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19
Deliverables: A Categorical Approach to Program Development in Type Theory
, 1992
"... This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's ..."
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Cited by 24 (1 self)
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This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigmatypes of the calculus enable us to achieve this. There are many similarities with the subset interpretation of MartinLof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...
Entailment Relations and Distributive Lattices
, 1998
"... . To any entailment relation [Sco74] we associate a distributive lattice. We use this to give a construction of the product of lattices over an arbitrary index set, of the Vietoris construction, of the embedding of a distributive lattice in a boolean algebra, and to give a logical description of ..."
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Cited by 18 (4 self)
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. To any entailment relation [Sco74] we associate a distributive lattice. We use this to give a construction of the product of lattices over an arbitrary index set, of the Vietoris construction, of the embedding of a distributive lattice in a boolean algebra, and to give a logical description of some spaces associated to mathematical structures. 1 Introduction Most spaces associated to mathematical structures: spectrum of a ring, space of valuations of a field, space of bounded linear functionals, . . . can be represented as distributive lattices. The key to have a natural definition in these cases is to use the notion of entailment relation due to Dana Scott. This note explains the connection between entailment relations and distributive lattices. An entailment relation may be seen as a logical description of a distributive lattice. Furthermore, most operations on distributive lattices are simpler when formulated as operations on entailment relations. A special kind of distribu...
Topological Completeness for HigherOrder Logic
 Journal of Symbolic Logic
, 1997
"... Using recent results in topos theory, two systems of higherorder logic are shown to be complete with respect to sheaf models over topological spacessocalled "topological semantics". The first is classical higherorder logic, with relational quantification of finitely high type; the second sy ..."
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Cited by 8 (3 self)
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Using recent results in topos theory, two systems of higherorder logic are shown to be complete with respect to sheaf models over topological spacessocalled "topological semantics". The first is classical higherorder logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.
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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Cited by 7 (3 self)
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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.
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 has a pla ..."
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Cited by 6 (0 self)
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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.
Minimal Invariant Spaces in Formal Topology
 The Journal of Symbolic Logic
, 1996
"... this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a ..."
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Cited by 4 (1 self)
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this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a suitable formal topology, and the "existence" of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is "similar in structure" to the topological proof [6, 8], but which uses a simple algebraic remark (proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements. 1 Construction of Minimal Invariant Subspace
Syntax and Semantics of the logic ...
, 1997
"... In this paper we study the logic L !! , which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck t ..."
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Cited by 4 (0 self)
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In this paper we study the logic L !! , which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of L !! with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting valued models. The logic L !! is the strongest for which Heyting valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes.
To appear in Structural Foundations of Quantum Gravity,
, 2004
"... General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n − 1)dimensional manifolds representing ‘space ’ and whose morphisms are ndimensiona ..."
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Cited by 3 (0 self)
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General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n − 1)dimensional manifolds representing ‘space ’ and whose morphisms are ndimensional cobordisms representing ‘spacetime’. Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe ‘states’, and whose morphisms are bounded linear operators used to describe ‘processes’. Moreover, the categories nCob and Hilb resemble each other far more than either resembles Set, the category whose objects are sets and whose morphisms are functions. In particular, both Hilb and nCob but not Set are ∗categories with a noncartesian monoidal structure. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat Hilb as analogous to Set rather than nCob, so that quantum theory will make more sense when regarded as part of a theory of spacetime. 1