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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 ..."
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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.
Finite dimensional vector spaces are complete for traced symmetric monoidal categories
 in: Pillars of Computer Science: Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday, Lecture Notes in Computer Science 4800 (2008
"... Abstract. We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two differe ..."
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Abstract. We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two different arrows in the free traced category there always exists astrongtracedfunctorintoFinVectk which distinguishes them. Therefore two arrows in the free traced category are the same if and only if they agree for all interpretations in FinVectk. 1
Axioms for Recursion in CallbyValue
 HIGHERORDER AND SYMBOLIC COMPUT
, 2001
"... We propose an axiomatization of fixpoint operators in typed callbyvalue programming languages, and give its justifications in two ways. First, it is shown to be sound and complete for the notion of uniform Tfixpoint operators of Simpson and Plotkin. Second, the axioms precisely account for Filins ..."
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Cited by 12 (5 self)
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We propose an axiomatization of fixpoint operators in typed callbyvalue programming languages, and give its justifications in two ways. First, it is shown to be sound and complete for the notion of uniform Tfixpoint operators of Simpson and Plotkin. Second, the axioms precisely account for Filinski's fixpoint operator derived from an iterator (infinite loop constructor) in the presence of firstclass continuations, provided that we define the uniformity principle on such an iterator via a notion of effectfreeness (centrality). We then explain how these two results are related in terms of the underlying categorical structures.
Using synthetic domain theory to prove operational properties of a polymorphic programming language based on strictness
 Manuscript
"... We present a simple and workable axiomatization of domain theory within intuitionistic set theory, in which predomains are (special) sets, and domains are algebras for a simple equational theory. We use the axioms to construct a relationally parametric settheoretic model for a compact but powerful ..."
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Cited by 12 (3 self)
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We present a simple and workable axiomatization of domain theory within intuitionistic set theory, in which predomains are (special) sets, and domains are algebras for a simple equational theory. We use the axioms to construct a relationally parametric settheoretic model for a compact but powerful polymorphic programming language, given by a novel extension of intuitionistic linear type theory based on strictness. By applying the model, we establish the fundamental operational properties of the language. 1.
Traced Premonoidal Categories
, 1999
"... Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating trace ..."
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Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating traces and Conway operators in cartesian categories.
Duality between CallbyName Recursion and CallbyValue Iteration
 IN PROC. COMPUTER SCIENCE LOGIC, SPRINGER LECTURE NOTES IN COMPUT. SCI
, 2001
"... We investigate the duality between callbyname recursion and callbyvalue iteration on the λµcalculi. The duality between callbyname and callbyvalue was first studied by Filinski, and Selinger has studied the categorytheoretic duality on the models of the callbyname λµcalculus and the cal ..."
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Cited by 11 (5 self)
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We investigate the duality between callbyname recursion and callbyvalue iteration on the λµcalculi. The duality between callbyname and callbyvalue was first studied by Filinski, and Selinger has studied the categorytheoretic duality on the models of the callbyname λµcalculus and the callbyvalue one. We extend the callbyname λµcalculus and the callbyvalue one with a fixedpoint operator and an iteration operator, respectively. We show that the dual translations constructed by Selinger can be expanded into our extended λµcalculi, and we also discuss their implications to practical applications.
From coalgebraic to monoidal traces
 Coalgebraic Methods in Computer Science (CMCS 2010), volume 264 of Elect. Notes in Theor. Comp. Sci
, 2010
"... The main result of this paper shows how coalgebraic traces, in suitable Kleisli categories, give rise to traced monoidal structure in those Kleisli categories, with finite coproducts as monoidal structure. At the heart of the matter lie partially additive monads inducing partially additive structure ..."
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Cited by 8 (2 self)
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The main result of this paper shows how coalgebraic traces, in suitable Kleisli categories, give rise to traced monoidal structure in those Kleisli categories, with finite coproducts as monoidal structure. At the heart of the matter lie partially additive monads inducing partially additive structure in their Kleisli categories. By applying the standard “Int ” construction one obtains compact closed categories for “bidirectional monadic computation”. 1
Semantics of value recursion for monadic input/output
 Journal of Theoretical Informatics and Applications
, 2002
"... Abstract. Monads have been employed in programming languages for modeling various language features, most importantly those that involve side effects. In particular, Haskell’s IO monad provides access to I/O operations and mutable variables, without compromising referential transparency. Cyclic defi ..."
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Cited by 7 (1 self)
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Abstract. Monads have been employed in programming languages for modeling various language features, most importantly those that involve side effects. In particular, Haskell’s IO monad provides access to I/O operations and mutable variables, without compromising referential transparency. Cyclic definitions that involve monadic computations give rise to the concept of valuerecursion, where the fixedpoint computation takes place only over the values, without repeating or losing effects. In this paper, we describe a semantics for a lazy language based on Haskell, supporting monadic I/O, mutable variables, usual recursive definitions, and value recursion. Our semantics is composed of two layers: A natural semantics for the functional layer, and a labeled transition semantics for the IO layer. Mathematics Subject Classification. 68N18, 68Q55, 18C15.
What are iteration theories
, 2007
"... “In the setting of algebraic theories enriched with an external fixedpoint operation, the notion of an iteration theory seems to axiomatize the equational properties of all the computationally interesting structures of this kind.” S. L. Bloom and Z. ´Esik (1996), see [3] We prove that iteration t ..."
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Cited by 7 (5 self)
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“In the setting of algebraic theories enriched with an external fixedpoint operation, the notion of an iteration theory seems to axiomatize the equational properties of all the computationally interesting structures of this kind.” S. L. Bloom and Z. ´Esik (1996), see [3] We prove that iteration theories can be introduced as algebras for the monad on the category of signatures assigning to every signature the rational tree signature. This supports the claim that iteration theories axiomatize precisely the equational properties of least fixed points in domain theory: is the monad of free rational theories and every rational theory has a continuous completion. 1.
An Equational Notion of Lifting Monad
 TITLE WILL BE SET BY THE PUBLISHER
, 2003
"... We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category ..."
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We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers. The representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right. 1