Generating Compressed Combinatory Proof Structures -- An Approach to Automated First-Order Theorem Proving

TL;DR

This paper introduces a novel approach to automated first-order theorem proving by representing proof trees as combinator terms, enabling efficient proof search and sharing of subproofs, with initial implementation and experimental results.

Contribution

It presents a new combinator-based proof structure method for first-order theorem proving, integrating features from connection calculus and implementing proof sharing.

Findings

Proof sharing via DAG representation improves efficiency.

Enumeration of combinator terms enables systematic proof search.

Initial experiments show promising results in proof discovery.

Abstract

Representing a proof tree by a combinator term that reduces to the tree lets subtle forms of duplication within the tree materialize as duplicated subterms of the combinator term. In a DAG representation of the combinator term these straightforwardly factor into shared subgraphs. To search for proofs, combinator terms can be enumerated, like clausal tableaux, interwoven with unification of formulas that are associated with nodes of the enumerated structures. To restrict the search space, the enumeration can be based on proof schemas defined as parameterized combinator terms. We introduce here this "combinator term as proof structure" approach to automated first-order proving, present an implementation and first experimental results. The approach builds on a term view of proof structures rooted in condensed detachment and the connection method. It realizes features known from the connection structure calculus, which has not been implemented so far.