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25
Notions of Computation Determine Monads
 Proc. FOSSACS 2002, Lecture Notes in Computer Science 2303
, 2002
"... We give semantics for notions of computation, also called computational effects, by means of operations and equations. We show that these generate several of the monads of primary interest that have been used to model computational effects, with the striking omission of the continuations monad, demo ..."
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Cited by 56 (7 self)
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We give semantics for notions of computation, also called computational effects, by means of operations and equations. We show that these generate several of the monads of primary interest that have been used to model computational effects, with the striking omission of the continuations monad, demonstrating the latter to be of a different character, as is computationally true. We focus on semantics for global and local state, showing that taking operations and equations as primitive yields a mathematical relationship that reflects their computational relationship.
Algebraic Operations and Generic Effects
 Applied Categorical Structures
, 2003
"... Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal Vcategory C with cotensors and a strong Vmonad T on C, we investigate axioms under which an ObCindexed family of operations of the form α_x : (Tx)^ν → (Tx)^ω provides ..."
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Cited by 34 (7 self)
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Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal Vcategory C with cotensors and a strong Vmonad T on C, we investigate axioms under which an ObCindexed family of operations of the form &alpha;_x : (Tx)^&nu; &rarr; (Tx)^&omega; provides semantics for algebraic operations on the computational &lambda;calculus. We recall a definition for which we have elsewhere given adequacy results, and we show that an enrichment of it is equivalent to a range of other possible natural definitions of algebraic operation. In particular, we define the notion of generic effect and show that to give a generic effect is equivalent to giving an algebraic operation. We further show how the usual monadic semantics of the computational &lambda;calculus extends uniformly to incorporate generic effects. We outline examples and nonexamples and we show that our definition also enriches one for callbyname languages with e#ects.
Nondeterminism and Probabilistic Choice: Obeying the Laws
 In Proc. 11th CONCUR, volume 1877 of LNCS
, 2000
"... In this paper we describe how to build semantic models that support both nondeterministic choice and probabilistic choice. Several models exist that support both of these constructs, but none that we know of satisfies all the laws one would like. Using domaintheoretic techniques, we show how models ..."
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Cited by 27 (2 self)
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In this paper we describe how to build semantic models that support both nondeterministic choice and probabilistic choice. Several models exist that support both of these constructs, but none that we know of satisfies all the laws one would like. Using domaintheoretic techniques, we show how models can be devised using the "standard model" for probabilistic choice, and then applying modified domaintheoretic models for nondeterministic choice. These models are distinguished by the fact that the expected laws for nondeterministic choice and probabilistic choice remain valid. We also describe some potential applications of our model to aspects of security.
Power Domain Constructions
 SCIENCE OF COMPUTER PROGRAMMING
, 1998
"... The variety of power domain constructions proposed in the literature is put into a general algebraic framework. Power constructions are considered algebras on a higher level: for every ground domain, there is a power domain whose algebraic structure is specified by means of axioms concerning the alg ..."
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Cited by 23 (9 self)
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The variety of power domain constructions proposed in the literature is put into a general algebraic framework. Power constructions are considered algebras on a higher level: for every ground domain, there is a power domain whose algebraic structure is specified by means of axioms concerning the algebraic properties of the basic operations empty set, union, singleton, and extension of functions. A host of derived operations is introduced and investigated algebraically. Every power construction is shown to be equipped with a characteristic semiring such that the resulting power domains become semiring modules. Power homomorphisms are introduced as a means to relate different power constructions. They also allow to define the notion of initial and final constructions for a fixed characteristic semiring. Such initial and final constructions are shown to exist for every semiring, and their basic properties are derived. Finally, the known power constructions are put into the general framewo...
Combining Computational Effects: Commutativity and Sum
, 2002
"... We begin to develop a unified account of modularity for computational effects. We use the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi's paradigm for modelling computational effects; we emphasise the importance here of the operations ..."
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Cited by 19 (4 self)
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We begin to develop a unified account of modularity for computational effects. We use the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi's paradigm for modelling computational effects; we emphasise the importance here of the operations that induce computational effects. Effects qua theories are then combined by appropriate bifunctors (on the category of theories). We give a theory of the commutative combination of effects, which in particular yields Moggi's sideeffects monad transformer (an application is the combination of sideeffects with nondeterminism). And we give a theory...
A Convenient Category of Domains
 GDP FESTSCHRIFT ENTCS, TO APPEAR
"... We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also su ..."
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Cited by 14 (3 self)
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We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.
Power Domains and Second Order Predicates
 THEORETICAL COMPUTER SCIENCE
, 1993
"... Lower, upper, sandwich, mixed, and convex power domains are isomorphic to domains of second order predicates mapping predicates on the ground domain to logical values in a semiring. The various power domains differ in the nature of the underlying semiring logic and in logical constraints on the seco ..."
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Cited by 13 (7 self)
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Lower, upper, sandwich, mixed, and convex power domains are isomorphic to domains of second order predicates mapping predicates on the ground domain to logical values in a semiring. The various power domains differ in the nature of the underlying semiring logic and in logical constraints on the second order predicates.
Countable Lawvere Theories and Computational Effects
, 2006
"... Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere ..."
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Cited by 12 (3 self)
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Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.
A domain equation for refinement of partial systems
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENC
"... ..."
A universal characterization of the closed euclidean interval (Extended Abstract)
 PROC. OF 16TH ANN. IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE, LICS'01
, 2001
"... We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basi ..."
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Cited by 12 (1 self)
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We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the