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Adequacy for algebraic effects
 In 4th FoSSaCS
, 2001
"... We present a logic for algebraic effects, based on the algebraic representation of computational effects by operations and equations. We begin with the acalculus, a minimal calculus which separates values, effects, and computations and thereby canonises the order of evaluation. This is extended to ..."
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Cited by 33 (17 self)
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We present a logic for algebraic effects, based on the algebraic representation of computational effects by operations and equations. We begin with the acalculus, a minimal calculus which separates values, effects, and computations and thereby canonises the order of evaluation. This is extended to obtain the logic, which is a classical firstorder multisorted logic with higherorder value and computation types, as in Levy’s callbypushvalue, a principle of induction over computations, a free algebra principle, and predicate fixed points. This logic embraces Moggi’s computational λcalculus, and also, via definable modalities, HennessyMilner logic, and evaluation logic, though Hoare logic presents difficulties. 1
Combining algebraic effects with continuations
, 2007
"... We consider the natural combinations of algebraic computational effects such as sideeffects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor ext ..."
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Cited by 9 (4 self)
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We consider the natural combinations of algebraic computational effects such as sideeffects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor extend, with effort, to include commonly used combinations of the various algebraic effects with continuations. Continuations also give rise to a third sort of combination, that given by applying the continuations monad transformer to an algebraic effect. We investigate the extent to which sum and tensor extend from algebraic effects to arbitrary monads, and the extent to which Felleisen et al.’s C operator extends from continuations to its combination with algebraic effects. To do all this, we use Dubuc’s characterisation of strong monads in terms of enriched large Lawvere theories.
Linearlyused state in models of callbyvalue
"... Abstract. We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fullycomplete statepassing translation from an impure callbyvalue language to a new linear type theory: the enriched callbyvalue calculus. The results are not specific to store, but can b ..."
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Cited by 5 (5 self)
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Abstract. We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fullycomplete statepassing translation from an impure callbyvalue language to a new linear type theory: the enriched callbyvalue calculus. The results are not specific to store, but can be applied to any computational effect expressible using algebraic operations, even to effects that are not usually thought of as stateful. There is a bijective correspondence between generic effects in the source language and state access operations in the enriched callbyvalue calculus. From the perspective of categorical models, the enriched callbyvalue calculus suggests a refinement of the traditional Kleisli models of effectful callbyvalue languages. The new models can be understood as enriched adjunctions. 1
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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Cited by 5 (2 self)
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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.
An axiomatization of arrays for Kleene algebra with tests
 In: Proc. 9th Int. Conf. Relational Methods in Computer Scienceand 4th Int. Workshop Applications of Kleene Algebra (RelMiCS/AKA’06). Volume 4136 of LNCS
, 2006
"... Abstract. The formal analysis of programs with arrays is a notoriously difficult problem due largely to aliasing considerations. In this paper we augment the rules of Kleene algebra with tests (KAT) with rules for the equational manipulation of arrays in the style of schematic KAT. These rules captu ..."
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Abstract. The formal analysis of programs with arrays is a notoriously difficult problem due largely to aliasing considerations. In this paper we augment the rules of Kleene algebra with tests (KAT) with rules for the equational manipulation of arrays in the style of schematic KAT. These rules capture and make explicit the essence of subscript aliasing, where two array accesses can be to the same element. We prove the soundness of our rules, as well as illustrate their usefulness with several examples, including a complete proof of the correctness of heapsort. 1
Monoidal indeterminates and categories of possible worlds
 In Proc. of MFPS XXV
, 2009
"... Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffin ..."
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Cited by 1 (0 self)
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Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffine category (symmetric monoidal category with initial unit) on a given small symmetric monoidal category. We then exhibit all the known categories of “possible worlds ” used to treat languages that allow for dynamic creation of “new ” variables, locations, or names as instances of this construction and explicate their associated universality properties. As an application of the resulting characterisation of O(W), Oles’s category of possible worlds, we present an O(W)indexed Lawvere theory of manysorted storage, generalizing the singlesorted one introduced by J. Power, and we describe explicitly an associated
Towards Effects in Mathematical Operational Semantics
"... In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effectfree coalgebraic operational semantics, given by a natural transformation of syntax over behaviour. The operational semantics of the extended language arises by distributi ..."
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In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effectfree coalgebraic operational semantics, given by a natural transformation of syntax over behaviour. The operational semantics of the extended language arises by distributing program syntax over effects, again inducing a coalgebraic operational semantics, but this time in the Kleisli category for the monad derived from the algebraic effects. The final coalgebra in this Kleisli category then serves as the denotational model. For it to exist, we ensure that the the Kleisli category is enriched over CPOs by considering the monad of possibly infinite terms, extended with a bottom element. Unlike the effectless setting, not all operational specifications give rise to adequate and compositional semantics. We give a proof of adequacy and compositionality provided the specifications can be described by evaluationincontext. We illustrate our techniques with a simple extension of (stateless) while programs with global store, i.e. variable lookup.