Generating Compressed Combinatory Proof Structures -- An Approach to Automated First-Order Theorem Proving
This work addresses automated theorem proving for logic and AI, presenting a novel method that is incremental as it builds on existing approaches like condensed detachment and the connection method.
The paper tackles the problem of automated first-order theorem proving by representing proof trees as combinator terms that reduce to trees, allowing duplication to appear as duplicated subterms that can be shared in a DAG representation. The result is an implementation with first experimental results, building on condensed detachment and the connection method to realize features from the unimplemented connection structure calculus.
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.