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40
A Proof Procedure Using Connection Graphs
 JACM
, 1975
"... ABSTRACT. Various deficiencies of resolution systems are investigated and a new theoremproving system designed to remedy those deficiencms is presented The system is notable for eliminating redundancies present in SLresolutlon, for incorporating preprocessing procedures, for liberahzing the order ..."
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Cited by 79 (5 self)
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ABSTRACT. Various deficiencies of resolution systems are investigated and a new theoremproving system designed to remedy those deficiencms is presented The system is notable for eliminating redundancies present in SLresolutlon, for incorporating preprocessing procedures, for liberahzing the order in which subgoals can be activated, for incorporating multidirectmnal searches, and for giving immediate access to pairs of clauses which resolve Examples of how the new system copes with the defic2encies of other theoremproving systems are chosen from the areas of predicate logic programming and language parsing. The paper emphasizes the historical development of the new system, beginning as a supplement to SLresolution in the form of classificatmn trees and incorporating an analogue of the Waltz algorithm for picture Interpretation The paper ends with a discussion of the opportunities for using lookahead to guide the search for proofs
A Science of Reasoning
, 1991
"... This paper addresses the question of how we can understand reasoning in general and mathematical proofs in particular. It argues the need for a highlevel understanding of proofs to complement the lowlevel understanding provided by Logic. It proposes a role for computation in providing this high ..."
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Cited by 78 (21 self)
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This paper addresses the question of how we can understand reasoning in general and mathematical proofs in particular. It argues the need for a highlevel understanding of proofs to complement the lowlevel understanding provided by Logic. It proposes a role for computation in providing this highlevel understanding, namely by the association of proof plans with proofs. Proof plans are defined and examples are given for two families of proofs. Criteria are given for assessing the association of a proof plan with a proof. 1 Motivation: the understanding of mathematical proofs The understanding of reasoning has interested researchers since, at least, Aristotle. Logic has been proposed by Aristotle, Boole, Frege and others as a way of formalising arguments and understanding their structure. There have also been psychological studies of how people and animals actually do reason. The work on Logic has been especially influential in the automation of reasoning. For instance, resolution...
Nonresolution theorem proving
 Artificial Intelligence
, 1977
"... This talk reviews those efforts in automatic theorem proving, during the past few years, which have emphasized techniques other than resolution. These include: knowledge bases, natural deduction, reduction, (rewrite rules), typing, procedures, advice, controlled forward chaining, algebraic simplific ..."
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Cited by 60 (3 self)
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This talk reviews those efforts in automatic theorem proving, during the past few years, which have emphasized techniques other than resolution. These include: knowledge bases, natural deduction, reduction, (rewrite rules), typing, procedures, advice, controlled forward chaining, algebraic simplification, builtin associativity and commutativity, models, analogy, and manmachine systems. Examples are given and suggestions are made for future work. 1.
Proof Planning with Multiple Strategies
 In Proc. of the First International Conference on Computational Logic
, 2000
"... . Humans have different problem solving strategies at their disposal and they can flexibly employ several strategies when solving a complex problem, whereas previous theorem proving and planning systems typically employ a single strategy or a hard coded combination of a few strategies. We introd ..."
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Cited by 53 (34 self)
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. Humans have different problem solving strategies at their disposal and they can flexibly employ several strategies when solving a complex problem, whereas previous theorem proving and planning systems typically employ a single strategy or a hard coded combination of a few strategies. We introduce multistrategy proof planning that allows for combining a number of strategies and for switching flexibly between strategies in a proof planning process. Thereby proof planning becomes more robust since it does not necessarily fail if one problem solving mechanism fails. Rather it can reason about preference of strategies and about failures. Moreover, our strategies provide a means for structuring the vast amount of knowledge such that the planner can cope with the otherwise overwhelming knowledge in mathematics. 1 Introduction The choice of an appropriate problem solving strategy is a crucial human skill and is typically guided by some metalevel reasoning. Trained mathematicia...
Proving Theorems about LISP Functions
, 1975
"... Program verification is the idea that properties of programs can be precisely stated and proved in the mathematical sense. In this paper, some simple heuristics combining evaluation and mathematical induction are described, which the authors have implemented in a program that automatically proves a ..."
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Cited by 53 (2 self)
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Program verification is the idea that properties of programs can be precisely stated and proved in the mathematical sense. In this paper, some simple heuristics combining evaluation and mathematical induction are described, which the authors have implemented in a program that automatically proves a wide variety of theorems about recursive LISP functions. The method the program uses to generate induction formulas is described at length. The theorems proved by the program include that REVERSE is its own inverse and that a particular SORT program is correct. A list of theorems proved by the program is given. key words and phrases: LISP, automatic theoremproving, structural induction, program verification cr categories: 3.64, 4.22, 5.21 1 Introduction We are concerned with proving theorems in a firstorder theory of lists, akin to the elementary theory of numbers. We use a subset of LISP as our language because recursive list processing functions are easy to write in LISP and because ...
The BoyerMoore Theorem Prover and Its Interactive Enhancement
, 1995
"... . The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. F ..."
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Cited by 32 (0 self)
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. The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. First we introduce the logic in which theorems are proved. Then we briefly describe the two mechanized theorem proving systems. Next, we present a simple but illustrative example in some detail in order to give an impression of how these systems may be used successfully. Finally, we give extremely short descriptions of a large number of applications of these systems, in order to give an idea of the breadth of their uses. This paper is intended as an informal introduction to systems that have been described in detail and similarly summarized in many other books and papers; no new results are reported here. Our intention here is merely to present Nqthm to a new audience. This research was supported in part by ONR Contract N...
A Theorem Prover for a Computational Logic
, 1990
"... We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of line ..."
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Cited by 26 (0 self)
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We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of linear resolution, rewriting, and arithmetic decision procedures. We describe some applications of the prover, including a proof of the correct implementation of a higher level language on a microprocessor defined at the gate level. We also describe the ongoing project of recoding the entire prover as an applicative function within its own logic.
The "Limit" Domain
 In
, 1998
"... Proof planning is an application of AIplanning in mathematical domains. As opposed to planning for domains such as blocks world or transportation, the domain knowledge for mathematical domains is dicult to extract. Hence proof planning requires clever knowledge engineering and representation ..."
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Cited by 21 (11 self)
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Proof planning is an application of AIplanning in mathematical domains. As opposed to planning for domains such as blocks world or transportation, the domain knowledge for mathematical domains is dicult to extract. Hence proof planning requires clever knowledge engineering and representation of the domain knowledge. We think that on the one hand, the resulting domain denitions that include operators, supermethods, controlrules, and constraint solver are interesting in itself. On the other hand, they can provide ideas for modeling other realistic domains and for means of search reduction in planning. Therefore, we present proof planning and an exemplary domain used for planning proofs of socalled limit theorems that lead to proofs that were beyond the capabilities of other current proof planners and theorem provers. 1 Introduction While humans can cope with long and complex proofs and have strategies to avoid less promising proof paths, classical automated theore...
The use of lemmas in the model elimination procedure
 Journal of Automated Reasoning
, 1997
"... When the Model Elimination (ME) procedure was rst proposed, a notion of lemma was put forth as a promising augmentation to the basic complete proof procedure. Here the lemmas that are used are also discovered by the procedure in the same proof run. Several implementations of ME now exist but only a ..."
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Cited by 14 (0 self)
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When the Model Elimination (ME) procedure was rst proposed, a notion of lemma was put forth as a promising augmentation to the basic complete proof procedure. Here the lemmas that are used are also discovered by the procedure in the same proof run. Several implementations of ME now exist but only a 1970's implementation explicitly examined this lemma mechanism, with indi erent results. We report on the successful use of lemmas using the METEOR implementation of ME. Not only does the lemma device permit METEOR to obtain proofs not otherwise obtainable by METEOR, or any other ME prover not using lemmas, but some wellknown challenge problems are solved. We discuss several of these more di cult problems, including two challenge problems for uniform generalpurpose provers, where METEOR was rst in obtaining the proof. The problems are not selected simply to show o the lemma device, but rather to understand it better. Thus, we choose problems with widely di erent characteristics, including one where very few lemmas are created automatically, the opposite of normal behavior. This selection points out the potential of, and the problems with, lemma use. The biggest problem normally is the selection of appropriate lemmas to retain from the large number generated. 1