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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 automated prover for ZermeloFraenkel set theory in Theorema
 In LMCS02
"... This paper presents some fundamental aspects of the design and the implementation of an automated prover for ZermeloFraenkel set theory within the wellknown Theorema system. The method applies the “ProveComputeSolve”paradigm as its major strategy for generating proofs in a natural style for sta ..."
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This paper presents some fundamental aspects of the design and the implementation of an automated prover for ZermeloFraenkel set theory within the wellknown Theorema system. The method applies the “ProveComputeSolve”paradigm as its major strategy for generating proofs in a natural style for statements involving constructs from set theory.
The Creation and Use of a Knowledge Base of Mathematical Theorems and Definitions
, 1995
"... IPR is an automatic theoremproving system intended particularly for use in higherlevel mathematics. It discovers the proofs of theorems in mathematics applying known theorems and definitions. Theorems and definitions are stored in the knowledge base in the form of sequents rather than formulas or ..."
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IPR is an automatic theoremproving system intended particularly for use in higherlevel mathematics. It discovers the proofs of theorems in mathematics applying known theorems and definitions. Theorems and definitions are stored in the knowledge base in the form of sequents rather than formulas or rewrite rules. Because there is more easilyaccessible information in a sequent than there is in the formula it represents, a simple algorithm can be used to search the knowledge base for the most useful theorem or definition to be used in the theoremproving process. This paper describes how the sequents in the knowledge base are formed from theorems stated by the user and how the knowledge base is used in the theoremproving process. An example of a theorem proved and the English proof output are also given. 1 Introduction The motivating goal behind this work is to develop a theoremproving system which will be useful to both an expert and a nonexpert in the attempt to prove theorems in ...
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Formalizing Automata II: Decidable Properties
"... Is it possible to create formal proofs of interesting mathematical theorems which are mechanically checked in every detail and yet are readable and even faithful to the best expositions of those results in the literature? This paper answers that question positively for theorems about decidable prope ..."
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Is it possible to create formal proofs of interesting mathematical theorems which are mechanically checked in every detail and yet are readable and even faithful to the best expositions of those results in the literature? This paper answers that question positively for theorems about decidable properties of nite automata. The exposition is from Hopcroft and Ullman's classic 1969 textbook Formal Languages and Their Relation to Automata. This paper describes a successful formalization which is faithful to that book. The requirement of being faithful to the book has unexpected consequences, namely that the underlying formal theory must include primitive notions of computability. This requirement makes a constructive formalization especially suitable. It also opens the possibility ofusingthe formal proofs to decide properties of automata. The paper shows how to do this. 1
SCHEMATA: THE CONCEPT OF SCHEMA IN THE HISTORY OF LOGIC
"... Abstract. Schemata have played important roles in logic since Aristotle’s Prior Analytics. The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 19 ..."
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Abstract. Schemata have played important roles in logic since Aristotle’s Prior Analytics. The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in firstorder number theory where Peano’s secondorder Induction Axiom is approximated by Herbrand’s InductionAxiom Schema [23]. Similarly, in firstorder set theory, Zermelo’s secondorder Separation Axiom is approximated by Fraenkel’s firstorder Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a templatetext or schemetemplate, a syntactic string composed of one or more “blanks ” and also possibly significant words and/or symbols. In accordance with a side condition the templatetext of a schema is used as a “template ” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argumenttexts, called instances of the schema. The side condition is a second component. The collection of instances may
ΩMKRP: A Proof Development Environment
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semigroups and automata [3] wi ..."
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In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semigroups and automata [3] with the firstorder theorem prover mkrp [11]. An important finding was that although current automated theorem provers have evidently reached the power to solve nontrivial problems, they do not provide sufficient assistance for proving the theorems contained in such a textbook. On account of this, we believe that significantly more support for proof development can be provided by a system with the following two features:  The system must provide a comfortable humanoriented problemsolving environment. In particular, a human user should be able to specify the problem to be solved in a natural way and communicate on proof
Adding Enrichments to Refined Interleavings: A New Model for the πCalculus
, 1999
"... The question of how to model πcalculus name passing has attracted significant interest. Here, this topic is approached with a new fully abstract interleaving model. Its central feature: Every semantic object contains all its transformations under injective name replacements. It is shown how this en ..."
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The question of how to model πcalculus name passing has attracted significant interest. Here, this topic is approached with a new fully abstract interleaving model. Its central feature: Every semantic object contains all its transformations under injective name replacements. It is shown how this enrichment can be used, in a systematic way, to obtain compositional interpretations of the constructors of the πcalculus. The theory of nonwellfounded sets serves as the mathematical basis. Moreover, category theory is used in the form of coalgebras of endofunctors. Not more is needed since transformations under name replacements are not regarded as arrows of a category of partial orders of (unenriched) semantic objects. This approach is a hallmark of previous interleaving models of the πcalculus. It seems to require a lot more category theory than is used here. Also, unlike other related work, the present one does not employ type theory.