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Continuous Previsions
"... We define strong monads of continuous (lower, upper) previsions, and of forks, modeling both probabilistic and nondeterministic choice. This is an elegant alternative to recent proposals by Mislove, Tix, Keimel, and Plotkin. We show that our monads are sound and complete, in the sense that they m ..."
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Cited by 9 (6 self)
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We define strong monads of continuous (lower, upper) previsions, and of forks, modeling both probabilistic and nondeterministic choice. This is an elegant alternative to recent proposals by Mislove, Tix, Keimel, and Plotkin. We show that our monads are sound and complete, in the sense that they model exactly the interaction between probabilistic and (demonic, angelic, chaotic) choice.
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.
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
Modular ChurchRosser Modulo: The Complete Picture
"... Abstract. In [19], Toyama proved that the union of two confluent termrewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous ca ..."
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Cited by 2 (0 self)
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Abstract. In [19], Toyama proved that the union of two confluent termrewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous cap and a so called alien, possibly heterogeneous substitution, was later substantially simplified in [8, 12]. In this paper 3, we present a further simplification of the proof of Toyama’s result for confluence, which shows that the crux of the problem lies on two different properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions and a modularity property of ordered completion that allows to pairwise match the caps and alien substitutions of two equivalent terms obtained from the cleaning lemma. The approach allows for arbitrary kinds of rules, and scales up to rewriting modulo arbitrary sets of equations. 1
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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Cited by 1 (0 self)
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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.
Modularity of Confluence Constructed
"... Abstract. We present a novel proof of Toyama’s famous modularity of confluence result for term rewriting systems. Apart from being short and intuitive, the proof is modular itself in that it factors through the decreasing diagrams technique for abstract rewriting systems, is constructive in that it ..."
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Abstract. We present a novel proof of Toyama’s famous modularity of confluence result for term rewriting systems. Apart from being short and intuitive, the proof is modular itself in that it factors through the decreasing diagrams technique for abstract rewriting systems, is constructive in that it gives a construction for the converging rewrite sequences given a pair of diverging rewrite sequences, and is flexible in that it extends to constructorsharing term rewriting systems. We show that for term rewrite systems with extra variables, confluence is not preserved under decomposition, and discuss whether for these systems confluence is preserved under composition. 1
Nordic Journal of Computing 10(2003), 290–312. REWRITING VIA COINSERTERS
"... Abstract. This paper introduces a semantics for rewriting that is independent of the data being rewritten and which, nevertheless, models key concepts such as substitution which are central to rewriting algorithms. We demonstrate the naturalness of this construction by showing how it mirrors the usu ..."
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Abstract. This paper introduces a semantics for rewriting that is independent of the data being rewritten and which, nevertheless, models key concepts such as substitution which are central to rewriting algorithms. We demonstrate the naturalness of this construction by showing how it mirrors the usual treatment of algebraic theories as coequalizers of monads. We also demonstrate its naturalness by showing how it captures several canonical forms of rewriting.
Abstract Composing Monads Using Coproducts
"... Monads are a useful abstraction of computation, as they model diverse computational effects such as stateful computations, exceptions and I/O in a uniform manner. Their potential to provide both a modular semantics and a modular programming style was soon recognised. However, in general, monads prov ..."
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Monads are a useful abstraction of computation, as they model diverse computational effects such as stateful computations, exceptions and I/O in a uniform manner. Their potential to provide both a modular semantics and a modular programming style was soon recognised. However, in general, monads proved difficult to compose and so research focused on special mechanisms for their composition such as distributive monads and monad transformers. We present a new approach to this problem which is general in that nearly all monads compose, mathematically elegant in using the standard categorical tools underpinning monads and computationally expressive in supporting a canonical recursion operator. In a nutshell, we propose that two monads should be composed by taking their coproduct. Although abstractly this is a simple idea, the actual construction of the coproduct of two monads is nontrivial. We outline this construction, show how to implement the coproduct within Haskell and demonstrate its usage with a few examples. We also discuss its relationship with other ways of combining monads, in particular distributive laws for monads and monad transformers.
Rewriting the Conditions in Conditional Rewriting
, 2000
"... Category theory has been used to provide a semantics for term rewriting systems at an intermediate level of abstraction between the actual syntax and the relational model. Recently we have developed a semantics for TRSs using monads which generalises the equivalence between algebraic theories and f ..."
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Category theory has been used to provide a semantics for term rewriting systems at an intermediate level of abstraction between the actual syntax and the relational model. Recently we have developed a semantics for TRSs using monads which generalises the equivalence between algebraic theories and finitary monads on the category Set. This semantics underpins the recent categorical proofs of stateoftheart results in modular rewriting. We believe that our methods can be applied to modularity for conditional rewriting where several open problems exist. Any results we achieve here would be highly significant as, for the first time, substantial open problems in rewriting would have been solved using categorical techniques. This paper reports on the first step in this project, namely the construction of a semantics for CTRS using monads.