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Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
The Fundamental Role of Entailment in Knowledge Representation and Reasoning
 JOURNAL OF COMPUTING AND INFORMATION
, 1996
"... The hot controversy about the role of logic in AI has been repeated so far and probably will continue on as usual. An important fact is that the "logic" as the center of the controversy is classical mathematical logic and/or its various extensions, though there are some "more logical& ..."
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Cited by 18 (16 self)
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The hot controversy about the role of logic in AI has been repeated so far and probably will continue on as usual. An important fact is that the "logic" as the center of the controversy is classical mathematical logic and/or its various extensions, though there are some "more logical" logic systems. Until recently, what is debated by the researchers working on the fundamentals of AI is, among other things, what role the classical mathematical logic and/or its various extensions plays in knowledge representation and reasoning. As a result, most work on the fundamentals of AI are directly or indirectly based on the classical mathematical logic. It is the time to investigate the role of logic in AI from a more general point of view. What notion plays the most fundamental role in knowledge representation and reasoning? This question is discussed in this paper from the viewpoint of logic. The proposition presented in the paper is that it is the notion of entailment that plays the most fundamental role in knowledge representation and reasoning because any reasoning must invoke it and many important issues in knowledge representation and reasoning have to find a calculus of entailment as the logical basis of their solutions. In order to establish a satisfactory logic calculus of entailment to underlie knowledge representation and reasoning, the paper proposes some new relevant logics and suggests some important research problems.
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 17 (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...
EnCal: An Automated Forward Deduction System for GeneralPurpose Entailment Calculus
, 1996
"... This paper presents the fundamental design ideas, working principles, and implementation of an automated forward deduction system for generalpurpose entailment calculus, named EnCal, shows its potential applications in knowledge acquisition, reasoning rule generation, and theorem finding, reports s ..."
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This paper presents the fundamental design ideas, working principles, and implementation of an automated forward deduction system for generalpurpose entailment calculus, named EnCal, shows its potential applications in knowledge acquisition, reasoning rule generation, and theorem finding, reports some current results of our experiments with EnCal, and suggests some important research problems.
Knowledge Representation and Classical Logic
, 2007
"... Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspe ..."
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Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of firstorder (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of firstorder logic; recent
Term rewriting: Some experimental results
 Journal of Symbolic Computation
, 1991
"... We discuss tenn rewriting in conjunction with sprfn, a Prologbased theorem prover. Two techniques for theorem proving that utilize tenn rewriting are presented. We demonstrate their effectiveness by exhibiting the results of our experiments in proving some theorems of von NeumannBemaysGodel set t ..."
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We discuss tenn rewriting in conjunction with sprfn, a Prologbased theorem prover. Two techniques for theorem proving that utilize tenn rewriting are presented. We demonstrate their effectiveness by exhibiting the results of our experiments in proving some theorems of von NeumannBemaysGodel set theory. Some outstanding problems associated with tenn rewriting are also addressed. Key words and phrases. Theorem proving, term rewriting, set theory. 1
On the Translation of HigherOrder Problems into FirstOrder Logic
 Proceedings of ECAI94
, 1994
"... . In most cases higherorder logic is based on the  calculus in order to avoid the infinite set of socalled comprehension axioms. However, there is a price to be paid, namely an undecidable unification algorithm. If we do not use the calculus, but translate higherorder expressions into firstor ..."
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. In most cases higherorder logic is based on the  calculus in order to avoid the infinite set of socalled comprehension axioms. However, there is a price to be paid, namely an undecidable unification algorithm. If we do not use the calculus, but translate higherorder expressions into firstorder expressions by standard translation techniques, we have to translate the infinite set of comprehension axioms, too. Of course, in general this is not practicable. Therefore such an approach requires some restrictions such as the choice of the necessary axioms by a human user or the restriction to certain problem classes. This paper will show how the infinite class of comprehension axioms can be represented by a finite subclass, so that an automatic translation of finite higherorder problems into finite firstorder problems is possible. This translation is sound and complete with respect to a Henkinstyle general model semantics. 1 Introduction Firstorder logic is a powerful tool for ...
An Equational ReEngineering of Set Theories
 Automated Deduction in Classical and NonClassical Logics, LNCS 1761 (LNAI
, 1998
"... New successes in dealing with set theories by means of stateoftheart theoremprovers may ensue from terse and concise axiomatizations, such as can be moulded in the framework of the (fully equational) TarskiGivant map calculus. In this paper we carry out this task in detail, setting the ground fo ..."
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New successes in dealing with set theories by means of stateoftheart theoremprovers may ensue from terse and concise axiomatizations, such as can be moulded in the framework of the (fully equational) TarskiGivant map calculus. In this paper we carry out this task in detail, setting the ground for a number of experiments. Key words: Set theory, relation algebras, firstorder theoremproving, algebraic logic. 1 Introduction Like other mature fields of mathematics, Set Theory deserves sustained efforts that bring to light richer and richer decidable fragments of it [5], general inference rules for reasoning in it [23, 2], effective proof strategies based on its domainknowledge, and so forth. Advances in this specialized area of automated reasoning tend, in spite of their steadiness, to be slow compared to the overall progress in the field. Many experiments with set theories have hence been carried out with standard theoremproving systems. Still today such experiments pose consider...
Map calculus: Initial application scenarios and experiments based on Otter
, 1998
"... Properties of a few familiar structures (natural numbers, nested lists, lattices) are formally specified in TarskiGivant's map calculus, with the aim of bringing to light new translation techniques that may bridge the gap between firstorder predicate calculus and the map calculus. It is also ..."
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Properties of a few familiar structures (natural numbers, nested lists, lattices) are formally specified in TarskiGivant's map calculus, with the aim of bringing to light new translation techniques that may bridge the gap between firstorder predicate calculus and the map calculus. It is also highlighted to what extent a stateoftheart theoremprover for firstorder logic, namely Otter, can be exploited not only to emulate, but also to reason about, map calculus.