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12
Finiteness spaces
 Mathematical Structures in Computer Science
, 1987
"... We investigate a new denotational model of linear logic based on the purely relational model. In this semantics, webs are equipped with a notion of “finitary ” subsets satisfying a closure condition and proofs are interpreted as finitary sets. In spite of a formal similarity, this model is quite dif ..."
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Cited by 53 (13 self)
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We investigate a new denotational model of linear logic based on the purely relational model. In this semantics, webs are equipped with a notion of “finitary ” subsets satisfying a closure condition and proofs are interpreted as finitary sets. In spite of a formal similarity, this model is quite different from the usual models of linear logic (coherence semantics, hypercoherence semantics, the various existing game semantics...). In particular, the standard fixpoint operators used for defining the general recursive functions are not finitary, although the primitive recursion operators are. This model can be considered as a discrete version of the Köthe space semantics introduced in a previous paper: we show how, given a field, each finiteness space gives rise to a vector space endowed with a linear topology, a notion introduced by Lefschetz in 1942, and we study the corresponding model where morphisms are linear continuous maps (a version of Girard’s quantitative semantics with coefficients in the field). We obtain in that way a new model of the recently introduced differential lambdacalculus. Notations. If S is a set, we denote by M(S) = N S the set of all multisets over S. If µ ∈ M(S), µ denotes the support of µ which is the set of all a ∈ S such that µ(a) ̸ = 0. A multiset is finite if it has a finite support. If a1,..., an are elements of some given set S, we denote by [a1,..., an] the corresponding multiset over S. The usual operations on natural numbers are extended to multisets pointwise. If (Si)i∈I are sets, we denote by πi the ith projection πi: ∏ j∈I Sj → Si.
Categories of Containers
 In Proceedings of Foundations of Software Science and Computation Structures
, 2003
"... Abstract. We introduce the notion of containers as a mathematical formalisation of the idea that many important datatypes consist of templates where data is stored. We show that containers have good closure properties under a variety of constructions including the formation of initial algebras and f ..."
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Cited by 40 (7 self)
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Abstract. We introduce the notion of containers as a mathematical formalisation of the idea that many important datatypes consist of templates where data is stored. We show that containers have good closure properties under a variety of constructions including the formation of initial algebras and final coalgebras. We also show that containers include strictly positive types and shapely types but that there are containers which do not correspond to either of these. Further, we derive a representation result classifying the nature of polymorphic functions between containers. We finish this paper with an application to the theory of shapely types and refer to a forthcoming paper which applies this theory to differentiable types. 1
Continuous functions on final coalgebras
, 2007
"... In a previous paper we have given a representation of continuous functions on streams, both discretevalued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We ga ..."
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Cited by 10 (1 self)
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In a previous paper we have given a representation of continuous functions on streams, both discretevalued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We gave also a combinator on the representations of stream processing functions that reflects composition. Streams are the simplest example of a nontrivial final coalgebras, playing in the coalgebraic realm the same role as do the natural numbers in the algebraic realm. Here we extend our previous results to cover the case of final coalgebras for a broad class of functors generalising (×A). The functors we deal with are those that arise from countable signatures of finiteplace untyped operators. These have many applications. The topology we put on the final coalgebra for such a functor is that induced by taking for basic neighbourhoods the set of infinite objects which share a common prefix, according to the usual definition of the final coalgebra as the limit of a certain inverse chain starting at �. 1
∂ for Data: Differentiating Data Structures
"... This paper and our conference paper (Abbott, Altenkirch, Ghani, and McBride, 2003b) explain and analyse the notion of the derivative of a data structure as the type of its onehole contexts based on the central observation made by McBride (2001). To make the idea precise we need a generic notion of ..."
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Cited by 7 (1 self)
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This paper and our conference paper (Abbott, Altenkirch, Ghani, and McBride, 2003b) explain and analyse the notion of the derivative of a data structure as the type of its onehole contexts based on the central observation made by McBride (2001). To make the idea precise we need a generic notion of a data type, which leads to the notion of a container, introduced in (Abbott, Altenkirch, and Ghani, 2003a) and investigated extensively in (Abbott, 2003). Using containers we can provide a notion of linear map which is the concept missing from McBride’s first analysis. We verify the usual laws of differential calculus including the chain rule and establish laws for initial algebras and terminal coalgebras.
Derivatives of containers
 of Lecture notes in Computer Science
, 2003
"... Abstract. We are investigating McBride’s idea that the type of onehole contexts are the formal derivative of a functor from a categorical perspective. Exploiting our recent work on containers we are able to characterise derivatives by a universal property and show that the laws of calculus includin ..."
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Cited by 7 (2 self)
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Abstract. We are investigating McBride’s idea that the type of onehole contexts are the formal derivative of a functor from a categorical perspective. Exploiting our recent work on containers we are able to characterise derivatives by a universal property and show that the laws of calculus including a rule for initial algebras as presented by McBride hold — hence the differentiable containers include those generated by polynomials and least fixpoints. Finally, we discuss abstract containers (i.e. quotients of containers) — this includes a container which plays the role of e x in calculus by being its own derivative. 1
Short Cut Fusion of Recursive Programs with Computational Effects
 SYMPOSIUM ON TRENDS IN FUNCTIONAL PROGRAMMING 2008 (TFP'08)
, 2008
"... Fusion is the process of improving the efficiency of modularly constructed programs by transforming them into monolithic equivalents. This paper defines a generalization of the standardbuild combinator which expresses uniform production of functorial contexts containing data of inductive types. It a ..."
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Cited by 3 (1 self)
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Fusion is the process of improving the efficiency of modularly constructed programs by transforming them into monolithic equivalents. This paper defines a generalization of the standardbuild combinator which expresses uniform production of functorial contexts containing data of inductive types. It also proves correct a fusion rule which generalizes thefold/build andfold/buildp rules from the literature, and eliminates intermediate data structures of inductive types without disturbing the contexts in which they are situated. An important special case arises when this context is monadic. When it is, a second rule for fusing combinations of producers and consumers via monad operations, rather than via composition, is also available. We give examples illustrating both rules, and consider their coalgebraic duals as well.
On Differential Interaction Nets and the Picalculus
 Preuves, Programmes et Systèmes
, 2006
"... We propose a translation of a finitary (that is, replicationfree) version of the picalculus into promotionfree differential interaction net structures, a linear logic version of the differential lambdacalculus (or, more precisely, of a resource lambdacalculus). For the sake of simplicity only, w ..."
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Cited by 1 (0 self)
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We propose a translation of a finitary (that is, replicationfree) version of the picalculus into promotionfree differential interaction net structures, a linear logic version of the differential lambdacalculus (or, more precisely, of a resource lambdacalculus). For the sake of simplicity only, we restrict our attention to a monadic version of the picalculus, so that the differential interaction net structures we consider need only to have exponential cells. We prove that the nets obtained by this translation satisfy an acyclicity criterion weaker than the standard Girard (or DanosRegnier) acyclicity criterion, and we compare the operational semantics of the picalculus, presented by means of an environment machine, and the reduction of differential interaction nets. Differential interaction net structures being of a logical nature, this work provides a CurryHoward interpretation of processes.
A quantum double construction in Rel
, 2010
"... We study bialgebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of ..."
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We study bialgebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of
21.1 GENERALIZING SHORT CUT FUSION 21.1.1 Introducing Short Cut Fusion
"... Abstract: Fusion is the process of improving the efficiency of modularly constructed programs by transforming them into monolithic equivalents. This paper defines a generalization of the standard build combinator which expresses uniform production of functorial contexts containing data of inductive ..."
Abstract
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Abstract: Fusion is the process of improving the efficiency of modularly constructed programs by transforming them into monolithic equivalents. This paper defines a generalization of the standard build combinator which expresses uniform production of functorial contexts containing data of inductive types. It also proves correct a fusion rule which generalizes the fold / build and fold/buildp rules from the literature, and eliminates intermediate data structures of inductive types without disturbing the contexts in which they are situated. An important special case arises when this context is monadic. When it is, a second rule for fusing combinations of producers and consumers via monad operations, rather than via composition, is also available. We give examples illustrating both rules, and consider their coalgebraic duals as well.
unknown title
, 2010
"... We study bialgebras and Hopf algebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras in Rel. In particular, for any group G, we ..."
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We study bialgebras and Hopf algebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras in Rel. In particular, for any group G, we