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16
The Meaning of Types  From Intrinsic to Extrinsic Semantics
 Department of Computer Science, University of Aarhus
, 2000
"... A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic " if all phrases have meanings that are independent of their t ..."
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A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic " if all phrases have meanings that are independent of their typings, while typings represent properties of these meanings. For a simply typed lambda calculus, extended with recursion, subtypes, and named products, we give an intrinsic denotational semantics and a denotational semantics of the underlying untyped language. We then establish a logical relations theorem between these two semantics, and show that the logical relations can be "bracketed" by retractions between the domains of the two semantics. From these results, we derive an extrinsic semantics that uses partial equivalence relations.
Categorical and domain theoretic models of parametric polymorphism
, 2005
"... We present a domaintheoretic model of parametric polymorphism based on admissible per’s over a domaintheoretic model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkin’s logic for parametricity, by the construction of an LAPLstructure as defined by the authors ..."
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Cited by 9 (6 self)
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We present a domaintheoretic model of parametric polymorphism based on admissible per’s over a domaintheoretic model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkin’s logic for parametricity, by the construction of an LAPLstructure as defined by the authors in [7, 5]. This construction gives formal proof of solutions to a large class of recursive domain equations, which we explicate. As an example of a computation in the model, we explicitly describe the natural numbers object obtained using parametricity. The theory of admissible per’s can be considered a domain theory for (impredicative) polymorphism. By studying various categories of admissible and chain complete per’s and their relations, we discover a picture very similar to that of domain theory. 1
What do Types Mean?  From Intrinsic to Extrinsic Semantics
, 2001
"... A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic" if all phrases have meanings that are independent of their ..."
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Cited by 7 (0 self)
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A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic" if all phrases have meanings that are independent of their typings, while typings represent properties of these meanings.
The Scott model of Linear Logic is the extensional collapse of its relational model
, 2011
"... We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus. ..."
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Cited by 3 (2 self)
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We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus.
Categorical semantics of linear logic
 In: Interactive Models of Computation and Program Behaviour, Panoramas et Synthèses 27, Société Mathématique de France 1–196
, 2009
"... Proof theory is the result of a short and tumultuous history, developed on the periphery of mainstream mathematics. Hence, its language is often idiosyncratic: sequent calculus, cutelimination, subformula property, etc. This survey is designed to guide the novice reader and the itinerant mathematic ..."
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Proof theory is the result of a short and tumultuous history, developed on the periphery of mainstream mathematics. Hence, its language is often idiosyncratic: sequent calculus, cutelimination, subformula property, etc. This survey is designed to guide the novice reader and the itinerant mathematician along a smooth and consistent path, investigating the symbolic mechanisms of cutelimination, and their algebraic transcription as coherence diagrams in categories with structure. This spiritual journey at the meeting point of linguistic and algebra is demanding at times, but also pleasantly rewarding: to date, no language (either formal or informal) has been studied by mathematicians as thoroughly as the language of proofs. We start the survey by a short introduction to proof theory (Chapter 1) followed by an informal explanation of the principles of denotational semantics (Chapter 2) which we understand as a representation theory for proofs – generating algebraic invariants modulo cutelimination. After describing in full detail the cutelimination procedure of linear logic (Chapter 3), we explain how to transcribe it into the language of categories with structure. We review
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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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.
Programming with a Quantum Stack
, 2007
"... This thesis presents the semantics of quantum stacks and a functional quantum programming language, LQPL. An operational semantics for LQPL based on quantum stacks in the form of a term logic is developed and used as an interpretation of quantum circuits. The operational semantics is then extend ..."
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This thesis presents the semantics of quantum stacks and a functional quantum programming language, LQPL. An operational semantics for LQPL based on quantum stacks in the form of a term logic is developed and used as an interpretation of quantum circuits. The operational semantics is then extended to handle recursion and algebraic datatypes. Recursion and datatypes are not concepts found in quantum circuits, but both are generally required for modern programming languages. The language LQPL is introduced in a discussion and example format. Various example programs using both classical and quantum algorithms are used to illustrate features of the language. Details of the language, including handling of qubits, general data types and classical data are covered. The quantum stack machine is then presented. Supporting data for operation of the machine are introduced and the transitions induced by the machine’s instructions are given.
A Completeness Theorem for Symmetric Product Phase Spaces
, 2000
"... In a previous work with Antonio Bucciarelli, we introduced indexed linear logic as a tool for studying and enlarging the denotational semantics of linear logic. In particular, we showed how to dene new denotational models of linear logic using symmetric product phase models (truthvalue models) of in ..."
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In a previous work with Antonio Bucciarelli, we introduced indexed linear logic as a tool for studying and enlarging the denotational semantics of linear logic. In particular, we showed how to dene new denotational models of linear logic using symmetric product phase models (truthvalue models) of indexed linear logic. We present here a sequent calculus of indexed linear logic which strictly extends the system LL(I) presented in [BE99] and for which the symmetric product phase spaces provide a complete semantics. We study the connection between this new system and LL(I).
Non uniform (hyper/multi)coherence spaces
, 2008
"... In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs. Intuitively, vertices represent results of computations and the edge relation witnesses the ability of being assembled into a same piece of data or a same (strongly) stable function, at arrow types. In (hyper)coherence semanti ..."
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In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs. Intuitively, vertices represent results of computations and the edge relation witnesses the ability of being assembled into a same piece of data or a same (strongly) stable function, at arrow types. In (hyper)coherence semantics, the argument of a (strongly) stable functional is always a (strongly) stable function. As a consequence, comparatively to the relational semantics, where there is no edge relation, some vertices are missing. Recovering these vertices is essential for the purpose of reconstructing proofs/terms from their interpretations. It shall also be useful for the comparison with other semantics, like game semantics. In [BE01], Bucciarelli and Ehrhard introduced a so called non uniform coherence space semantics where no vertex is missing. By constructing the cofree exponential we set a new version of this last semantics, together with non uniform versions of hypercoherences and multicoherences, a new semantics where an edge is a finite multiset. Thanks to the cofree construction, these non uniform semantics are deterministic in the sense that the intersection of a clique and of an anticlique contains at most one vertex, a result of interaction, and extensionally collapse onto the corresponding uniform semantics.