An order-algebraic definition of knuthian semantics

Chirica, Laurian M.; Martin, David F.

doi:10.1007/bf01744285

Cited by 63 publications

(31 citation statements)

References 17 publications

(47 reference statements)

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“…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). This generalises the result on the equivalence between such attribute grammars formulated in a domain-theoretic setting (Chirica and Martin 1979).…”

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

confidence: 79%

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

confidence: 79%

On traced monoidal closed categories

Hasegawa

2009

Math. Struct. Comp. Sci.

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“…For our tree transformation strategy, where re-evaluations may be restricted to the restructured area, we extend the classical attribute grammar framework by allowing a set of values to be correct for each attribute instance. Each value of such a set should be an approximation of the correct value according to the classical attribute grammar definition [5,[8][9][10]14].…”

Section: Iteration Of Evaluation and Tree Transformation Phasesmentioning

confidence: 99%

Iteration of transformation passes over attributed program trees

Alblas

1989

Acta Informatica

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Summary.Transformations of attributed program trees form an essential part of compiler optimizations. A strategy of repeatedly applying alternate attribute evaluation and tree transformation phases is discussed. An attribute evaluation phase consists of a sequence of passes over the tree. A tree transformation phase consists of a single pass, which is never interrupted to carry out a re-evaluation. Both phases can be performed in parallel. This strategy requires a distinction between consistent (i.e., correct) and approximate attribute values. Tree transformations can be considered safe if they guarantee that the attribute values everywhere in the program tree will remain consistent or will become at least approximations of the consistent values, so that subsequent transformations can be applied correctly.This attribute evaluation and tree transformation strategy shows similarities with the evaluation methods for circular attribute grammars.

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“…We show how this semantics allows us to prove results about attribute grammars in a calculational style. In particular, we give a new proof of Chirica and Martin's result [6], that the attribute values can be computed by a structural recursion over the tree. We also derive a new definedness test, which encompasses the traditional closure and circularity tests.…”

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confidence: 99%

A Functional Semantics of Attribute Grammars

Backhouse

2002

Tools and Algorithms for the Construction and Analysis of Systems

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Abstract.A definition of the semantics of attribute grammars is given, using the lambda calculus. We show how this semantics allows us to prove results about attribute grammars in a calculational style. In particular, we give a new proof of Chirica and Martin's result [6], that the attribute values can be computed by a structural recursion over the tree. We also derive a new definedness test, which encompasses the traditional closure and circularity tests. The test is derived by abstract interpretation.

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An order-algebraic definition of knuthian semantics

Cited by 63 publications

References 17 publications

On traced monoidal closed categories

On traced monoidal closed categories

Iteration of transformation passes over attributed program trees

A Functional Semantics of Attribute Grammars

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