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14
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
The Algebraic LambdaCalculus
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2009
"... We introduce an extension of the pure lambdacalculus by endowing the set of terms with a structure of vector space, or more generally of module, over a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with value ..."
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Cited by 6 (1 self)
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We introduce an extension of the pure lambdacalculus by endowing the set of terms with a structure of vector space, or more generally of module, over a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vector space. We then study a natural extension of betareduction in this setting: we prove it is confluent, then discuss consistency and conservativity over the ordinary lambdacalculus. We also provide normalization results for a simple type system.
Events, Causality and Symmetry
, 2008
"... The article discusses causal models, such as Petri nets and event structures, how they have been rediscovered in a wide variety of recent applications, and why they are fundamental to computer science. A discussion of their present limitations leads to their extension with symmetry. The consequences ..."
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Cited by 4 (2 self)
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The article discusses causal models, such as Petri nets and event structures, how they have been rediscovered in a wide variety of recent applications, and why they are fundamental to computer science. A discussion of their present limitations leads to their extension with symmetry. The consequences, actual and potential, are discussed.
When Is a Container a Comonad?
"... Abstract. Abbott, Altenkirch, Ghani and others have taught us that many parameterized datatypes (set functors) can be usefully analyzed via container representations in terms of a set of shapes and a set of positions in each shape. This paper builds on the observation that datatypes often carry addi ..."
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Cited by 2 (2 self)
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Abstract. Abbott, Altenkirch, Ghani and others have taught us that many parameterized datatypes (set functors) can be usefully analyzed via container representations in terms of a set of shapes and a set of positions in each shape. This paper builds on the observation that datatypes often carry additional structure that containers alone do not account for. We introduce directed containers to capture the common situation where every position in a datastructure determines another datastructure, informally, the subdatastructure rooted by that position. Some natural examples are nonempty lists and nodelabelled trees, and datastructures with a designated position (zippers). While containers denote set functors via a fullyfaithful functor, directed containers interpret fullyfaithfully into comonads. But more is true: every comonad whose underlying functor is a container is represented by a directed container. In fact, directed containers are the same as containers that are comonads. We also describe some constructions of directed containers. We have formalized our development in the dependently typed programming language Agda. 1
Transport of finiteness structures and applications
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... ..."
Differential Linear Logic and Polarization
"... We study an extension of EhrhardRegnier's differential linear logic along the lines of Laurent's polarization. We show that a particular object of the wellknown relational model of linear logic provides a denotational semantics for this new system, which canonically extends the semantics of both ..."
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Cited by 1 (1 self)
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We study an extension of EhrhardRegnier's differential linear logic along the lines of Laurent's polarization. We show that a particular object of the wellknown relational model of linear logic provides a denotational semantics for this new system, which canonically extends the semantics of both differential and polarized linear logics: this justifies our choice of cut elimination rules. Then we show this new system models the recently introduced convolution _*ucalculus, the same as linear logic decomposes calculus.
Di erential linear logic and polarization
 in Curien (2009
"... We extend Ehrhard Regnier's di erential linear logic along the lines of Laurent's polarization. We provide a denotational semantics of this new system in the wellknown relational model of linear logic, extending canonically the semantics of both di erential and polarized linear logics: this justi e ..."
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Cited by 1 (0 self)
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We extend Ehrhard Regnier's di erential linear logic along the lines of Laurent's polarization. We provide a denotational semantics of this new system in the wellknown relational model of linear logic, extending canonically the semantics of both di erential and polarized linear logics: this justi es our choice of cut elimination rules. Then we show this polarized di erential linear logic re nes the recently introduced convolution ¯λµcalculus, the same as linear logic decomposes λcalculus. 1
A NonUniform Finitary Relational Semantics of System T
, 2009
"... We study iteration and recursion operators in the denotational semantics of typed λcalculi derived from the multiset relational model of linear logic. Although these operators are defined as fixpoints of typed functionals, we prove them finitary in the sense of Ehrhard’s finiteness spaces. 1 ..."
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We study iteration and recursion operators in the denotational semantics of typed λcalculi derived from the multiset relational model of linear logic. Although these operators are defined as fixpoints of typed functionals, we prove them finitary in the sense of Ehrhard’s finiteness spaces. 1
The Computational Meaning of Probabilistic Coherence Spaces
"... Abstract—We study the probabilistic coherent spaces — a denotational semantics interpreting programs by power series with non negative real coefficients. We prove that this semantics is adequate for a probabilistic extension of the untyped λcalculus: the probability that a term reduces to a head no ..."
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Abstract—We study the probabilistic coherent spaces — a denotational semantics interpreting programs by power series with non negative real coefficients. We prove that this semantics is adequate for a probabilistic extension of the untyped λcalculus: the probability that a term reduces to a head normal form is equal to its denotation computed on a suitable set of values. The result gives, in a probabilistic setting, a quantitative refinement to the adequacy of Scott’s model for untyped λcalculus. I.
MFPS 2009 Continuous Functions on Final Coalgebras
"... In a previous paper we gave a representation of, and simultaneously a way of programming with, continuous functions on streams, whether discretevalued functions, or functions between streams. We also defined a combinator on the representations of such continuous functions that reflects composition. ..."
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In a previous paper we gave a representation of, and simultaneously a way of programming with, continuous functions on streams, whether discretevalued functions, or functions between streams. We also defined a combinator on the representations of such continuous functions that reflects composition. Streams are one of the simplest examples of nontrivial final coalgebras. Here we extend our previous results to cover the case of final coalgebras for a broader class of functors than that giving rise to streams. Among the functors we can 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’, a la Baire space. The datatype of prefixes is defined together with the set of ‘growth points ’ in a prefix, simultaneously. This we call beheading. To program and reason about representations of continuous functions requires a language whose type system incorporates the dependent function and pair types, inductive definitions at types Set, I → Set and (Σ I: Set) Set I, coinductive definitions at types Set and I → Set, as well as universal arrows for such definitions. Keywords: Continuous functions, final coalgebras, containers