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Possible Worlds and Resources: The Semantics of BI
 THEORETICAL COMPUTER SCIENCE
, 2003
"... The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to a ..."
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The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to arise from two main perspectives. On the one hand, from prooftheoretic or categorical concerns and, on the other, from a possibleworlds semantics based on preordered (commutative) monoids. This semantics may be motivated from a basic model of the notion of resource. We explain BI's prooftheoretic, categorical and semantic origins. We discuss in detail the question of completeness, explaining the essential distinction between BI with and without ? (the unit of _). We give an extensive discussion of BI as a semantically based logic of resources, giving concrete models based on Petri nets, ambients, computer memory, logic programming, and money.
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
Classifying Toposes for First Order Theories
 Annals of Pure and Applied Logic
, 1997
"... By a classifying topos for a firstorder theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) firstorder theory has a classifying topos in this sense, but we characterize those which ..."
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By a classifying topos for a firstorder theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) firstorder theory has a classifying topos in this sense, but we characterize those which do by an appropriate `smallness condition', and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every firstorder theory has a conservative extension to one which possesses a classifying topos, and we obtain a Heytingvalued completeness theorem for infinitary firstorder logic.
Lambda Definability with Sums via Grothendieck Logical Relations
, 1999
"... . We introduce a notion of Grothendieck logical relation and use it to characterise the definability of morphisms in stable bicartesian closed categories by terms of the simplytyped lambda calculus with finite products and finite sums. Our techniques are based on concepts from topos theory, how ..."
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. We introduce a notion of Grothendieck logical relation and use it to characterise the definability of morphisms in stable bicartesian closed categories by terms of the simplytyped lambda calculus with finite products and finite sums. Our techniques are based on concepts from topos theory, however our exposition is elementary. Introduction The use of logical relations as a tool for characterising the definable elements in a model of the simplytyped calculus originated in the work of Plotkin [10], who obtained such a characterisation of the definable elements in the full type hierarchy using a notion of Kripke logical relation. Subsequently, the more general notion of a Kripke logical relation of varying arity was developed by Jung and Tiuryn, and shown to characterise the definable elements in any Henkin model [4]. Although not emphasised in [4], relations of varying arity are powerful enough to characterise relative definability with respect to any given set of elements con...
An Elementary Definability Theorem for First Order Logic
"... this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S ae M (i.e., a ..."
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Cited by 4 (1 self)
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this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S ae M (i.e., a subset S = fa j M j= '(a)g defined by some formula ') is invariant under all automorphisms of M . The same is of course true for subsets of M
A Topological Model of Ultrafilters
, 1997
"... this paper is what should be a formal description of non separable topological spaces. This question seems to apply as well to the description of StoneCech compactification described in [2]. This note is a further instance of the use of topological models in prooftheoretic problems (see [7] for an ..."
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this paper is what should be a formal description of non separable topological spaces. This question seems to apply as well to the description of StoneCech compactification described in [2]. This note is a further instance of the use of topological models in prooftheoretic problems (see [7] for another example). We hope to have shown that this can be seen as an illustration of Hilbert's program [12], reformulated in a constructive framework: nonprincipal ultrafilters are ideal objects, that can be eliminated, here by using suitable topological models, in any given proof of a concrete statement. Acknowledgement
A Boolean Model of Ultrafilters
"... We introduce the notion of Boolean measure algebra. It can be described shortly using some standard notations and terminology. If B is any Boolean algebra, let B N denote the algebra of sequences (xn); xn 2 B: Let us write pk 2 B N the sequence such that pk (i) = 1 if i k and pk (i) = 0 if k ! ..."
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We introduce the notion of Boolean measure algebra. It can be described shortly using some standard notations and terminology. If B is any Boolean algebra, let B N denote the algebra of sequences (xn); xn 2 B: Let us write pk 2 B N the sequence such that pk (i) = 1 if i k and pk (i) = 0 if k ! i: If x 2 B, denote by x 2 B N the constant sequence x = (x; x; x; : : :): We define a Boolean measure algebra to be a Boolean algebra B with an operation : B N ! B such that (pk ) = 0 and (x ) = x. Any Boolean measure algebra can be used to model non principal ultrafilters in a suitable sense. Also, we can build effectively the initial Boolean measure algebra. This construction is related to the closed open Ramsey Theorem [8]. AMS Class. 03C90 (03F65 05D10 06E 54A05) Keywords: ultrafilters, boolean algebras, boolean models, Ramsey Theorem Introduction Nonprincipal ultrafilters over natural numbers constitute a typical example of objects that cannot be described effecti...
Parametric Sheaves for modelling Store Locality
"... In this paper, we bring together two important ideas in the semantics of Algollike imperative programming languages. One is that program phrases act on fixed sets of storage locations. The second is that the information of local variables is hidden from client programs. This involves combining ..."
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In this paper, we bring together two important ideas in the semantics of Algollike imperative programming languages. One is that program phrases act on fixed sets of storage locations. The second is that the information of local variables is hidden from client programs. This involves combining sheaf theory and parametricity to produce new classes of sheaves. We define the semantics of an Algollike language using such sheaves and discuss the reasoning principles validated by the semantics.