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A Uniform Axiomatic View of Lists, Multisets, and Sets, and the Relevant Unification Algorithms
, 1998
"... . The rstorder theories of lists, multisets, compact lists (i.e., lists where the number of contiguous occurrences of each element is immaterial), and sets are introduced via axioms. Such axiomatizations are shown to be very wellsuited for the integration with free functor symbols governed by the ..."
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Cited by 24 (15 self)
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. The rstorder theories of lists, multisets, compact lists (i.e., lists where the number of contiguous occurrences of each element is immaterial), and sets are introduced via axioms. Such axiomatizations are shown to be very wellsuited for the integration with free functor symbols governed by the classical Clark's axioms in the context of (Constraint) Logic Programming. Adaptations of the extensionality principle to the various theories taken into account is then exploited in the design of unication algorithms for the considered data structures. All the theories presented can be combined providing frameworks to deal with We acknowledge partial support from C.N.R. Grant 97.02426.CT12, C.N.R. project SETA, and from the MURST project \Tecniche formali per la specica, l'analisi, la verica, la sintesi e la trasformazione di sistemi software". 202 Dovier, Policriti, and Rossi / A uniform axiomatic view of lists, multisets, and sets several of the proposed data structures simultan...
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 19 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
NuPRL’s class theory and its applications
 Foundations of Secure Computation, NATO ASI Series, Series F: Computer & System Sciences
, 2000
"... This article presents a theory of classes and inheritance built on top of constructive type theory. Classes are defined using dependent and very dependent function types that are found in the Nuprl constructive type theory. Inheritance is defined in terms of a general subtyping relation over the und ..."
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Cited by 15 (7 self)
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This article presents a theory of classes and inheritance built on top of constructive type theory. Classes are defined using dependent and very dependent function types that are found in the Nuprl constructive type theory. Inheritance is defined in terms of a general subtyping relation over the underlying types. Among the basic types is the intersection type which plays a critical role in the applications because it provides a method of composing program components. The class theory is applied to defining algebraic structures such as monoids, groups, rings, etc. and relating them. It is also used to define communications protocols as infinite state automata. The article illustrates the role of these formal automata in defining the services of a distributed group communications system. In both applications the inheritance mechanisms allow reuse of proofs and the statement of general properties of system composition. 1
A Formal Theory for KnowledgeBased Product Model Representation
, 1996
"... The field of design science attempts to place engineering design on a more formal, rigorous footing. This paper introduces recent work by the author in this area. ArtifactCentered Modeling (ACM) is a general framework intended to partition the design endeavor in manageable sections. A fundamental p ..."
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Cited by 10 (6 self)
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The field of design science attempts to place engineering design on a more formal, rigorous footing. This paper introduces recent work by the author in this area. ArtifactCentered Modeling (ACM) is a general framework intended to partition the design endeavor in manageable sections. A fundamental part of ACM is the representation of information about products being designed. The Axiomatic Information Model for Design (AIMD) is a formal theory about product information based on axiomatic set theory. AIMD provides formal bases for quantities, features, parts and assemblies, systems, and subassemblies; these are all notions essential to design. It is not a product modeling system per se, but rather a logic of product structure whose axioms define criteria for determining the logical validity of product models. A previous version of the theory has been found to contain logical inconsistencies; the version presented herein addresses those problems. A complete axiomatization of the new th...
Comparing Expressiveness of Set Constructor Symbols
, 2000
"... In this paper we consider the relative expressive power of two very common operators applicable to sets and multisets: the with and the union operators. For such operators we prove that they are not mutually expressible by means of existentially quantified formulae. In order to prove our results ..."
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Cited by 10 (7 self)
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In this paper we consider the relative expressive power of two very common operators applicable to sets and multisets: the with and the union operators. For such operators we prove that they are not mutually expressible by means of existentially quantified formulae. In order to prove our results, canonical forms for settheoretic and multisettheoretic formulae are established and a particularly natural axiomatization of multisets is given and studied.
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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Cited by 9 (2 self)
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
Word and objects
 Noûs
, 2002
"... The aim of this essay is to show that the subjectmatter of ontology is richer than one might have thought. Our route will be indirect. We will argue that there are circumstances under which standard firstorder regimentation is unacceptable, and that more appropriate varieties of regimentation lead ..."
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Cited by 8 (6 self)
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The aim of this essay is to show that the subjectmatter of ontology is richer than one might have thought. Our route will be indirect. We will argue that there are circumstances under which standard firstorder regimentation is unacceptable, and that more appropriate varieties of regimentation lead to unexpected kinds of ontological commitment. Quine has taught us that ontological inquiry—inquiry as to what there is—can be separated into two distinct tasks. 1 On the one hand, there is the problem of determining the ontological commitments of a given theory; on the other, the problem of deciding what theories to accept. The objects whose existence we have reason to believe in are then the ontological commitments of the theories we have reason to accept. Regarding the former of these two tasks, Quine maintains that a firstorder theory is committed to the existence of an object satisfying a certain predicate if and only if some object satisfying that predicate must be admitted among the values of the theory’s variables in order for the theory to be true. Quine’s criterion is extremely attractive, but it applies only to theories that are couched in firstorder languages. Offhand this is not a serious constraint, because most of our theories have straightforward firstorder regimentations. But here we shall see that there is a special kind of tension between regimenting our discourse in a firstorder language and allowing our quantifiers to range over absolutely everything. 2 We will proceed on the assumption that absolutely unrestricted quantification is possible, and show that an important class of English sentences resists firstorder regimentation. This will lead us to develop alternate languages of regimentation, languages containing plural
Constructive Negation and Constraint Logic Programming with Sets
 New Generation Computing
"... The aim of this paper is to extend the Constructive Negation technique to the case of CLP (SET ), a Constraint Logic Programming (CLP ) language based on hereditarily (and hybrid) finite sets. The challenging aspects of the problem originate from the fact that the structure on which CLP (SET ) is ba ..."
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Cited by 8 (5 self)
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The aim of this paper is to extend the Constructive Negation technique to the case of CLP (SET ), a Constraint Logic Programming (CLP ) language based on hereditarily (and hybrid) finite sets. The challenging aspects of the problem originate from the fact that the structure on which CLP (SET ) is based is not admissible closed, and this does not allow to reuse the results presented in the literature concerning the relationships between CLP and constructive negation. We propose a new constraint satisfaction algorithm, capable of correctly handling constructive negation for large classes of CLP (SET ) programs; we also provide a syntactic characterization of such classes of programs. The resulting algorithm provides a novel constraint simplification procedure to handle constructive negation, suitable to theories where unification admits multiple most general unifiers. We also show, using a general result, that it is impossible to construct an interpreter...
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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Cited by 8 (4 self)
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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 ...