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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 70 (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
Reconstructing Proofs at the Assertion Level
, 1994
"... Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND pr ..."
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Cited by 69 (9 self)
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Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND proofs are easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. To obtain proofs similar to those found in mathematical textbooks, we propose a new formalism, called ND style proofs at the assertion level , where derivations are mostly justified by the application of a definition or a theorem. After characterizing the structure of compound ND proof segments allowing assertion level justification, we show that the same derivations can be achieved by domainspecific inference rules as well. Furthermore, these rules can be represented compactly in a tre structure. Finally, we describe a system called PROVERB , which substantially sh...
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 28 (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
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 ..."
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Cited by 21 (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...
System description: TPS: A theorem proving system for type theory
 Automated Deduction — CADE17; 17th Int. Conf. on Automated Deduction
, 2000
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Translating MachineGenerated Resolution Proofs into NDProofs at the Assertion Level
 IN PROC. OF PRICAI96
, 1996
"... Most automated theorem provers suffer from the problem that the resulting proofs are difficult to understand even for experienced mathematicians. An effective communication between the system and its users, however, is crucial for many applications, such as in a mathematical assistant system. Theref ..."
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Cited by 13 (1 self)
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Most automated theorem provers suffer from the problem that the resulting proofs are difficult to understand even for experienced mathematicians. An effective communication between the system and its users, however, is crucial for many applications, such as in a mathematical assistant system. Therefore, efforts have been made to transform machine generated proofs (e.g. resolution proofs) into natural deduction (ND) proofs. The stateoftheart procedure of proof transformation follows basically its completeness proof: the premises and the conclusion are decomposed into unit literals, then the theorem is derived by multiple levels of proofs by contradiction. Indeterminism is introduced by heuristics that aim at the production of more elegant results. This indeterministic character entails not only a complex search, but also leads to unpredictable results. In this paper we first study resolution proofs in terms of meaningful operations employed by human mathematicians, and thereby esta...
Verification using Satisfiability Checking, Predicate Abstraction, and Craig Interpolation
, 2008
"... not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the sponsoring institutions, the U.S. Government or any other entity. Keywords: Formal methods, model checking, abstraction, refinement, bounded model checking, Boolean satisfiabilit ..."
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Cited by 4 (1 self)
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not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the sponsoring institutions, the U.S. Government or any other entity. Keywords: Formal methods, model checking, abstraction, refinement, bounded model checking, Boolean satisfiability, nonclausal SAT solvers, DPLL, general matings, unsatisfiable core, craig interpolation, proofs of unsatisfiability, linear diophantine equations, linear modular equations (linear congruences), linear diophantine Automatic verification of hardware and software implementations is crucial for building reliable computer systems. Most verification tools rely on decision procedures to check the satisfiability of various formulas that are generated during the verification process. This thesis develops new techniques for building efficient decision procedures and adds new capabilities to the existing decision procedures for certain logics. Boolean satisfiability (SAT) solvers are used heavily in verification tools as decision procedures for propositional logic. Most stateoftheart SAT solvers are
Transformation of MachineFound Proofs into Assertion Level Proofs
, 2000
"... Most automated theorem provers suffer on the problem that the proofs they produce are difficult to understand even for experienced users. Therefore, many efforts have been made to transform, abstract and restructure machinefound proofs to produce proof formats better understandable for humans. One ..."
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Cited by 2 (1 self)
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Most automated theorem provers suffer on the problem that the proofs they produce are difficult to understand even for experienced users. Therefore, many efforts have been made to transform, abstract and restructure machinefound proofs to produce proof formats better understandable for humans. One of the most prefered target formats is the natural deduction proof. Current approaches suffer mainly on two problems. First, the stateofthe art transformation procedures generate very often natural deduction proofs with many indirect parts since they translate at the literal level steps from the machinefound proofs into the natural deduction proofs. Secondly, the natural deduction calculus itself is not eligible for presenting mathematical proofs. The problem is that inferences in the natural deduction calculus are still on the level of syntactical manipulations of logical connectives and quantifiers. They are not on the level of theorem or definition applications as we have in mathematic...
Structuring of ComputerGenerated Proofs by Cut Introduction
"... . As modern Automated Deduction systems rely heavily on the use of a machineoriented representation of a given problem, together with sophisticated redundancyavoiding techniques, a major task in convincing human users of the correctness of automatically generated proofs is the intelligible rep ..."
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. As modern Automated Deduction systems rely heavily on the use of a machineoriented representation of a given problem, together with sophisticated redundancyavoiding techniques, a major task in convincing human users of the correctness of automatically generated proofs is the intelligible representation of these proofs. In this paper, we propose the use of the cutrule in the humanoriented presentation of computergenerated proofs. The intelligent application of cuts enables the integration of essential lemmata and therefore shortens and structures proof presentation. We show that many translation techniques in Automated Deduction, such as antiprenexing and some forms of normal form translations, can be described as cuts and are indeed part of the deductive solution of a problem. Furthermore, we demonstrate the connection between symmetric simplification, quantorial extension principles and the application of the cutrule. 1 Introduction Most of today's Automated Ded...