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Models of Sharing Graphs: A Categorical Semantics of let and letrec
, 1997
"... To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sha ..."
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Cited by 60 (9 self)
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To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is firstorder acyclic sharing graphs represented by letsyntax, and others are extensions with higherorder constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced
The maximality of the typed lambda calculus and of cartesian closed categories
 Publ. Inst. Math. (N.S
"... From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here ..."
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Cited by 17 (2 self)
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From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson.
Finite dimensional vector spaces are complete for traced symmetric monoidal categories
 in: Pillars of Computer Science: Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday, Lecture Notes in Computer Science 4800 (2008
"... Abstract. We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two differe ..."
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Cited by 7 (0 self)
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Abstract. We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two different arrows in the free traced category there always exists astrongtracedfunctorintoFinVectk which distinguishes them. Therefore two arrows in the free traced category are the same if and only if they agree for all interpretations in FinVectk. 1
The Maximality of Cartesian Categories
, 1999
"... It is proved that equations between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equation in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, w ..."
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It is proved that equations between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equation in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, which may lack a terminal object. The proof is based on a coherence result for cartesian categories, which is related to modeltheoretical methods of normalization. The equations between arrows assumed for cartesian categories are maximal in the sense that extending them with new equations collapses the categories into preorders (i.e. categories in which between any two objects there is at most one arrow). The equations envisaged for the extension are in the language of free cartesian categories generated by sets of objects, and variables for arrows don’t occur in them. If such an equation doesn’t hold in the free cartesian category generated by a set of objects, then any cartesian category in which this equation holds is a preorder. An analogous result is provable for categories with binary products, which differ from cartesian categories in not necessarily having a terminal object. The proof of these results, which we are going to present below, is based on a coherence property of cartesian categories. This coherence, which is ultimately inspired by the geometric modelling of categories of [3], is related to modeltheoretic methods of normalization. It permits to establish uniqueness of normal form for arrow terms without proceeding via the ChurchRosser property for reductions. It also yields an easy decision procedure for the commuting of diagrams in free cartesian categories.
Conditions for the Completeness of Functional and Algebraic Equational Reasoning
, 1998
"... We consider the following question: in the simplytyped lcalculus with algebraic operations, is the set of equations valid in a particular model exactly those provable from (b), (h), and the set of algebraic equations, E, that are valid in the model? We find conditions for determining whether bhEe ..."
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We consider the following question: in the simplytyped lcalculus with algebraic operations, is the set of equations valid in a particular model exactly those provable from (b), (h), and the set of algebraic equations, E, that are valid in the model? We find conditions for determining whether bhEequational reasoning is complete. We demonstrate the utility of the results by presenting a number of simple corollaries for particular models. 1 Introduction The two axioms of the lcalculus, (b) ((lx: M) N) =M[x := N] (h) (lx: M x) =M; if x not free in M lie at the heart of reasoning about functional programs: (b) explains function application syntactically, and (h) states that the meaning of functions can be based solely on their meaning under application. The (b) and (h) axioms turn out to be fundamental: not only are they sound, they also are complete for proving equations that hold in all models of the simplytyped lcalculus [Friedman, 1975]. In other words, an equation between sim...
Topological Representation of the &ambda;Calculus
, 1998
"... The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is al ..."
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The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a "minimal" topological model, in which every continuous function is definable. These results subsume earlier ones using cartesian closed categories, as well as those employing socalled Henkin and Kripke models. Introduction The calculus originates with Church [6]; it is intended as a formal calculus of functional application and specification. In this paper, we are mainly interested in the version known as simply typed calculus ; as is now wellknown, the untyped version can be treated as a special case of this ([17]). We present here a topological representation of the calculus: types are represented by cert...
Bicartesian Coherence
, 2007
"... Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free ..."
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Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free
Bicartesian coherence revisited
, 2008
"... A survey is given of results about coherence for categories with finite products and coproducts. For these results, which were published previously by the authors in several places, some formulations and proofs are here corrected, and matters are updated. The categories investigated in this paper fo ..."
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A survey is given of results about coherence for categories with finite products and coproducts. For these results, which were published previously by the authors in several places, some formulations and proofs are here corrected, and matters are updated. The categories investigated in this paper formalize equality of proofs in classical and intuitionistic conjunctivedisjunctive logic without distribution of conjunction over disjunction.
Coherent bicartesian and sesquicartesian categories
, 2006
"... Coherence is here demonstrated for sesquicartesian categories, which are categories with nonempty finite products and arbitrary finite sums, including the empty sum, where moreover the first and the second projection from the product of the initial object with itself are the same. (Every ..."
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Coherence is here demonstrated for sesquicartesian categories, which are categories with nonempty finite products and arbitrary finite sums, including the empty sum, where moreover the first and the second projection from the product of the initial object with itself are the same. (Every