Iteration Theories

Bloom, Stephen L.; Labella, Anna; Ésik, Zoltán; Manes, Ernest G.

doi:10.1007/978-3-642-78034-9_7

Cited by 48 publications

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“…X diagonal property Another equivalent axiomatisation of Conway fixed-point operators is by the Bekić property, which says that the fixed point of f : † We should note that mathematically equivalent observations had been made by several authors before the notion of traced monoidal categories was introduced, in particular by Bloom andÉsik (Bloom andÉsik 1993) and Ş tefǎnescu (Ş tefǎnescu 2000).…”

Section: Trace-fixpoint Correspondencementioning

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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Section: Trace-fixpoint Correspondencementioning

confidence: 99%

On traced monoidal closed categories

Hasegawa

2009

Math. Struct. Comp. Sci.

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“…Other applications of the equational logic of fixed points include proofs of Kleene's theorem and its generalizations [4] (see also [20,21,5], where the presentation is not fully based on equational reasoning), and of Greibach's normal form theorem for context-free grammars [9]. The methods employed in the papers [19,8] even indicate that one can embed the proof of the Krohn-Rhodes decomposition theorem [11] for finite automata and semigroups within equational logic.…”

Section: For Every µ-Term T There Exists a Rational Term R Such That mentioning

confidence: 99%

A Fully Equational Proof of Parikh's Theorem

Aceto

Ésik

Ingólfsdóttir

2002

RAIRO-Theor. Inf. Appl.

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“…The word "total" is due to Filinski [4], though in his original definition h v is asked to be a value rather than a central term. 1 Remark 2. The expressive power of an iterator is not so weak, as we can derive a general feedback operator feedback σ,τ : (σ → σ + τ ) → σ → τ from an iterator using sums and first-class controls, which satisfies (with a syntax sugar for sums)…”

Section: Recursion From Iterationmentioning

confidence: 99%

“…In particular, we note that the dinaturality loop F f g)). These can be seen axiomatizing the call-by-value counterpart of the Conway theories [1,9]. One may further consider the call-by-value version of the Bekic property (another equivalent axiomatization of these properties [9]) along this line, which could be used for reasoning about mutual recursion.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

“…While the equational theories of fixpoint operators in call-by-name programming languages and in domain theory have been extensively studied and now there are some canonical axiomatizations (including the iteration theories [1] and Conway theories, equivalently traced cartesian categories [9] -see [18] for the latest account), there seems no such widely-accepted result in the context of call-by-value (cbv) programming languages. In this paper we propose a candidate of such an axiomatization, which consists of three simple axioms.…”

Section: Introductionmentioning

confidence: 99%

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Axioms for Recursion in Call-by-Value

Hasegawa

Kakutani

2001

Lecture Notes in Computer Science

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Abstract.We propose an axiomatization of fixpoint operators in typed call-by-value programming languages, and give its justifications in two ways. First, it is shown to be sound and complete for the notion of uniform T -fixpoint operators of Simpson and Plotkin. Second, the axioms precisely account for Filinski's fixpoint operator derived from an iterator (infinite loop constructor) in the presence of first-class controls, provided that we define the uniformity principle on such an iterator via a notion of effect-freeness (centrality). We also investigate how these two results are related in terms of the underlying categorical models.

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Iteration Theories

Cited by 48 publications

References 0 publications

On traced monoidal closed categories

On traced monoidal closed categories

A Fully Equational Proof of Parikh's Theorem

Axioms for Recursion in Call-by-Value

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