Functorial Boxes in String Diagrams

Melliès, Paul-André

doi:10.1007/11874683_1

Cited by 42 publications

(60 citation statements)

References 39 publications

(53 reference statements)

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“…Since Adm is traced and the linear exponential comonad of Adm is idempotent, the statement holds by Remark 5.1 in [17]. For the proof, see [17,21].…”

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confidence: 84%

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A Modified GoI Interpretation for a Linear Functional Programming Language and Its Adequacy

Hoshino

2011

Foundations of Software Science and Computational Structures

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Abstract. Geometry of Interaction (GoI) introduced by Girard provides a semantics for linear logic and its cut elimination. Several extensions of GoI to programming languages have been proposed, but it is not discussed to what extent they capture behaviour of programs as far as the author knows. In this paper, we study GoI interpretation of a linear functional programming language (LFP). We observe that we can not extend the standard GoI interpretation to an adequate interpretation of LFP, and we propose a new adequate GoI interpretation of LFP by modifying the standard GoI interpretation. We derive the modified interpretation from a realizability model of LFP. We also relate the interpretation of recursion to cyclic computation (the trace operator in the category of sets and partial maps) in the realizability model.

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“…Since Adm is traced and the linear exponential comonad of Adm is idempotent, the statement holds by Remark 5.1 in [17]. For the proof, see [17,21].…”

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confidence: 84%

“…The following theorem shown in [17,21] relates a trace operator on a linear category to an interpretation of recursion. In this paper, we are interested in the traced symmetric monoidal category Pfn of sets and partial maps and the compact closed category Int(Pfn).…”

Section: Definitionmentioning

confidence: 99%

A Modified GoI Interpretation for a Linear Functional Programming Language and Its Adequacy

Hoshino

2011

Foundations of Software Science and Computational Structures

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“…We start by defining the message passing circuits we use in the compilation. These circuits may be understood as a particular instance of string diagrams for monoidal categories [13]. The are also related to proof nets, see [13] for a discussion.…”

Section: Reducing Upper Class To Working Classmentioning

confidence: 99%

Functional Programming in Sublinear Space

Lago

Schöpp²

2010

Programming Languages and Systems

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Abstract. We consider the problem of functional programming with data in external memory, in particular as it appears in sublinear space computation. Writing programs with sublinear space usage often requires one to use special implementation techniques for otherwise easy tasks, e.g. one cannot compose functions directly for lack of space for the intermediate result, but must instead compute and recompute small parts of the intermediate result on demand. In this paper, we study how the implementation of such techniques can be supported by functional programming languages.Our approach is based on modeling computation by interaction using the Int construction of Joyal, Street & Verity. We derive functional programming constructs from the structure obtained by applying the Int construction to a term model of a given functional language. The thus derived functional language is formulated by means of a type system inspired by Baillot & Terui's Dual Light Affine Logic. We assess its expressiveness by showing that it captures LOGSPACE.

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“…Using the box notation (Cockett and Seely 1999;Melliès 2006), this f † can be nicely expressed as follows:…”

Section: Linear Fixed-points: Recursion From Cyclic Sharing Revisitedmentioning

confidence: 99%

“…More concretely, consider the compact closed category Rel of sets and relations, with the powerset comonad as the linear exponential comonad -this has a linear fixed-point operator (Hasegawa 1997;1999) but no non-linear one (cf. Melliès (2006)). In Katsumata's work, the adjunction N U of Theorem 4.1 provides the equivalence between attribute grammars and synthesised attribute grammars (attribute grammars with just bottom-up information flow).…”

Section: Linear Fixed-points: Recursion From Cyclic Sharing Revisitedmentioning

confidence: 99%

On traced monoidal closed categories

Hasegawa

2009

Math. Struct. Comp. Sci.

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The structure theorem of Joyal, Street and Verity says that every traced monoidal category C arises as a monoidal full subcategory of the tortile monoidal category Int C. In this paper we focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into Int C has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixed-point computation. To make the paper more self-contained, we also include various background results for traced monoidal categories. IntroductionIn Joyal et al. (1996), Joyal, Street and Verity introduced the notion of traced monoidal categories. They showed that every traced monoidal category C fully faithfully embeds in a tortile monoidal category Int C, and that this Int-construction gives a left biadjoint of the forgetful 2-functor from the 2-category of tortile monoidal categories to that of traced monoidal categories. This remarkable result attracted much attention from theoretical computer scientists, particularly in connection with linear logic (Girard 1987) and the Geometry of Interaction (Girard 1989;Abramsky and Jagadeesan 1994;Abramsky 1996;Haghverdi 2000;Abramsky et al. 2002;Haghverdi and Scott 2006), where the special case of traced symmetric monoidal categories and compact closed categories (Kelly and Laplaza 1980) is of interest.In this paper we shall see that the monoidal closed structure can be tied with the Int-construction in an unexpected manner. Namely, we show the following result (Theorem 4.1). Theorem.A traced monoidal category C is closed if and only if the embedding from C into Int C has a right adjoint.Despite its simplicity, to the best of our knowledge, this fact has not previously been pointed out in the literature. Perhaps this is partly because the Int-construction works too nicely: tortile monoidal categories are closed, therefore every traced monoidal category embeds into a closed one just via the Int-construction. So it seems that for this reason it was not felt that traced monoidal closed categories were themselves interesting (an exception being Coccia et al. (2002), where traced monoidal closed categories with extra structure are used for modelling higher-order cyclic shared structures). However, the tortile structure (or compact closed structure in the symmetric case) just describes a very special http://journals.cambridge.org M. Hasegawa 218 kind of closedness, and not every traced monoidal closed category is tortile. Our result says that the relation between tortile structure and general closed structure is still in harmony: every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. The embedding from C to Int C may not preserve the closed structure (see Remark 4.2 for a counterexample), but the closed structure on C is determined by the closed struct...

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Functorial Boxes in String Diagrams

Cited by 42 publications

References 39 publications

A Modified GoI Interpretation for a Linear Functional Programming Language and Its Adequacy

A Modified GoI Interpretation for a Linear Functional Programming Language and Its Adequacy

Functional Programming in Sublinear Space

On traced monoidal closed categories

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