The Category-Theoretic Solution of Recursive Domain Equations

Smyth, Michael B.; Plotkin, Gordon

doi:10.1137/0211062

Cited by 370 publications

(96 citation statements)

References 29 publications

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“…This initial-final coincidence is familiar from domain theoretic models of recursive types (see, e.g., Smyth and Plotkin [23]), and has previously been observed as a feature of guarded recursive types by Birkedal et al [6].…”

Section: Final Coalgebras From Initial Algebras Guards and Clockssupporting

confidence: 56%

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Productive coprogramming with guarded recursion

Atkey

McBride

2013

SIGPLAN Not.

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Total functional programming offers the beguiling vision that, just by virtue of the compiler accepting a program, we are guaranteed that it will always terminate. In the case of programs that are not intended to terminate, e.g., servers, we are guaranteed that programs will always be productive. Productivity means that, even if a program generates an infinite amount of data, each piece will be generated in finite time. The theoretical underpinning for productive programming with infinite output is provided by the category theoretic notion of final coalgebras. Hence, we speak of coprogramming with non-well-founded codata, as a dual to programming with wellfounded data like finite lists and trees.Systems that offer facilities for productive coprogramming, such as the proof assistants Coq and Agda, currently do so through syntactic guardedness checkers, which ensure that all self-recursive calls are guarded by a use of a constructor. Such a check ensures productivity. Unfortunately, these syntactic checks are not compositional, and severely complicate coprogramming.Guarded recursion, originally due to Nakano, is tantalising as a basis for a flexible and compositional type-based approach to coprogramming. However, as we show, guarded recursion by itself is not suitable for coprogramming due to the fact that there is no way to make finite observations on pieces of infinite data. In this paper, we introduce the concept of clock variables that index Nakano's guarded recursion. Clock variables allow us to "close over" the generation of infinite codata, and to make finite observations, something that is not possible with guarded recursion alone.

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Section: Final Coalgebras From Initial Algebras Guards and Clockssupporting

confidence: 56%

“…We are guaranteed to be able to obtain such a D by standard results about the category DCPO ⊥ (see, e.g., Smyth and Plotkin [23]). The five components of the right hand side of this equation will be used to carry functions, products, the left and right injections for sum types and the unit value respectively.…”

Section: Semantics Of Termsmentioning

confidence: 98%

Productive coprogramming with guarded recursion

Atkey

McBride

2013

SIGPLAN Not.

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“…This concept was mentioned in example 2.7, and is also used by Goguen andBurstall (1980, 1984), Hoare and He (1988) and Moggi (1989), among other places, and is mentioned in Smyth and Plotkin (1982). This section mentions some further categorical constructions, about each of which one might express surprise at how many examples there are in computing science.…”

Section: Further Topicsmentioning

confidence: 97%

A categorical manifesto

Goguen

1991

Math. Struct. Comp. Sci.

155

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This paper tries to explain why and how category theory is useful in computing science, by giving guidelines for applying seven basic categorical concepts: category, functor, natural transformation, limit, adjoint, colimit and comma category. Some examples, intuition, and references are given for each concept, but completeness is not attempted. Some additional categorical concepts and some suggestions for further research are also mentioned. The paper concludes with some philosophical discussion. IntroductionIn this paper we will try to explain why category theory is useful in computing science. The basic answer is that computing science is a young field that is growing rapidly, is poorly organised, and needs all the help it can get, and that category theory can provide help with at least the following:-Formulating definitions and theories. In computing science, it is often more difficult to formulate concepts and results than to give a proof. The seven guidelines of this paper can help with formulation; the guidelines can also be used to measure the elegance and coherence of existing formulations. -Carrying out proofs. Once basic concepts have been correctly formulated in a categorical language, it often seems that proofs 'just happen': at each step, there is a 'natural' thing to try, and it works. Diagram chasing (see section 1.2) provides many examples of this. It could almost be said that the purpose of category theory is to reduce all proofs to such simple calculations. -Discovering and exploiting relations with other fields. Sufficiently abstract formulationscan reveal surprising connections. For example, an analogy between Petri nets and the A-calculus might suggest looking for a closed category structure on the category of Petri nets (Meseguer and Montanari, 1988).-Dealing with abstraction and representation independence. In computing science, more abstract viewpoints are often more useful, because of the need to achieve independence from the often overwhelmingly complex details of how things are represented or implemented. A corollary of the first guideline (given in section 1) is that two objects are 'abstractly the same' iff they are isomorphic; see section 1.1. Moreover, universal

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“…Proof. F t satisfies the co-completeness condition required in order to have a minimal fixed point [32], i.e. F t (colim(F n t (0), F n t (0 F t (0) )) )#colim F t ((F n t (0), F n t (0 F t (0) )) ).…”

Section: A Denotational Account Of Kleene Starmentioning

confidence: 99%

Models of Nondeterministic Regular Expressions

Corradini

Nicola

Labella

1999

Journal of Computer and System Sciences

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Nondeterminism is a direct outcome of interactions and is, therefore a central ingredient for modelling concurrent systems. Trees are very useful for modelling nondeterministic behaviour. We aim at a tree-based interpretation of regular expressions and study the effect of removing the idempotence law X+X=X and the distribution law X v (Y+Z)=X vY+X vZ from Kleene algebras. We show that the free model of the new set of axioms is a class of trees labelled over A. We also equip regular expressions with a two-level behavioural semantics. The basic level is described in terms of a class of labelled transition systems that are detailed enough to describe the number of equal actions a system can perform from a given state. The abstract level is based on a so-called resource bisimulation preorder that permits ignoring uninteresting details of transition systems. The three proposed interpretations of regular expressions (algebraic, denotational, and behavioural ) are proven to coincide. When dealing with infinite behaviours, we rely on a simple version of the |-induction and obtain a complete proof system also for the full language of nondeterministic regular expressions.

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The Category-Theoretic Solution of Recursive Domain Equations

Cited by 370 publications

References 29 publications

Productive coprogramming with guarded recursion

Productive coprogramming with guarded recursion

A categorical manifesto

Models of Nondeterministic Regular Expressions

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