Results 11  20
of
118
A Resolution Theorem Prover for Intuitionistic Logic
 Proceedings of CADE13
, 1996
"... We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is show ..."
Abstract

Cited by 44 (4 self)
 Add to MetaCart
(Show Context)
We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is shown to be a decision procedure for a new syntactically described decidable class of intuitionistic logic. We compare the search strategies suitable for the resolution method with strategies suitable for the tableau method. The performance of our prover is compared with the performance of a tableau prover for intuitionistic logic presented in [17].
The TPS theorem proving system
 9th International Conference on Automated Deduction, Argonne, Illinois
, 1988
"... TPS is a theorem proving system for first and higherorder logic which runs in Common Lisp and can operate in automatic, semiautomatic, and interactive modes. As its logical language TPS uses the typed Acalculus [6], in which most theorems of mathematics can be expressed very directly. TPS can be ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
TPS is a theorem proving system for first and higherorder logic which runs in Common Lisp and can operate in automatic, semiautomatic, and interactive modes. As its logical language TPS uses the typed Acalculus [6], in which most theorems of mathematics can be expressed very directly. TPS can be used to search for an expansion proof [10, 11] of a theorem, which represents in a nonredtmdant way the basic combinatorial information required to construct a proof of
A FineGrained Notation for Lambda Terms and Its Use in Intensional Operations
 Journal of Functional and Logic Programming
, 1996
"... We discuss issues relevant to the practical use of a previously proposed notation for lambda terms in contexts where the intensions of such terms have to be manipulated. This notation uses the `nameless' scheme of de Bruijn, includes expressions for encoding terms together with substitutions to ..."
Abstract

Cited by 25 (9 self)
 Add to MetaCart
We discuss issues relevant to the practical use of a previously proposed notation for lambda terms in contexts where the intensions of such terms have to be manipulated. This notation uses the `nameless' scheme of de Bruijn, includes expressions for encoding terms together with substitutions to be performed on them and contains a mechanism for combining such substitutions so that they can be effected in a common structure traversal. The combination mechanism is a general one and consequently difficult to implement. We propose a simplification to it that retains its functionality in situations that occur commonly in fireduction. We then describe a system for annotating terms to determine if they can be affected by substitutions generated by external ficontractions. These annotations can lead to a conservation of space and time in implementations of reduction by permitting substitutions to be performed trivially in certain situations. The use of the resulting notation in the reduction...
Some Uses of HigherOrder Logic in Computational Linguistics
 In 24st Annual Meeting of the Association for Computational Linguistics
, 1986
"... Consideration of the question of meaning in the framework of linguistics often requires an allusion to sets and other higherorder notions. The traditional approach to representing and reasoning about meaning in a computational setting has been to use knowledge representation systems that are either ..."
Abstract

Cited by 23 (10 self)
 Add to MetaCart
(Show Context)
Consideration of the question of meaning in the framework of linguistics often requires an allusion to sets and other higherorder notions. The traditional approach to representing and reasoning about meaning in a computational setting has been to use knowledge representation systems that are either based on firstorder logic or that use mechanisms whose formal justifications are to be provided after the fact. In this paper we shall consider the use of a higherorder logic for this task. We first present a version of definite clauses (positive Horn clauses) that is based on this logic. Predicate and function variables may occur in such clauses and the terms in the language are the typed terms. Such term structures have a richness that may be exploited in representing meanings. We also describe a higherorder logic programming language, called Prolog, which represents programs as higherorder definite clauses and interprets them using a depthfirst interpreter. A virtue of this languag...
An extension of dependency pair method for proving termination of higherorder rewrite systems
 IEICE Trans. on Information and Systems
, 2001
"... Abstract. This paper explores how to extend the dependency pair technique for proving termination of higherorder rewrite systems. In the first order case, the termination of term rewriting systems are proved by showing the nonexistence of an infinite Rchain of the dependency pairs. However, the t ..."
Abstract

Cited by 22 (2 self)
 Add to MetaCart
(Show Context)
Abstract. This paper explores how to extend the dependency pair technique for proving termination of higherorder rewrite systems. In the first order case, the termination of term rewriting systems are proved by showing the nonexistence of an infinite Rchain of the dependency pairs. However, the termination and the nonexistence of an infinite Rchain do not coincide in the higherorder case. We introduce a new notion of dependency forest that characterize infinite reductions and infinite Rchains, and show that the termination property of higherorder rewrite systems R can be checked by showing the nonexistence of an infinite Rchain, if R is strongly linear or nonnested. 1
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
Abstract

Cited by 19 (0 self)
 Add to MetaCart
Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
HigherOrder Semantics and Extensionality
 Journal of Symbolic Logic
, 2004
"... Abstract. In this paper we reexamine the semantics of classical higherorder logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higherorder models with respect to various combinations of Boolean extensionality and three forms of func ..."
Abstract

Cited by 18 (10 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we reexamine the semantics of classical higherorder logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higherorder models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of (machineoriented) higherorder calculi with respect to these model classes. §1. Motivation. In classical firstorder predicate logic, it is rather simple to assess the deductive power of a calculus: firstorder logic has a wellestablished and intuitive settheoretic semantics, relative to which completeness can easily be verified using, for instance, the abstract consistency method (cf. the introductory textbooks [6, 22]). This well understood metatheory has supported the development of calculi adapted to special applications—such as automated theorem proving (cf. [16, 47] for an overview). In higherorder logics, the situation is rather different: the intuitive settheoretic standard semantics cannot give a sensible notion of completeness, since it does
TPS: A TheoremProving System for Classical Type Theory
, 1996
"... . This is description of TPS, a theoremproving 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 higherorder logic, and searching for proofs of such wffs interactively or automatically, or in a ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
. This is description of TPS, a theoremproving 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 higherorder 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 higherorder logic. AMS Subject Classification: 0304, 68T15, 03B35, 03B15, 03B10. Key words: higherorder logic, type theory, mating, connection, expansion proof, natural deduction. 1. Introduction TPS is a theoremproving 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...