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Monads and Modular Term Rewriting
, 1997
"... . Monads can be used to model term rewriting systems by generalising the wellknown equivalence between universal algebra and monads on the category Set. In [Lu96], the usefulness of this semantics was demonstrated by giving a purely categorical proof of the modularity of confluence for the disjoint ..."
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Cited by 20 (13 self)
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. Monads can be used to model term rewriting systems by generalising the wellknown equivalence between universal algebra and monads on the category Set. In [Lu96], the usefulness of this semantics was demonstrated by giving a purely categorical proof of the modularity of confluence for the disjoint union of term rewriting systems (Toyama's theorem). This paper provides further support for the use of monads in term rewriting by giving a categorical proof of the most general theorem concerning the modularity of strong normalisation. In the process, we also improve upon some technical aspects of the earlier work. 1 Introduction Term rewriting systems (TRSs) are widely used throughout computer science as they provide an abstract model of computation while retaining a relatively simple syntax and semantics. Reasoning about large term rewriting systems can be very difficult and an alternative is to define structuring operations which build large term rewriting systems from smaller ones. Of...
Categorical Term Rewriting: Monads and Modularity
 University of Edinburgh
, 1998
"... Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting syste ..."
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Cited by 11 (6 self)
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Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, thatis,ifthe components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking. This thesis posits that part of the problem is the usual, concrete and syntaxoriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from
Compositional Term Rewriting: An Algebraic Proof of Toyama's Theorem
 Rewriting Techniques and Applications, 7th International Conference, number 1103 in Lecture Notes in Computer Science
, 1996
"... This article proposes a compositional semantics for term rewriting systems, i.e. a semantics preserving structuring operations such as the disjoint union. The semantics is based on the categorical construct of a monad, adapting the treatment of universal algebra in category theory to term rewriting ..."
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Cited by 9 (2 self)
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This article proposes a compositional semantics for term rewriting systems, i.e. a semantics preserving structuring operations such as the disjoint union. The semantics is based on the categorical construct of a monad, adapting the treatment of universal algebra in category theory to term rewriting systems. As an example, the preservation of confluence under the disjoint union of two term rewriting systems is shown, obtaining an algebraic proof of Toyama's theorem, generalised slightly to term rewriting systems introducing variables on the righthand side of the rules.
Geometric and higher order logic in terms of abstract Stone duality
 THEORY AND APPLICATIONS OF CATEGORIES
, 2000
"... The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this ..."
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Cited by 8 (0 self)
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The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.
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.
Subspaces in abstract Stone duality
 Theory and Applications of Categories
, 2002
"... ABSTRACT. By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a selfadjoint exponential Σ (−) on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idemp ..."
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Cited by 6 (5 self)
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ABSTRACT. By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a selfadjoint exponential Σ (−) on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idempotents that play a role similar to that of nuclei in locale theory. Paré showed that any elementary topos has this duality, and we prove it intuitionistically for the category of locally compact locales. The paper is largely concerned with the construction of such a category out of one that merely has powers of some fixed object Σ. It builds on Sober Spaces and Continuations, where the related but weaker notion of abstract sobriety was considered. The construction is done first by formally adjoining certain equalisers that Σ (−) takes to coequalisers, then using Eilenberg–Moore algebras, and finally presented as a lambda calculus similar to the axiom of comprehension in set theory. The comprehension calculus has a normalisation theorem, by which every type can
Iterated distributive laws
, 2007
"... We give a framework for combining n monads on the same category via distributive laws satisfying YangBaxter equations, extending the classical result of Barr and Wells which combines two monads via one distributive law. We show that this corresponds to iterating ntimes the process of taking the 2 ..."
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Cited by 6 (1 self)
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We give a framework for combining n monads on the same category via distributive laws satisfying YangBaxter equations, extending the classical result of Barr and Wells which combines two monads via one distributive law. We show that this corresponds to iterating ntimes the process of taking the 2category of monads in a 2category, extending the result of Street characterising distributive laws. We show that this framework can be used to construct the free strict ncategory monad on ndimensional globular sets; we first construct for each i a monad for composition along bounding icells, and then we show that the interchange laws define distributive laws between these monads, satisfying the necessary YangBaxter equations.
Colimits, StanleyReisner Algebras, and Loop Spaces
, 2003
"... We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: rightangled Artin and Coxeter groups (and their complex analogues, which ..."
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Cited by 4 (3 self)
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We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: rightangled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); StanleyReisner algebras and coalgebras; Davis and Januszkiewicz’s spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend wellknown results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops ΩDJ(K). We define homotopy colimits for diagrams of topological monoids and topological groups, and show they commute with the formation of classifying spaces in a suitably generalised sense. We deduce that the homotopy colimit of the appropriate diagram of topological groups is a model for ΩDJ(K) for an arbitrary complex K, and that the natural projection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible.
Geometry of abstraction in quantum computation
"... Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction i ..."
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Cited by 2 (2 self)
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Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum algorithms provide feasible solutions of important hard problems, such as factoring and discrete log (which are the building blocks of modern cryptography). It is of a great practical interest to precisely characterize the computational resources needed to execute such quantum algorithms. There are many ideas how to build a quantum computer. Can we prove some necessary conditions? Categorical semantics help with such questions. We show how to implement an important family of quantum algorithms using just abelian groups and relations.
Problems in Rewriting Applied to Categorical Concepts By the Example of a Computational Comonad
 Proceedings of the Sixth International Conference on Rewriting Techniques and Applications
, 1995
"... . We present a canonical system for comonads which can be extended to the notion of a computational comonad [BG92] where the crucial point is to find an appropriate representation. These canonical systems are checked with the help of the Larch Prover [GG91] exploiting a method by G. Huet [Hue90a] to ..."
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. We present a canonical system for comonads which can be extended to the notion of a computational comonad [BG92] where the crucial point is to find an appropriate representation. These canonical systems are checked with the help of the Larch Prover [GG91] exploiting a method by G. Huet [Hue90a] to represent typing within an untyped rewriting system. The resulting decision procedures are implemented in the programming language Elf [Pfe89] since typing is directly supported by this language. Finally we outline an incomplete attempt to solve the problem which could be used as a benchmark for rewriting tools. 1 Introduction The starting point of this work was to provide methods for checking the commutativity of diagrams arising in category theory. Diagrams in this context are used as a visual description of equations between morphisms. To check the commutativity of a diagram amounts to check the equality of the morphisms involved. One way to support this task is to solve the uniform wor...