This is a description of TPS, a theorem proving system for classical type theory (Church’s 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 which TPS can prove completely automatically are given to illustrate certain aspects of TPS’s behavior and problems of theorem proving in higher-order logic. 7

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Contents i Introduction ........................................ 613 1.1 Some useful notation ................................ 614 1.2 Equality. The first axiomatization ......................... 615 1.3 Paramodulation and its refinements ........................ 616 1.4 Equality in sequent systems ............................. 622 1.5 Overview of calculi introduced in this chapter ................... 628 2 Translation of logic with equality into logic without equality ............. 628 2.1 Modification method ................................. 630 2.2 Constrained equality elimination .......................... 633 3 Free variable systems ................................... 637 3.1 Free variable tableaux ................................ 637 3.2 Matings ........................................ 640 3.3 The inverse method ................................. 641 4 Early history ........................................ 644 5 Simultaneous rigid E-unification ............................. 646 5

We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cut-elimination procedure. This gives us a “Boolean ” category, which is not a poset. With the semiring of natural numbers, we obtain a sound semantics for classical logic, in which fewer proofs are identified. Though a “real” sequentialization theorem is missing, these proof nets have a grip on complexity issues. In both cases the cut elimination procedure is closely related to its equivalent in the calculus of structures.

. This is description of TPS, a theorem-proving system for classical type theory (Church's typed #-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 TPS's behavior and problems of theorem proving in higher-order logic. AMS Subject Classification: 03-04, 68T15, 03B35, 03B15, 03B10. Key words: higher-order logic, type theory, mating, connection, expansion proof, natural deduction. 1. Introduction TPS is a theorem-proving system for classical type theory ## (Church's typed #-calculus [20]) which has been under development at Carnegie Mellon University for a number years. This paper gives a general...

In its most general meaning, a Boolean category is to categories what a Boolean algebra is to posets. In a more specific meaning a Boolean category should provide the abstract algebraic structure underlying the proofs in Boolean Logic, in the same sense as a Cartesian closed category captures the proofs in intuitionistic logic and a *-autonomous category captures the proofs in linear logic. However, recent work has shown that there is no canonical axiomatisation of a Boolean category. In this work, we will see a series (with increasing strength) of possible such axiomatisations, all based on the notion of *-autonomous category. We will particularly focus on the medial map, which has its origin in an inference rule in KS, a cut-free deductive system for Boolean logic in the calculus of structures. Finally, we will present a category proof nets as a particularly well-behaved example of a Boolean category.

A new methodology/data structure, the clause tree, is developed for automated reasoning based on resolution in first order logic. A clause tree T on a set S of clauses is a 4-tuple <N,E,L,M>, where N is a set of nodes, divided into clause nodes and atom nodes, E is a set of edges, each of which joins a clause node to an atom node, L is a labeling of N E which assigns to each clause node a clause of S, to each atom node an instance of an atom of some clause of S, and to each edge either + or -. The edge joining a clause node to an atom node is labeled by the sign of the corresponding literal in the clause. A resolution is represented by unifying two atom nodes of different clause trees which represent complementary literals. The merge of two identical literals is represented by placing the path joining the two corresponding atom nodes into the set M of chosen merge paths. The tail of the merge path becomes a closed leaf, while the head remains an open leaf which can be resolved on. Th...

A new way of handling equality in the connection method is described. Our method is called equality elimination. It does not require rigid E-unification which is the basis of most known extension procedures. Equality elimination uses efficient ordering strategies via basic superposition and simplification. In addition, we present a new treatment of the connection method via named matrices which gives a number of advantages. Section 1 Introduction We propose an approach to adding equality to extension proof methods which are known under two different names connection method [BS 75, Bib 87] and the method of matings [And 76, And 81, Gal 90, Gal 92]. Both methods express the same idea going back to the fundamental Herbrand theorem [Her 30, Min 66, Min 67a]. According to these approaches, the proof-search can be considered as a the problem of verifying that each path through a matrix of the goal formula is complementary (in the non-equality case it means that each path contains a litera...

The first part of this paper introduces an extension for a variant of Huet's higher-order resolution calculus [Hue72, Hue73] based upon classical type theory (Church's typed -calculus [Chu40]) in order to obtain a calculus which is complete with respect to Henkin models [Hen50]. The new rules connect higher-order pre-unification with the general refutation process in an appropriate way to establish full extensionality for the whole system. The general idea of the calculus is discussed on different examples. The second part introduces the Leo system which implements the discussed extensional higher-order resolution calculus. This part mainly focus on the embedding of the new extensionality rules into the refutation process and the treatment of higher-order unification. 1 Introduction Many mathematical problems can be expressed shortly and elegantly in higher order logic whereas they often lead to unnatural and inflated formulations in first-order logic, e.g., when coding them into axio...

Proof procedures are an essential part of logic applied to artificial intelligence tasks, and form the basis for logic programming languages. As such, many of the chapters throughout this handbook utilize, or study, proof procedures. The study of proof procedures that are useful in artificial intelligence would require a large book so we focus on proof procedures that relate to logic programming. We begin with the resolution procedures that influenced the definition of SLD-resolution, the procedure upon which Prolog is built. Starting with the general resolution procedure we move through linear resolution to a very restricted linear resolution, SLresolution, which actually is not a resolution restriction, but a variant using an augmented logical form. (SL-resolution actually is a derivative of the Model Elimination procedure, which was developed independently of resolution.) We then consider logic programming itself, reviewing SLD-resolution and then describing a general criterion for ...