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15
Introducing OBJ
, 1993
"... This is an introduction to the philosophy and use of OBJ, emphasizing its operational semantics, with aspects of its history and its logical semantics. Release 2 of OBJ3 is described in detail, with many examples. OBJ is a wide spectrum firstorder functional language that is rigorously based on ..."
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Cited by 121 (30 self)
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This is an introduction to the philosophy and use of OBJ, emphasizing its operational semantics, with aspects of its history and its logical semantics. Release 2 of OBJ3 is described in detail, with many examples. OBJ is a wide spectrum firstorder functional language that is rigorously based on (order sorted) equational logic and parameterized programming, supporting a declarative style that facilitates verification and allows OBJ to be used as a theorem prover.
Semantics of Local Variables
, 1992
"... This expository article discusses recent progress on the problem of giving sufficiently abstract semantics to localvariable declarations in Algollike languages, especially work using categorical methods. ..."
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Cited by 35 (4 self)
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This expository article discusses recent progress on the problem of giving sufficiently abstract semantics to localvariable declarations in Algollike languages, especially work using categorical methods.
Granulation for Graphs
 Spatial Information Theory. Cognitive and Computational Foundations of Geographic Information Science. International Conference COSIT'99, volume 1661 of Lecture Notes in Computer Science
, 1999
"... . In multiresolution data handling, a less detailed structure is often derived from a more detailed one by amalgamating elements which are indistinguishable at the lower level of detail. This gathering together of indistinguishable elements is called a granulation of the more detailed structure ..."
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Cited by 10 (6 self)
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. In multiresolution data handling, a less detailed structure is often derived from a more detailed one by amalgamating elements which are indistinguishable at the lower level of detail. This gathering together of indistinguishable elements is called a granulation of the more detailed structure. When handling spatial data at several levels of detail the granulation of graphs is an important topic. The importance of graphs arises from their widespread use in modelling networks, and also from the use of dual graphs of spatial partitions. This paper demonstrates that there are several quite different kinds of granulation for graphs. Four kinds are described in detail, and situations where some of these may arise in spatial information systems are indicated. One particular kind of granulation leads to a new formulation of the boundarysensitive approach to qualitative location developed by Bittner and Stell. Vague graphs and their connection with granulation are also discusse...
A Topos for Algebraic Quantum Theory
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2009
"... The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C ..."
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Cited by 8 (1 self)
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The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topostheoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on �, and selfadjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topostheoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
Semantics of Dynamic Variables in Algollike Languages
, 1997
"... A denotational semantic model of an Algollike programming language with local variables, providing fully functional dynamic variable manipulation is presented. Along with the other usual language features, the standard operations with pointers, that is reattachement and dereferencing, and dynamic v ..."
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Cited by 5 (1 self)
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A denotational semantic model of an Algollike programming language with local variables, providing fully functional dynamic variable manipulation is presented. Along with the other usual language features, the standard operations with pointers, that is reattachement and dereferencing, and dynamic variables, that is creation and assignment, are explicated using a possible worlds, functor category, location oriented model. It is shown that the model used to explicate local variables in Algollike languages can be extended to dynamic variables and pointers. Such a model allows for an analytic comparison of the properties of local and dynamic variables and, at the same time, validates several equivalences that, by common computational and operational intuition, are expected to hold. Two fundamental types of equivalences for linked data structures created using pointers are defined, observational equivalence and aeisomorphism, and it is contended that they are both the extensional respect...
Morphisms and Semantics for Higher Order Parameterized Programming
, 2002
"... Parameterized programming is extended to higher order modules, by extending views, which fit actual parameters to formal parameters in a flexible way, to morphisms, with higher order module expressions to compose modules into systems. A category theoretic semantics is outlined, and examples in BOBJ ..."
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Cited by 1 (0 self)
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Parameterized programming is extended to higher order modules, by extending views, which fit actual parameters to formal parameters in a flexible way, to morphisms, with higher order module expressions to compose modules into systems. A category theoretic semantics is outlined, and examples in BOBJ show the power of morphisms.
Categories: A Free Tour
"... Category theory plays an important role as a unifying agent in a rapidly expanding universe of mathematics. In this paper, an introduction is given to the basic denitions of category theory, as well as to more advanced concepts such as adjointness, factorization systems and cartesian closedness. ..."
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Category theory plays an important role as a unifying agent in a rapidly expanding universe of mathematics. In this paper, an introduction is given to the basic denitions of category theory, as well as to more advanced concepts such as adjointness, factorization systems and cartesian closedness. In the past decades, the subject of mathematics has experienced an explosive increase both in diversity and in the sheer amount of published material. (E.g., the Mathematical Reviews volume of 1950 features 766 pages of reviews, compared to a total of 4550 pages in the six volumes for the rst half of 2000.) It has thus become inevitable that this growth, taking place in numerous and increasingly disconnected branches, be complemented by some form of unifying theory. There have been attempts at such unications in the past, such as Birkhostyle universal algebra or the encyclopedic work of Bourbaki. However, the most successful and universal approach so far is certainly the theory of cat...
Cover semantics for quantified lax logic
 Journal of Logic and Computation
"... Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of LawvereTierneyGrothendieck topologies on topoi. This paper provides a complete semantics for quantified la ..."
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Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of LawvereTierneyGrothendieck topologies on topoi. This paper provides a complete semantics for quantified lax logic by combining the BethKripkeJoyal cover semantics for firstorder intuitionistic logic with the classical relational semantics for a “diamond ” modality. The main technique used is the lifting of a multiplicative closure operator (nucleus) from a Heyting algebra to its MacNeille completion, and the representation of an arbitrary locale as the lattice of “propositions ” of a suitable cover system. In addition, the theory is worked out for certain constructive versions of the classical logics K and S4. An alternative completeness proof is given for (nonmodal) firstorder intuitionistic logic itself with respect to the cover semantics, using a simple and explicit Henkinstyle construction of a characteristic model whose points are principal theories rather than prime saturated ones. The paper provides further evidence that there is more to intuitionistic modal logic than the generalisation of properties of boxes and diamonds from Boolean modal logic.
Functional Monadic Bounded Algebras
, 2010
"... The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ∃ reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomor ..."
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The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ∃ reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions. We show that FMBA is characterised by the disjunction of the equations ∃E = 1 and ∃E = 0. We also define a weaker notion of “relatively functional ” algebra, and show that every member of MBA is isomorphic to a relatively functional one. In [1], an equationally defined class MBA of monadic bounded algebras was introduced. Each of these algebras comprises a Boolean algebra B with a distinguished element E, thought of as an existence predicate, and an operator ∃ on B reflecting the properties of the existential quantifier in logic without existence assumptions. MBA was shown to be generated by a certain proper