Guard Your Daggers and Traces: Properties of Guarded (Co-)recursion

Milius, Stefan; Litak, Tadeusz

doi:10.3233/fi-2017-1475

Cited by 8 publications

(7 citation statements)

References 33 publications

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“…Remark 23. Proposition 22 connects our approach to previous work based precisely on the assumptions of the proposition [28] (in fact, the term guarded traced category is already used there, with different meaning). A limitation of the approach via a functor arises from the need to fix globally, so that, e.g., the ideal guarded structure on metric spaces (Example 21) is not coveredcapturing contractivity via requires fixing a single global contraction factor.…”

Section: Guardedness Via Guarded Idealsmentioning

confidence: 81%

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Guarded Traced Categories

Goncharov

Schröder

2018

Lecture Notes in Computer Science

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Notions of guardedness serve to delineate the admissibility of cycles, e.g. in recursion, corecursion, iteration, or tracing. We introduce an abstract notion of guardedness structure on a symmetric monoidal category, along with a corresponding notion of guarded traces, which are defined only if the cycles they induce are guarded. We relate structural guardedness, determined by propagating guardedness along the operations of the category, to geometric guardedness phrased in terms of a diagrammatic language. In our setup, the Cartesian case (recursion) and the co-Cartesian case (iteration) become completely dual, and we show that in these cases, guarded tracedness is equivalent to presence of a guarded Conway operator, in analogy to an observation on total traces by Hasegawa and Hyland. Moreover, we relate guarded traces to unguarded categorical uniform fixpoint operators in the style of Simpson and Plotkin. Finally, we show that partial traces based on Hilbert-Schmidt operators in the category of Hilbert spaces are an instance of guarded traces.

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Section: Guardedness Via Guarded Idealsmentioning

confidence: 81%

“…Guarded approach (cf. [28]): Extend Cpo to a guarded category: f : X Ŷ Ñ pr 2 Z iff f P {g pid ˆηq | g : X ˆYK Ñ Z} (see Proposition 22), and define a guarded recursion operator sending f " g pid ˆηq : Y ˆX Ñ pr 2 X to f : " g id, f : Y Ñ X with f pyq P X K calculated as the least fixpoint of λz. ηgpy, zq.…”

Section: Guarded Vs Unguarded Recursionmentioning

confidence: 99%

Guarded Traced Categories

Goncharov

Schröder

2018

Lecture Notes in Computer Science

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“…As the above overview makes clear, the area has grown too large to allow an adequate summary in this paper. See [Lit14] for more information and [ML17] for an overview of models of guarded (co)recursion, i.e., from our point of view, categorical models for proof systems for fragments of such logics. Our question here is whether the Lewis arrow naturally occurs in this context.…”

Section: Modalities For Guarded (Co)recursionmentioning

confidence: 99%

“…Let us note here that Lemma 4.19 implies that any semantics for i-KM.lin a must make Box valid: in other words, i-KM.lin a can be just seen as another syntactic presentation of i-KM.lin 2 . However, Lemma 4.19 requires all the axioms of i-KM.lin 2 and when studying broader classes of models of guarded (co)recursion [ML17], more flexibility in adding is possible. Open Question 7.4.…”

Section: Modalities For Guarded (Co)recursionmentioning

confidence: 99%

Lewis meets Brouwer: Constructive strict implication

Litak¹,

Visser

2018

Indagationes Mathematicae

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C. I. Lewis invented modern modal logic as a theory of "strict implication" . Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than P and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction has been discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of intuitionistic in terms of preservativity in extensions of HA, i.e., Heyting's Arithmetic. 65The basic option is to read § § 2-4 to get the basics of motivational background, the Kripke semantics and an impression of possible reasoning systems. ? The reader who wants more solid treatment of Kripke semantics can extend the basic option with § 6. | The computer science package consists of the basic option and § 7.= The reader who wants to go somewhat more deeply into the history of the subject can extend the basic option with Appendix D. « The reader who wants to understand the basics of arithmetical interpretations can extend the basic option with § 5. » An extended package for arithmetical interpretations combines « with § 8.-The full arithmetical package extends » with Appendices A, B and C. The rise and fall of the house of Lewis "The error of philosophers"We are reflecting on L.E.J. Brouwer's heritage half a century after his passing. Given his negative views on the rôle of logic and formalisms in mathematics, it seems somewhat paradoxical that these days the name of intuitionism survives mostly in the context of intuitionistic logic. 3 One is reminded in this context of what Nietzsche called the error of philosophers:The philosopher believes that the value of his philosophy lies in the whole, in the structure. Posterity finds it in the stone with which he built and with which, from that time forth, men will build oftener and better-in other words, in the fact that the structure may be destroyed and yet have value as material. 4

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“…Semantic models accommodate this idea in various ways, e.g. from a modal [36,2,33], (ultra-)metric [13,28], and a unifying topos-theoretic perspective [5,10].…”

Section: Introductionmentioning

confidence: 99%

A Metalanguage for Guarded Iteration

Goncharov

Rauch

Schröder

2018

Theoretical Aspects of Computing – ICTAC 2018

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Notions of guardedness serve to delineate admissible recursive definitions in various settings in a compositional manner. In recent work, we have introduced an axiomatic notion of guardedness in symmetric monoidal categories, which serves as a unifying framework for various examples from program semantics, process algebra, and beyond. In the present paper, we propose a generic metalanguage for guarded iteration based on combining this notion with the fine-grain callby-value paradigm, which we intend as a unifying programming language for guarded and unguarded iteration in the presence of computational effects. We give a generic (categorical) semantics of this language over a suitable class of strong monads supporting guarded iteration, and show it to be in touch with the standard operational behaviour of iteration by giving a concrete big-step operational semantics for a certain specific instance of the metalanguage and establishing soundness and (computational) adequacy for this case.

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Guard Your Daggers and Traces: Properties of Guarded (Co-)recursion

Cited by 8 publications

References 33 publications

Guarded Traced Categories

Guarded Traced Categories

Lewis meets Brouwer: Constructive strict implication

A Metalanguage for Guarded Iteration

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