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Generic Models for Computational Effects
"... A Freydcategory is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in callbyvalue programming languages, such as the computational λcalculus, with computational effects. We develop the theory of Freydcategories with that in min ..."
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A Freydcategory is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in callbyvalue programming languages, such as the computational λcalculus, with computational effects. We develop the theory of Freydcategories with that in mind. We first show that any countable Lawvere theory, hence any signature of operations with countable arity subject to equations, directly generates a Freydcategory. We then give canonical, universal embeddings of Freydcategories into closed Freydcategories, characterised by being free cocompletions. The combination of the two constructions sends a signature of operations and equations to the Kleisli category for the monad on the category Set generated by it, thus refining the analysis of computational effects given by monads. That in turn allows a more structural analysis of the λccalculus. Our leading examples of signatures arise from sideeffects, interactive input/output and exceptions. We extend our analysis to an enriched setting in order to account for recursion and for computational effects and signatures that inherently involve it, such as partiality, nondeterminism and probabilistic nondeterminism. Key words: Freydcategory, enriched Yoneda embedding, conical colimit completion, canonical model
Semantics for Local Computational Effects
, 2006
"... Starting with Moggi’s work on monads as refined to Lawvere theories, we give a general construct that extends denotational semantics for a global computational effect canonically to yield denotational semantics for a corresponding local computational effect. Our leading example yields a construction ..."
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Starting with Moggi’s work on monads as refined to Lawvere theories, we give a general construct that extends denotational semantics for a global computational effect canonically to yield denotational semantics for a corresponding local computational effect. Our leading example yields a construction of the usual denotational semantics for local state from that for global state. Given any Lawvere theory L, possibly countable and possibly enriched, we first give a universal construction that extends L, hence the global operations and equations of a given effect, to incorporate worlds of arbitrary finite size. Then, making delicate use of the final comodel of the ordinary Lawvere theory L, we give a construct that uniformly allows us to model block, the universality of the final comodel yielding a universal property of the construct. We illustrate both the universal extension of L and the canonical construction of block by seeing how they work in the case of state.
Tensors of Comodels and Models for Operational Semantics
"... In seeking a unified study of computational effects, in particular in order to give a general operational semantics agreeing with the standard one for state, one must take account of the coalgebraic structure of state. Axiomatically, one needs a countable Lawvere theory L, a comodel C, typically the ..."
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In seeking a unified study of computational effects, in particular in order to give a general operational semantics agreeing with the standard one for state, one must take account of the coalgebraic structure of state. Axiomatically, one needs a countable Lawvere theory L, a comodel C, typically the final one, and a model M, typically free; one then seeks a tensor C ⊗ M of the comodel with the model that allows operations to flow between the two. We describe such a tensor implicit in the abstract category theoretic literature, explain its significance for computational effects, and calculate it in leading classes of examples, primarily involving state.
Logic for Computational Effects: work in progress
"... Abstract We outline a possible logic that will allow us to give a unified approach to reasoning about computational effects. The logic is given by extending Moggi's computational *calculus by basic types and a signature, the latter given by constant symbols, function symbols, and operation symbols, ..."
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Abstract We outline a possible logic that will allow us to give a unified approach to reasoning about computational effects. The logic is given by extending Moggi's computational *calculus by basic types and a signature, the latter given by constant symbols, function symbols, and operation symbols, and by including a _ operator. We give both syntax and semantics for the logic except for _. We consider a number of sound and complete classes of models, all given in categorytheoretic terms. We illustrate the ideas with some of our leading examples of computational effects, and we observe that operations give rise to natural modalities.
Some Varieties of Equational Logic (Extended Abstract), Algebra
 Meaning, and Computation, Essays Dedicated to Joseph A. Goguen on the Occasion of His 65th Birthday
, 2006
"... been a major theme of Joseph Goguen’s research, perhaps even the major theme. One strand of this work concerns algebraic datatypes. Recently there has been some interest in what one may call algebraic computation types. As we will show, these are also given by equational theories, if one only unders ..."
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been a major theme of Joseph Goguen’s research, perhaps even the major theme. One strand of this work concerns algebraic datatypes. Recently there has been some interest in what one may call algebraic computation types. As we will show, these are also given by equational theories, if one only understands the notion of equational logic in somewhat broader senses than usual. One moral of our work is that, suitably considered, equational logic is not tied to the usual firstorder syntax of terms and equations. Standard equational logic has proved a useful tool in several branches of computer science, see, for example, the RTA conference series [9] and textbooks, such as [1]. Perhaps the possibilities for richer varieties of equational logic discussed here will lead to further applications. We begin with an explanation of computation types. Starting around 1989, Eugenio Moggi introduced the idea of monadic notions of computation [11, 12]
Finitary construction of free algebras for equational systems
, 2008
"... The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications. Key words ..."
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The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications. Key words: Equational system; algebra; free construction; monad. 1
Equational Systems and Free Constructions (Extended Abstract)
"... Abstract. The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specif ..."
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Abstract. The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specifically to the theory of substitution in the presence of variable binding and to models of namepassing process calculi. 1
Two Probabilistic Powerdomains in Topological Domain Theory
"... We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given b ..."
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We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given by an observationally induced approach, following Schröder and Simpson. We show the two constructions coincide when restricted to ωcontinuous dcppos, in which case they yield the space of (continuous) probability valuations equipped with the Scott topology. Thus either construction generalises the classical domaintheoretic probabilistic powerdomain. On more general spaces, the constructions differ, and the second seems preferable. Indeed, for countablybased spaces, we characterise the observationally induced powerdomain as the space of probability valuations with weak topology. However, we show that such a characterisation does not extend to non countablybased spaces.
Presheaf models for the πcalculus
 In Proc. CTCS’97, volume 1290 of LNCS
, 1997
"... Abstract. The finite πcalculus has an explicit settheoretic functorcategory model that is known to be fully abstract for strong late bisimulation congruence. We characterize this as the initial free algebra for an appropriate set of operations and equations in the enriched Lawvere theories of Plo ..."
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Abstract. The finite πcalculus has an explicit settheoretic functorcategory model that is known to be fully abstract for strong late bisimulation congruence. We characterize this as the initial free algebra for an appropriate set of operations and equations in the enriched Lawvere theories of Plotkin and Power. Thus we obtain a novel algebraic description for models of the πcalculus, and validate an existing construction as the universal such model. The algebraic operations are intuitive, covering name creation, communication of names over channels, and nondeterminism; the equations then combine these features in a modular fashion. We work in an enriched setting, over a “possible worlds ” category of sets indexed by available names. This expands significantly on the classical notion of algebraic theories, and in particular allows us to use nonstandard arities that vary as processes evolve. Based on our algebraic theory we describe a category of models for the πcalculus, and show that they all preserve bisimulation congruence. We develop a direct construction of free models in this category; and generalise previous results to prove that all freealgebra models are fully abstract. 1
Language semantics Syntax
, 2010
"... ‣ continually evolving new versions of existing languages new language designs 2 Background Programming language specifications ‣ program syntax – formal lexical symbols phrase structure ‣ program semantics – informal static (compiletime) behaviour dynamic (runtime) behaviour 3 Formal semant ..."
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‣ continually evolving new versions of existing languages new language designs 2 Background Programming language specifications ‣ program syntax – formal lexical symbols phrase structure ‣ program semantics – informal static (compiletime) behaviour dynamic (runtime) behaviour 3 Formal semantics Problems (for major languages): ‣ reuse ‣ coevolution ‣ tool support 4