Results 1  10
of
11
Combining effects: sum and tensor
"... We seek a unified account of modularity for computational effects. We begin by reformulating Moggi’s monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations ..."
Abstract

Cited by 29 (4 self)
 Add to MetaCart
We seek a unified account of modularity for computational effects. We begin by reformulating Moggi’s monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular yields Moggi’s exceptions monad transformer and an interactive input/output monad transformer. We further give a theory of the commutative combination of effects, their tensor, which yields Moggi’s sideeffects monad transformer. Finally we give a theory of operation transformers, for redefining operations when adding new effects; we derive explicit forms for the operation transformers associated to the above monad transformers.
Computational Effects and Operations: An Overview
, 2004
"... We overview a programme to provide a unified semantics for computational effects based upon the notion of a countable enriched Lawvere theory. We define the notion of countable enriched Lawvere theory, show how the various leading examples of computational effects, except for continuations, give ris ..."
Abstract

Cited by 26 (8 self)
 Add to MetaCart
We overview a programme to provide a unified semantics for computational effects based upon the notion of a countable enriched Lawvere theory. We define the notion of countable enriched Lawvere theory, show how the various leading examples of computational effects, except for continuations, give rise to them, and we compare the definition with that of a strong monad. We outline how one may use the notion to model three natural ways in which to combine computational effects: by their sum, by their commutative combination, and by distributivity. We also outline a unified account of operational semantics. We present results we have already shown, some partial results, and our plans for further development of the programme.
Semantics for Algebraic Operations
 Proc. MFPS 17, Electronic Notes in Thoeret. Comp. Sci
, 2001
"... Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V category C with cotensors and a strong V monad T on C, we investigate axioms under which an ObC indexed family of operations of the form #x : (Tx) v # (Tx) w provides semantics for algebraic ope ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V category C with cotensors and a strong V monad T on C, we investigate axioms under which an ObC indexed family of operations of the form #x : (Tx) v # (Tx) w provides semantics for algebraic operations, which may be used to extend the usual monadic semantics of the computational #calculus uniformly. 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. We outline examples and nonexamples and we show that our definition also enriches one for callbyname languages with e#ects. 1
On the construction of free algebras for equational systems
 IN: SPECIAL ISSUE FOR AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2007). VOLUME 410 OF THEORETICAL COMPUTER SCIENCE
, 2009
"... The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applica ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.
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 ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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
ON THE MATHEMATICAL SYNTHESIS OF EQUATIONAL LOGICS
"... Birkhoff [1935] initiated the general study of algebraic structure. Importantly for our concerns here, his development was from (universal) algebra to (equational) logic. Birkhoff’s starting point was the informal conception of algebra based on familiar concrete examples. Abstracting from these, he ..."
Abstract
 Add to MetaCart
Birkhoff [1935] initiated the general study of algebraic structure. Importantly for our concerns here, his development was from (universal) algebra to (equational) logic. Birkhoff’s starting point was the informal conception of algebra based on familiar concrete examples. Abstracting from these, he introduced the concepts of signature and equational presentation,