Peter B. Andrews

Theorem Proving via General Matings

An approach to automaUc theorem proving using matmgs of arbitrary sentences is discussed No use is made of conjunctive normal form (clauses) or prenex normal form, since these forms tend to introduce superfluous redundancy, complicate the search for a proof, and impede analysis of the essential logical structure of the proposed theorem. A complete exposition of the logical foundations of theorem proving via general matmgs is given, starting with proofs of appropriate versions of Herbrands Theorem. It is shown that one may restrict quantifier duphcat,on to outermost quanUfiers without loss of completeness, though with possible loss of efficmncy. General matmgs could be used as the basis for a variety of theorem-proving procedures, and there are many opportunmes for research m this area. A procedure using the criterion of path acceptability for mattngs is discussed. This criterion ~s easily VlSUahzed m terms of a two-dimensional format for formulas. An implementation by Eve Cohen has yielded encouraging preliminary results. Some implementation issues are discussed.

TPS: A theorem-proving system for classical type theory

This is description of TPS, a theorem-proving system for classical type theory (Churchs typed λ-calculus). TPS has been designed to be a general research tool for manipulating wffs of first- and higher-order logic, and searching for proofs of such wffs interactively or automatically, or in a combination of these modes. An important feature of TPS is the ability to translate between expansion proofs and natural deduction proofs. Examples of theorems that TPS can prove completely automatically are given to illustrate certain aspects of TPSs behavior and problems of theorem proving in higher-order logic.

Transforming Matings into Natural Deduction Proofs

A procedure is given for transforming refutation matings into natural deduction proofs. Thus a theorem proving system which establishes the validity of a theorem by the general matings approach can apply this procedure to obtain a comprehensible proof of the theorem without further search. This illuminates the close relationship between matings and proofs, and serves as a step toward a synthesis between apparently quite different approaches to automated theorem proving.

Resolution With Merging

A refinement of the resolution method for mechanical theorem proving is presented. A resolvent <italic>C</italic> of clauses <italic>A</italic> and <italic>B</italic> is called a <italic>merge</italic> if literals from <italic>A</italic> and <italic>B</italic> merge together to form some literal of <italic>C</italic>. It is shown that the resolution method remains complete if it is required that two noninitial clauses which are not merges never be resolved with one another. It is also shown that this strategy can be combined with the set-of-support strategy.

On connections and higher-order logic

This is an expository introduction to an approach to theorem proving in higher-order logic based on establishing appropriate connections between subformulas of an expanded form of the theorem to be proved. Expansion trees and expansion proofs play key roles.

TPS: A hybrid automatic-interactive system for developing proofs☆

Abstract The theorem proving system Tps provides support for constructing proofs using a mix of automation and user interaction, and for manipulating and inspecting proofs. Its library facilities allow the user to store and organize work. Mathematical theorems can be expressed very naturally in Tps using higher-order logic. A number of proof representations are available in Tps , so proofs can be inspected from various perspectives.

The TPS Theorem Proving System

When one is seeking an expansion proof for a theorem of higher-order logic, not all necessary substitution terms can be generated by unification of formulas already present, so certain expansion options [5] are applied, and then a search for a p-acceptable mating [2] is made, using Huets higher-order unification algorithm [8] to generate all remaining substitution terms. The expansion options consist of quantifier duplications and projective and primitive substitutions (such as

System Description: TPS: A Theorem Proving System for Type Theory

This is a brief update on the Tps automated theorem proving system for classical type theory, which was described in [3]. Manuals and information about obtaining Tps can be found at http://gtps.math.cmu.edu/tps.html.

Selectively Instantiating Definitions

When searching for proofs of theorems which contain definitions, it is a significant problem to decide which instances of the definitions to instantiate. We describe a method called dual instantiation, which is a partial solution to the problem in the context of the connection method; the same solution may also be adaptable to other search procedures. Dual instantiation has been implemented in TPS, a theorem prover for classical type theory, and we provide some examples of theorems that have been proven using this method. Dual instantiation has the desirable properties that the search for a proof cannot possibly fail due to insufficient instantiation of definitions, and that the natural deduction proof which results from a successful search will contain no unnecessary instantiations of definitions. Furthermore, the time taken by a proof search using dual instantiation is in general comparable to the time taken by a search in which exactly the required instances of each definition have been instantiated. We also describe how this technique can be applied to the problem of instantiating set variables.

A Look at TPS

Certain aspects of the theorem proving system TPS are described. Type theory with λ-abstraction has been chosen as the logical language of TPS so that statements from many fields of mathematics and other disciplines can be expressed in terms accessible to the system.