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Higherorder logic programming
 HANDBOOK OF LOGIC IN AI AND LOGIC PROGRAMMING, VOLUME 5: LOGIC PROGRAMMING. OXFORD (1998
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Unification under a mixed prefix
 Journal of Symbolic Computation
, 1992
"... Unification problems are identified with conjunctions of equations between simply typed λterms where free variables in the equations can be universally or existentially quantified. Two schemes for simplifying quantifier alternation, called Skolemization and raising (a dual of Skolemization), are pr ..."
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Cited by 124 (13 self)
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Unification problems are identified with conjunctions of equations between simply typed λterms where free variables in the equations can be universally or existentially quantified. Two schemes for simplifying quantifier alternation, called Skolemization and raising (a dual of Skolemization), are presented. In this setting where variables of functional type can be quantified and not all types contain closed terms, the naive generalization of firstorder Skolemization has several technical problems that are addressed. The method of searching for preunifiers described by Huet is easily extended to the mixed prefix setting, although solving flexibleflexible unification problems is undecidable since types may be empty. Unification problems may have numerous incomparable unifiers. Occasionally, unifiers share common factors and several of these are presented. Various optimizations on the general unification search problem are as discussed. 1.
Unification: A multidisciplinary survey
 ACM Computing Surveys
, 1989
"... The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the ..."
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Cited by 103 (0 self)
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The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the paper include examples that highlight particular uses
tps: A theorem proving system for classical type theory
 Journal of Automated Reasoning
, 1996
"... This is a description of TPS, a theorem proving system for classical type theory (Church’s typed λcalculus). 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 comb ..."
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Cited by 71 (6 self)
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This is a description of TPS, a theorem proving system for classical type theory (Church’s typed λcalculus). 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 which TPS can prove completely automatically are given to illustrate certain aspects of TPS’s behavior and problems of theorem proving in higherorder logic. 7
Natural Deduction as HigherOrder Resolution
 Journal of Logic Programming
, 1986
"... An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. ..."
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Cited by 54 (8 self)
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An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause.
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 ..."
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Cited by 27 (5 self)
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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
Combining HigherOrder and FirstOrder Computation Using ρcalculus: Towards a Semantics of ELAN
 In Frontiers of Combining Systems 2
, 1999
"... The ρcalculus permits to express in a uniform and simple way firstorder rewriting, λcalculus and nondeterministic computations as well as their combination. In this paper, we present the main components of the ρcalculus and we give a full firstorder presentation of this rewriting calculus using ..."
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Cited by 20 (9 self)
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The ρcalculus permits to express in a uniform and simple way firstorder rewriting, λcalculus and nondeterministic computations as well as their combination. In this paper, we present the main components of the ρcalculus and we give a full firstorder presentation of this rewriting calculus using an explicit substitution setting, called ρσ, that generalizes the λσcalculus. The basic properties of the nonexplicit and explicit substitution versions are presented. We then detail how to use the ρcalculus to give an operational semantics to the rewrite rules of the ELAN language. 1
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 comb ..."
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Cited by 16 (0 self)
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. 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...
Extensional higherorder resolution
 In Kirchner and Kirchner [KK98
, 1998
"... Abstract. In this paper we present an extensional higherorder resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary for the completeness result – by specialized (goaldirected) inference rules is of practical applicabilit ..."
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Cited by 11 (6 self)
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Abstract. In this paper we present an extensional higherorder resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary for the completeness result – by specialized (goaldirected) inference rules is of practical applicability, as an implentation of the calculus in the LeoSystem shows. Furthermore, we prove the longstanding conjecture, that it is sufficient to restrict the order of primitive substitutions to the order of input formulae. 1