions beyond prior work.
• To demonstrate the feasibility of using HOL-based tools for meta-theoretic analysis, such as verifying correspondence results between axioms and semantic conditions.

Proposed method

• The paper formalizes conditional logics using selection function semantics, where a selection function maps each world and formula to a set of 'most similar' worlds.
• It embeds conditional logic formulas into classical higher-order logic (HOL) by encoding the selection function semantics as higher-order functions over sets of states.
• The embedding is proven sound and complete with respect to selection function semantics, ensuring that validity and entailment are preserved.
• The approach enables the use of existing off-the-shelf higher-order theorem provers (e.g., LEO-II, Satallax) and model finders for reasoning within conditional logics.
• It supports automated verification of correspondence results between conditional logic axioms (e.g., ID, MP, CEM, CS) and their semantic conditions via HOL reasoning systems.
• The method is evaluated through experiments on benchmark problems, including the automatic generation of countermodels for ill-formed conjectures.

Experimental results

Research questions

• RQ1 Can conditional logics be soundly and completely embedded into classical higher-order logic using selection function semantics?
• RQ2 To what extent can off-the-shelf higher-order theorem provers and model finders be used to automate reasoning in conditional logics?
• RQ3 Can the embedding support automated verification of correspondence results between axioms and semantic conditions in conditional logics?
• RQ4 How does the performance of HOL-based reasoning compare to direct provers for conditional logics or first-order encodings?
• RQ5 Can the embedding be extended to reason about meta-theoretical properties such as inclusion and equivalence between different conditional logic systems?

Key findings

• The embedding of conditional logics into HOL is both sound and complete with respect to selection function semantics, ensuring logical correctness.
• The approach enables automated reasoning within conditional logics using standard higher-order reasoning systems, including theorem proving and model generation.
• Several correspondence results between prominent conditional logic axioms (e.g., CEM, CS, ID) and their semantic conditions were automatically verified using HOL provers.
• The method successfully generated countermodels for ill-formulated or incorrect correspondence claims, demonstrating its utility in debugging and meta-theoretic analysis.
• The approach outperformed first-order encodings and direct provers in certain benchmark cases, particularly in meta-theoretic reasoning tasks.
• The framework is extensible and supports reasoning about first-order conditional logics and systematic analysis of inclusion and equivalence relations between logic systems.