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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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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.
Coalgebraic Monads
, 2002
"... This paper introduces coalgebraic monads as a unified model of term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary coalgebraic monads which allows us to generalise the notion of rat ..."
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Cited by 7 (5 self)
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This paper introduces coalgebraic monads as a unified model of term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary coalgebraic monads which allows us to generalise the notion of rational term and term graph to categories other than Set. As an application we sketch part of the correctness of the the term graph implementation of functional programming languages.
Monads and Modularity
"... This paper argues that the core of modularity problems is an understanding of how individual components of a large system interact with each other, and that this interaction can be described by a layer structure. We propose a uniform treatment of layers based upon the concept of a monad. The combina ..."
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Cited by 7 (5 self)
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This paper argues that the core of modularity problems is an understanding of how individual components of a large system interact with each other, and that this interaction can be described by a layer structure. We propose a uniform treatment of layers based upon the concept of a monad. The combination of different systems can be described by the coproduct of monads.
Algebras, Coalgebras, Monads and Comonads
, 2001
"... Whilst the relationship between initial algebras and monads is wellunderstood, the relationship between nal coalgebras and comonads is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as comonads and dually, that initial a ..."
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Cited by 7 (3 self)
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Whilst the relationship between initial algebras and monads is wellunderstood, the relationship between nal coalgebras and comonads is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as comonads and dually, that initial algebras form both monads and comonads. In developing these theories we strive to provide them with an associated notion of syntax. In the case of initial algebras and monads this corresponds to the standard notion of algebraic theories consisting of signatures and equations: models of such algebraic theories are precisely the algebras of the representing monad. We attempt to emulate this result for the coalgebraic case by defining a notion cosignature and coequation and then proving the models of this syntax are precisely the coalgebras of the representing comonad.
COPRODUCTS OF IDEAL MONADS
, 2004
"... The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by ..."
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Cited by 6 (1 self)
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The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by
Lifting of Operations in Modular Monadic Semantics
, 2009
"... Monads have become a fundamental tool for structuring denotational semantics and programs by abstracting a wide variety of computational features such as sideeffects, input/output, exceptions, continuations and nondeterminism. In this setting, the notion of a monad is equipped with operations that ..."
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Cited by 4 (1 self)
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Monads have become a fundamental tool for structuring denotational semantics and programs by abstracting a wide variety of computational features such as sideeffects, input/output, exceptions, continuations and nondeterminism. In this setting, the notion of a monad is equipped with operations that allow programmers to manipulate these computational effects. For example, a monad for sideeffects is equipped with operations for setting and reading the state, and a monad for exceptions is equipped with operations for throwing and handling exceptions. When several effects are involved, one can employ the incremental approach to modular monadic semantics, which uses monad transformers to build up the desired monad one effect at a time. However, a limitation of this approach is that the effectmanipulating operations need to be manually lifted to the resulting monad, and consequently, the lifted operations are nonuniform. Moreover, the number of liftings needed in a system grows as the product of the number of monad transformers and operations involved. This dissertation proposes a theory of uniform lifting of operations that extends the incremental approach to modular monadic semantics with a principled technique for lifting operations. Moreover the theory is generalized from monads to monoids in a monoidal category, making it possible to apply it to structures other than monads. The extended theory is taken to practice with the implementation of a new extensible monad transformer library in Haskell, and with the use of modular monadic semantics to obtain modular operational semantics. i No hay ejercicio intelectual que no sea finalmente inútil. Una doctrina es al principio una descripción verosímil del universo; giran los años y es un mero capítulo—cuando no un párrafo o un nombre—de la historia de la filosofía. There is no exercise of the intellect which is not, in the final analysis, useless. A philosophical doctrine begins as a plausible description of the universe; with the passage of the years it becomes a mere chapter—if not a paragraph or a name—in the history of philosophy.
Abstract Modularity
"... Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning ab ..."
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Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning about the modularity of specific properties for specific ways of combining specific forms of rewriting. This paper is, we believe, the first to ask a much more general question. Namely, what can be said about modularity independently of the specific form of rewriting, combination and property at hand. A priori there is no reason to believe that anything can actually be said about modularity without reference to the specifics of the particular systems etc. However, this paper shows that, quite surprisingly, much can indeed be said.
Path Rewriting and Combined Word Problems
, 2000
"... . We give an algorithm solving combined word problems (over non necessarily disjoint signatures) based on rewriting of equivalence classes of terms. The canonical rewriting system we introduce consists of few transparent rules and is obtained by applying KnuthBendix completion procedure to presenta ..."
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. We give an algorithm solving combined word problems (over non necessarily disjoint signatures) based on rewriting of equivalence classes of terms. The canonical rewriting system we introduce consists of few transparent rules and is obtained by applying KnuthBendix completion procedure to presentations of pushouts among categories with products. It applies to pairs of theories which are both constructible over their common reduct (on which we do not make any special assumption) . Lavoro svolto nell'ambito del progetto MURST \Logica". 1 1 Introduction An essential problem in automated deduction consists in integrating theorem provers which are able to perform separated tasks. In the eld of equational logic, this leads in particular to the following question: suppose you are able to solve word problems for theories T 1 ; T 2 ; can you solve word problem for T 1 [ T 2 ? Better, can you design an algorithm taking as input two arbitrary algorithms for word problems for T 1 and T 2...