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Tensors of Comodels and Models for Operational Semantics
"... In seeking a unified study of computational effects, in particular in order to give a general operational semantics agreeing with the standard one for state, one must take account of the coalgebraic structure of state. Axiomatically, one needs a countable Lawvere theory L, a comodel C, typically the ..."
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In seeking a unified study of computational effects, in particular in order to give a general operational semantics agreeing with the standard one for state, one must take account of the coalgebraic structure of state. Axiomatically, one needs a countable Lawvere theory L, a comodel C, typically the final one, and a model M, typically free; one then seeks a tensor C ⊗ M of the comodel with the model that allows operations to flow between the two. We describe such a tensor implicit in the abstract category theoretic literature, explain its significance for computational effects, and calculate it in leading classes of examples, primarily involving state.
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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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
Towards a Definition of an Algorithm
, 2005
"... We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are “essentially ” the same program. The set of all equivalence classes is the category of all algorithms. I ..."
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We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are “essentially ” the same program. The set of all equivalence classes is the category of all algorithms. In order to explore these ideas, the set of primitive recursive functions is considered. Each primitive recursive function can be described by many labeled binary trees that show how the function is built up. Each tree is like a program that shows how to compute a function. We give relations that say when two such trees are “essentially” the same. An equivalence class of such trees will be called an algorithm.
A CATEGORICAL OUTLOOK ON CELLULAR AUTOMATA
"... Abstract. In programming language semantics, it has proved to be fruitful to analyze contextdependent notions of computation, e.g., dataflow computation and attribute grammars, using comonads. We explore the viability and value of similar modeling of cellular automata. We identify local behaviors o ..."
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Abstract. In programming language semantics, it has proved to be fruitful to analyze contextdependent notions of computation, e.g., dataflow computation and attribute grammars, using comonads. We explore the viability and value of similar modeling of cellular automata. We identify local behaviors of cellular automata with coKleisli maps of the exponent comonad on the category of uniform spaces and uniformly continuous functions and exploit this equivalence to conclude some standard results about cellular automata as instances of basic categorytheoretic generalities. In particular, we recover CeccheriniSilberstein and Coornaert’s version of the CurtisHedlund theorem. 1.
On comonadicity of the extensionofscalars functors
 J. Algebra
"... Abstract. A criterion for comonadicity of the extensionof scalars functor associated to an extension of (not necessarily commutative) rings is given. As an application of this criterion, some known results on the comonadicity of such functors are obtained. 1. ..."
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Abstract. A criterion for comonadicity of the extensionof scalars functor associated to an extension of (not necessarily commutative) rings is given. As an application of this criterion, some known results on the comonadicity of such functors are obtained. 1.
Monadic Style Control Constructs For Inference Systems
, 2002
"... Recent advances in programming languages study and design have established a standard way of grounding computational systems representation in category theory. These formal results led to a better understanding of issues of control and sidee#ects in functional and imperative languages.
(http://www. ..."
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Recent advances in programming languages study and design have established a standard way of grounding computational systems representation in category theory. These formal results led to a better understanding of issues of control and sidee#ects in functional and imperative languages.
(http://www.lccapital.com/~jmc/articles/monadic1.pdf)
STRENGTHENING TRACK THEORIES
, 2003
"... and on the secondary cohomology operations [4] is based on the “calculus of tracks”. One of the main tricks in [4] is to make some track theories strong. The aim of this and some subsequent papers is to shed more light on this procedure. In this paper we prove that ..."
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and on the secondary cohomology operations [4] is based on the “calculus of tracks”. One of the main tricks in [4] is to make some track theories strong. The aim of this and some subsequent papers is to shed more light on this procedure. In this paper we prove that
INTERNAL COHOMOMORPHISMS FOR OPERADS
, 2007
"... In this paper we construct internal cohomomorphism objects in various categories of operads (ordinary, cyclic, modular, properads...) and algebras over operads. We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” structur ..."
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In this paper we construct internal cohomomorphism objects in various categories of operads (ordinary, cyclic, modular, properads...) and algebras over operads. We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” structures. We give also a unified axiomatic treatment of operads as functors on labeled graphs. Finally, we extend internal cohomomorphism constructions to more general categorical contexts.
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.
MONADIC STYLE CONTROL CONSTRUCTS FOR INFERENCE SYSTEMS
, 2002
"... Abstract. Recent advances in programming languages study and design have established a standard way of grounding computational systems representation in category theory. These formal results led to a better understanding of issues of control and sideeffects in functional and imperative languages. A ..."
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Abstract. Recent advances in programming languages study and design have established a standard way of grounding computational systems representation in category theory. These formal results led to a better understanding of issues of control and sideeffects in functional and imperative languages. Another benefit is a better way of modelling computational effects in logical frameworks. With this analogy in mind, we embark on an investigation of inference systems based on considering inference behaviour as a form of computation. We delineate a categorical formalisation of control constructs in inference systems. This representation emphasises the parallel between the modular articulation of the categorical building blocks (triples) used to account for the inference architecture and the modular composition of cognitive processes.