Natural Deduction as Higher-Order Resolution

TL;DR

This paper explores representing logical inference rules in Isabelle using higher-order resolution, enabling more flexible proof strategies and potential applications in logic programming.

Contribution

It introduces a novel approach of modeling inference rules as Horn clauses in Isabelle, integrating higher-order resolution with theorem proving.

Findings

Resolution supports a broad class of logics in Isabelle.

Higher-order unification is essential for representing logical syntax.

Potential for logic programming with depth-first subgoaling in higher-order resolution.

Abstract

An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. Resolution gives both forwards and backwards proof, supporting a large class of logics. Isabelle has been used to prove theorems in Martin-L\"of's Constructive Type Theory. Quantifiers pose several difficulties: substitution, bound variables, Skolemization. Isabelle's representation of logical syntax is the typed lambda-calculus, requiring higher- order unification. It may have potential for logic programming. Depth-first subgoaling along inference rules constitutes a higher-order Prolog.