S Ishtiaq and DJ Pym. Kripke Resource Models of a Dependently-Typed, Bunched -calculus. In M Rodrguez-Artalejo J Flum, editor, Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 235-249. Springer-Verlag, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is uniform. The basic idea, inspired by [71] is that there be a surjection from consequences : X; J 1 ( 1 ) Jm ( m ) L J( in L, where is the proof object in L, to consequences X ; y 1 : J 1 ( 1 ) y m : Jm ( m ) L M : J( in L . We can observe [141, 86, 85] that linear logic and other relevant (or substructural) logics cannot be uniformly encoded in LF. To obtain such an encoding it is necessary to move 11 to the RLF logical framework which, as described above, uses the calculus, together with judgements as types. s linear dependent function ....

S.S. Ishtiaq and D.J. Pym. Kripke resource models of a dependently-typed, bunched - calculus. In Preparation.

....propositions as types correspondence between BI and the type theory. In x 3, we describe the Kripke resource models of the type theory. x 5 concludes the paper by constructing a class of Kripke resource models from the category of families of sets. This paper, the content of which was sketched in [14], continues the work rst reported in Ishtiaq and Pym [16] see also [15] The reader is referred to that paper for a full syntactic study of the type theory, together with its use as a language in the logical framework RLF. The ideas presented there were rst considered by one of us in [29] 2 ....

S Ishtiaq and DJ Pym. Kripke Resource Models of a Dependently-Typed, Bunched -calculus. In M Rodrguez-Artalejo J Flum, editor, Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 235-249. Springer-Verlag, 1999.

....one 3 A CCC is bi cartesian closed if it is also bi cartesian, i.e. has co products as well as products. 4 X Y 6 F (X) X F (Y ) Y ( Fig. 2. Fibred Models of Proofs of which is bi cartesian. At the predicate level, the analysis is less clear. The calculus [14,13,15,16] is a dependently typed calculus which provides a partial analysis, being both in the spirit of BI and yet somewhat reliant on the presence of a form of Dereliction. Nevertheless, can be interpreted in the general bred framework sketched in Figure 2; Classical logic: the and ....

.... t : can be interpreted, subject to various structural conditions. All of these examples, and many more besides, can be seen as tting into 5 a model theoretic framework based on a formulation of Kripke Beth Joyal semantics [19] in a general setting based on bred (or indexed) categories [25,6,15,16]. The basic idea is that a model, M = hStruct; i consists of structure, Struct, together with an interpretation, of the syntax of the logic in the structure. The structure is given by a functor, Struct : W; B op ; V] in which W is a (small) category of worlds, B is ....

Ishtiaq, S. and D. Pym, Kripke resource models of a dependently-typed, bunched -calculus (Extended Abstract), LNCS 1683 235-249 1999.

....logic which has multiplicative quantifiers. That is, apart from Ambler s system for the existential only [2] the formulation of which is somewhat more complex than ours. Also relevant is the theory of multiplicative dependent function types in [20] together with its fibrational semantics [19]; incidentally, it can be regarded as relying on a version of bunches appropriate to dependent type theory. 242 PETER W. O HEARN AND DAVID J. PYM Acknowledgments. We are grateful to Pablo Armelin, Guy McCusker, Peter Schroeder Heister and Edmund Robinson for useful discussions about this work. ....

S. S. Ishtiaq and D. J. Pym, Kripke resource models of a dependently-typed, bunched #-calculus, in preparation, available at http://www.dcs.qmw.ac.uk/ ~ pym.

....the semantics of storage, where sharing rather than duplication is the major concern [12, 13, 14, 15] The demands of the specific models there require a logic different from extant linear or relevant logics. A similar view, from a type theoretic, or logical frameworks, perspective, can be seen in [16, 17]. The other, more foundational, motivation comes from a proposal of resource as a primitive concept, among others, in the emerging discipline of informatics, which can be defined as the science of the structure, complexity and communication of information [3] The treatment of resource in our ....

....and to BI. So, if substructural logics give an impoverished method of talking about resource consumption, where does that leave us (Again we emphasize that we are speaking of logic here, not type theory, where the consumptive aspect has played a useful role in the discovery of interesting models [37, 17] or type theories [38, 16] This brings us to permanent or long lived resources. Examples include files, storage and external devices, where sharing rather than consumption is the operative concept. Notice how temporary and permanent resources are sometimes linked: the state of the store is ....

S. S. Ishtiaq and D. J. Pym. Kripke resource models of a dependently-typed, bunched -calculus. In Proceedings of CSL'99, pages 235--249. Springer Verlag, LNCS 1683, 1999.

....eld College University of London fsi,pymg dcs.qmw.ac. uk October 12, 2000 Abstract This document contains corrections to errors discovered to date in, and also some remarks upon, both Samin Ishtiaq s thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document contains corrections to errors discovered to date in both Samin Ishtiaq s thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from ....

....[Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document contains corrections to errors discovered to date in both Samin Ishtiaq s thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. The postscript version of the thesis, available http: www.dcs.qmw.ac.uk si, is the source of the hard bound copies submitted to the libraries of the University of London (Senate House) and Queen Mary and West eld College. A copy of this document ....

[Article contains additional citation context not shown here]

S Ishtiaq and DJ Pym. Kripke Resource Models of a DependentlyTyped, Bunched -calculus. In M Rodrguez-Artalejo J Flum, editor, Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 235-249. Springer-Verlag, 1999.

....the semantics of storage, where sharing rather than duplication is the major concern [11, 12, 13, 14] The demands of the specific models there require a logic different from extant linear or relevant logics. A similar view, from a type theoretic, or logical frameworks, perspective, can be seen in [15, 16]. In this paper, the spatial view of semantics arises in the more abstract setting of topological models of BI (q.v. x 5) The other, more foundational, motivation comes from a proposal of resource as a primitive concept, among others, in the emerging discipline of informatics, which can be ....

....and to BI. So, if substructural logics give an impoverished method of talking about resource consumption, where does that leave us (Again we emphasize that we are speaking of logic here, not type theory, where the consumptive aspect has played a useful role in the discovery of interesting models [29, 16] or type theories [30, 15] This brings us to permanent or long lived resources. Examples include files, storage and external devices, where sharing rather than consumption is the operative concept. Notice how temporary and permanent resources are sometimes linked: the state of the store is ....

S. S. Ishtiaq and D. J. Pym. Kripke resource models of a dependently-typed, bunched -calculus. In Proceedings of CSL'99, pages 235--249. Springer Verlag, LNCS 1683, 1999.

....idea, inspired by [71] is that there be a surjection from consequences ffi : X; J 1 (OE 1 ) Jm (OE m ) L J(OE) in L, where ffi is the proof object in L, to consequences Gamma X ; y 1 : J 1 (OE 1 ) y m : Jm (OE m ) Sigma L M ffi : J(OE) in Sigma L . We can observe [141, 86, 85] that linear logic and other relevant (or substructural) logics cannot be uniformly encoded in LF. To obtain such an encoding it is necessary to move to the RLF logical framework which, as described above, uses the calculus, together with judgements as types. s linear dependent function space ....

S.S. Ishtiaq and D.J. Pym. Kripke resource models of a dependently-typed, bunched - calculus. In Preparation.

....logic which has multiplicative quantifiers. That is, apart from Ambler s system for the existential only [2] the formulation of which is somewhat more complex than ours. Also relevant is the theory of multiplicative dependent function types in [21] together with its fibrational semantics [20]; incidentally, it can be regarded as relying on a version of bunches appropriate to dependent type theory. Acknowledgments We are grateful to Pablo Armelin, Guy McCusker, Peter Schroeder Heister and Edmund Robinson for useful discussions about this work. The partial support of the UK EPSRC is ....

S.S. Ishtiaq and D.J. Pym. Kripke resource models of a dependently-typed, bunched -calculus. In Preparation. Available at http://www.dcs.qmw.ac.uk/~pym.

....worked out predicate logic with multiplicative quantifiers, apart from Ambler s system for the existential only [2] the formulation of which is somewhat more complex than ours. Also relevant is the theory of multiplicative dependent function types in [15] together with its fibrational semantics [14]; it can be regarded as relying on a version of bunches appropriate to dependent type theory. Linear logic has a number of computational readings, including: the number of uses reading, based on the original coherence space model; an eager and lazy evaluation reading, based on a strict function ....

S.S. Ishtiaq and D.J. Pym. Kripke resource models of a dependently-typed, bunched -calculus. Manuscript, 1999. http://www.dcs.qmw.ac.uk/~pym.

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