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Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is noth ..."
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Cited by 114 (22 self)
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In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
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. ¡ ¢
Maintenance Of Geometric Representations Through Space Decompositions
, 1997
"... The ability to transform between distinct geometric representations is the key to success of multiplerepresentation modeling systems. But the existing theory of geometric modeling does not directly address or support construction, conversion, and comparison of geometric representations. A study of ..."
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Cited by 10 (4 self)
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The ability to transform between distinct geometric representations is the key to success of multiplerepresentation modeling systems. But the existing theory of geometric modeling does not directly address or support construction, conversion, and comparison of geometric representations. A study of classical problems of CSG $ brep conversions, CSG optimization, and other representation conversions suggests a natural relationship between a representation scheme and an appropriate decomposition of space. We show that a hierarchy of space decompositions corresponding to different representation schemes can be used to enhance the theory and to develop a systematic approach to maintenance of geometric representations. 1. Motivation 1.1. Modern theory of representations The modern field of solid modeling owes much of its success to the theoretical foundations laid by members of the Production Automation Project at the University of Rochester in the 1970's. The history of these development...
Solid modeling
 Handbook of Computer Aided Geometric Design
, 2002
"... inversion for a global shear velocity ..."
Approximation Operators in Qualitative Data Analysis
 PROCEEDINGS OF THE 2002 IEEE INTERNATIONAL CONFERENCE ON DATA MINING
, 2002
"... ... In his paper, we present various forms of set approximations via the unifying concept of modalstyle operators. Two examples indicate the usefulness of the approach. ..."
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Cited by 6 (0 self)
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... In his paper, we present various forms of set approximations via the unifying concept of modalstyle operators. Two examples indicate the usefulness of the approach.
A cutfree sequent calculus for biintuitionistic logic: extended version
, 2007
"... Abstract. Biintuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Biintuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus for BiInt has recently been s ..."
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Cited by 5 (1 self)
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Abstract. Biintuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Biintuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus for BiInt has recently been shown by Uustalu to fail cutelimination. We present a new cutfree sequent calculus for BiInt, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose. 1
Propositional quantification in the topological semantics for S4
 Notre Dame Journal of Formal Logic
, 1997
"... quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S ..."
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Cited by 3 (1 self)
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quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub S4πt, isstrictly weaker than its Kripkean counterpart. I prove here that secondorder arithmetic can be recursively embedded in S4πt. Inthe course of the investigation, I also sketch a proof of Fine’s and Kripke’s results that the Kripkean system S4π+ is recursively isomorphic to secondorder logic. 1Introduction One way to extend a propositional logic to a language with propositional quantifiers is to begin with a semantics for the logic; extract from the semantics a notion of a proposition; and interpret the quantifiers as ranging over the propositions. Thus, Fine [4] extends the Kripke semantics for modal logics to propositionally quantified systems S5π+, S4π+, S4.2π+, and such: given a Kripke frame, the quantifiers range over all sets of possible worlds. S5π+ is decidable ([4] and Kaplan [14]). In later unpublished work, Fine and Kripke independently showed that S4π+, S4.2π+, K4π+, Tπ+, Kπ+, and Bπ+ and others are recursively isomorphic to full secondorder classical logic. (Fine informs me that he later proved this stronger result. Kripke informs me that he too proved this stronger result in the early 1970s. A proof of this result occurs in Kaminski and Tiomkin [13], who use techniques similar to those used in Kremer [16] and to those used below. These techniques do not apply to S4.3π+. But according to Kaminski and Tiomkin, work of Gurevich and Shelah ([9], [10], and [39]) implies that secondorder arithmetic is interpretable in S4.3π+ and furthermore that, under
Mass problems and intuitionism
, 2007
"... Let Pw be the lattice of Muchnik degrees of nonempty Π 0 1 subsets of 2 ω. The lattice Pw has been studied extensively in previous publications. In this note we prove that the lattice Pw is not Brouwerian. 1 ..."
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Cited by 3 (2 self)
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Let Pw be the lattice of Muchnik degrees of nonempty Π 0 1 subsets of 2 ω. The lattice Pw has been studied extensively in previous publications. In this note we prove that the lattice Pw is not Brouwerian. 1
Algebraic Polymodal Logic: A Survey
 LOGIC JOURNAL OF THE IGPL
, 2000
"... This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with ..."
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Cited by 2 (0 self)
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This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with
On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond
, 2008
"... Département de mathématiques Faculté des sciences ..."