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34
Algebraic properties of program integration
 Science of Computer Programming
, 1991
"... Abstract. The need to integrate several versions of a program into a common one arises frequently, but it is a tedious and time consuming task to merge programs by hand. The programintegration algorithm proposed by Horwitz, Prins, and Reps provides a way to create a semanticsbased tool for integra ..."
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Cited by 20 (4 self)
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Abstract. The need to integrate several versions of a program into a common one arises frequently, but it is a tedious and time consuming task to merge programs by hand. The programintegration algorithm proposed by Horwitz, Prins, and Reps provides a way to create a semanticsbased tool for integrating a base program with two or more variants. The integration algorithm is based on the assumption that any change in the behavior, rather than the text, of a program variant is significant and must be incorporated in the merged program. An integration system based on this algorithm will determine whether the variants incorporate interfering changes, and, if they do not, create an integrated program that includes all changes as well as all features of the base program that are preserved in all variants. To determine this information, the algorithm employs a program representation that is similar to the program dependence graphs that have been used previously in vectorizing and parallelizing compilers. This paper studies the algebraic properties of the programintegration operation, such as whether there are laws of associativity and distributivity. (For example, in this context associativity means: “If three variants of a given base are to be integrated by a pair of twovariant integrations, the same result is produced no matter which two variants are integrated first.”) To answer such questions, we reformulate the HorwitzPrinsReps integration algorithm as an operation in a Brouwerian algebra constructed from sets of dependence graphs. (A Brouwerian algebra is a distributive lattice with an operation a. − b characterized by a. ¡ ¢
Subtractive Logic
, 1999
"... This paper is the first part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambdacalculi, intuitionistic and classical logics) from syntactic, semantical and computational viewpoints. We start with category theory and we show that any ..."
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This paper is the first part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambdacalculi, intuitionistic and classical logics) from syntactic, semantical and computational viewpoints. We start with category theory and we show that any bicartesian closed category with coexponents is degenerated (i.e. there is at most one arrow between two objects). The remainder of the paper is devoted to logical issues. We examine the propositional calculus underlying the type system of bicartesian closed categories with coexponents and we show that this calculus corresponds to subtractive logic: a conservative extension of intuitionistic logic with a new connector (subtraction) dual to implication. Eventually, we consider first order subtractive logic and we present an embedding of classical logic into subtractive logic. Introduction This paper is the first part of a work whose purpose is to investigate duality in some related ...
On an Intuitionistic Modal Logic
 Studia Logica
, 2001
"... . In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models ..."
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Cited by 19 (4 self)
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. In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models of IS4, not in the framework of Kripke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability. 1. Introduction Modal logics are traditionally extensions of classical logic with new operators, or modalities, whose operation is intensional. Modal logics are most commonly justified by the provision of an intuitive semantics based upon `possible worlds', an idea originally due to Kripke. Kripke also provided a possible worlds semantics for intuitionistic logic, and so it is natural to consider intuitionistic logic extended with intensional modalities...
Lambda Terms for Natural Deduction, Sequent Calculus and Cut Elimination
"... It is wellknown that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequ ..."
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Cited by 13 (3 self)
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It is wellknown that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a oneone map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally equivalent type assignment systems for untyped lambda terms, one corresponding to natural deduction (N) and the other to sequent calculus (L). These two systems constitute different grammars for generating the same (type assignment relation for untyped) lambda terms. The second grammar is ambiguous, but the first one is not. This fact explains the manyone correspondence mentioned above. Moreover, the second type assignment system has a `cutfree' fragment (L cf ). This fragment generates exactly the typeable lambda terms in normal form. The cut elimination theorem becomes a simple consequence of the fact that typed lambda terms posses a normal form.
Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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Cited by 10 (4 self)
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
The Semantics of Entailment Omega
, 2002
"... This paper discusses the relation between the minimal positive relevant logic B+ and intersection and union type theories. There is a marvellous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the twea ..."
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Cited by 4 (2 self)
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This paper discusses the relation between the minimal positive relevant logic B+ and intersection and union type theories. There is a marvellous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking B intersect T of B+ , which saves implication and conjunction but drops disjunction. The filter models of the lambdacalculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors then become theory models of these calculi. (The logician's Theory is the algebraist's Filter.) The coincidence extends to a dual interpretation of key particles  the subtype translates to provable >, type intersection to conjunction, function space > to implication and whole domain omega to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper and to be expected that type union U should correspond to the logical disjunction \/ of B+ . But the simulation of functional application by a fusion (or modus ponens product) operation o on theories leaves the key Bubbling lemma of work on ITD unprovable for the \/prime theories now appropriate for the modelling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of lambda and CL.
Delaying Commitment in Plan Recognition Using Combinatory Categorial Grammars
"... This paper presents a new algorithm for plan recognition called ELEXIR (Engine for LEXicalized Intent Recognition). ELEXIR represents the plans to be recognized with a grammatical formalism called Combinatory Categorial Grammar(CCG). We show that representing plans with CCGs can allow us to prevent ..."
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Cited by 3 (0 self)
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This paper presents a new algorithm for plan recognition called ELEXIR (Engine for LEXicalized Intent Recognition). ELEXIR represents the plans to be recognized with a grammatical formalism called Combinatory Categorial Grammar(CCG). We show that representing plans with CCGs can allow us to prevent early commitment to plan goals and thereby reduce runtime. 1
Combining Conjunction with Disjunction
 Proceedings of the 2nd Indian International Conference on Artificial Intelligence (IICAI 2005
, 2005
"... Abstract. In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following probl ..."
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Cited by 3 (3 self)
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Abstract. In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following problem arises: if we combine these logics in a straightforward way, distributivity holds. On the other hand, distributivity does not arise if we use the usual notion of extension between consequence relations. A detailed discussion about this phenomenon, as well as some possible solutions for it, are given. 1
Logic of subtyping
 Theoretical Computer Science
, 2005
"... We introduce new modal logical calculi that describe subtyping properties of Cartesian product and disjoint union type constructors as well as mutuallyrecursive types defined using those type constructors. Basic Logic of Subtyping S extends classical propositional logic by two new binary modalities ..."
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Cited by 3 (2 self)
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We introduce new modal logical calculi that describe subtyping properties of Cartesian product and disjoint union type constructors as well as mutuallyrecursive types defined using those type constructors. Basic Logic of Subtyping S extends classical propositional logic by two new binary modalities ⊗ and ⊕. An interpretation of S is a function that maps standard connectives into settheoretical operations (intersection, union, and complement) and modalities into Cartesian product and disjoint union type constructors. This allows S to capture many subtyping properties of the above type constructors. We also consider logics Sρ and S ω ρ that incorporate into S mutuallyrecursive types over arbitrary and wellfounded universes correspondingly. The main results are completeness of the above three logics with respect to appropriate type universes. In addition, we prove Cut elimination theorem for S and establish decidability of S and S ω ρ.