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Proof of the Decidability of the Uniform Word Problem for Monads Assisted by Elf
, 1994
"... . We present a full proof of a canonical system for adjunctions as already suggested in [Cur93]. Termination can be shown in a similar style to [HL86]. Confluence is shown by checking all critical pairs. This is done firstly in the Larch Prover [GG91] and secondly more correctly in the programming l ..."
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. We present a full proof of a canonical system for adjunctions as already suggested in [Cur93]. Termination can be shown in a similar style to [HL86]. Confluence is shown by checking all critical pairs. This is done firstly in the Larch Prover [GG91] and secondly more correctly in the programming language Elf [Pfe89]. Exploiting theorems from category theory [BW85] this system can be used to solve the uniform word problem for monads. The resulting decision procedure is finally implemented in Elf. 1 Introduction A general problem in category theory is to check the commutativity of certain diagrams where diagrams are nothing but a compact encoding and visualization of equations involving morphisms. In [FS90] one can even find a suitable graphical language. Checking the commutativity means to check the equality of morphisms in a given category. To support this process one can solve the uniform word problem for this category which is of course not always possible having the halting probl...
Homological Algebra of Racks and Quandles
"... Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expan ..."
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Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expansions . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Factor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Abelian extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Quandle extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Involutory extensions . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Modules 29 2.1 Rack modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Beck modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 A digression on Gmodules . . . . . . . . . . . . . . . . . . . . . 41 2.4 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 The rack
An Elementary Theory of Various Categories of Spaces in Topology
, 2005
"... In Abstract Stone Duality the topology on a space X is treated, not as an infinitary lattice, but as an exponential space ΣX. This has an associated lambda calculus, in which monadicity of the selfadjunction Σ − a Σ − makes all spaces sober and gives subspaces the subspace topology, and the Euclid ..."
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In Abstract Stone Duality the topology on a space X is treated, not as an infinitary lattice, but as an exponential space ΣX. This has an associated lambda calculus, in which monadicity of the selfadjunction Σ − a Σ − makes all spaces sober and gives subspaces the subspace topology, and the Euclidean principle Fσ ∧ σ = F> ∧ σ makes Σ the classifier for open subspaces. Computably based locally compact locales provide the leading model for these axioms, although the methods are also applicable to CCDop (constructively completely distributive lattices). In this paper we recover the textbook theories, using the additional axiom that the subcategory of overt discrete objects have a coreflection, the “underlying set ” functor. This subcategory is then a topos, and the whole category is characterised in the minimal situation as that of locally compact locales over that topos. However, by adding further axioms regarding the existence of equalisers and injectivity of Σ, we find the category of sober spaces or of locales over the topos as a reflective subcategory, whilst the whole category is cartesian closed and has all finite limits and colimits.
Categorical Term Rewriting:
, 1997
"... Abstract 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 rewrit ..."
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Abstract 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, that is, if the 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 the term structure, such as substitutions, layers, redexes etc. Drawing on the concepts of category theory, such a semantics is proposed, based on the concept of a monad, generalising the very elegant treatment of equational presentations in category theory. The theoretical basis of this work is the theory of enriched monads. It is shown how structuring operations are modelled on the level of monads, and that the semantics is compositional (it preserves the structuring operations). Modularity results can now be obtained directly at the level of combining monads without recourse to the syntax at all. As an application and demonstration of the usefulness of this approach, two modularity results for the disjoint union of two term rewriting systems are proven, the modularity of confluence (Toyama's theorem) and the modularity of strong normalization for a particular class of term rewriting systems (noncollapsing term rewriting systems). The proofs in the categorical setting provide a mild generalisation of these results.
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"... Abstract. 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 ..."
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Abstract. 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. 1
Categorical Term Rewriting:
, 1997
"... Abstract 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 rewrit ..."
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Abstract 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, that is, if the 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 the term structure, such as substitutions, layers, redexes etc. Drawing on the concepts of category theory, such a semantics is proposed, based on the concept of a monad, generalising the very elegant treatment of equational presentations in category theory. The theoretical basis of this work is the theory of enriched monads. It is shown how structuring operations are modelled on the level of monads, and that the semantics is compositional (it preserves the structuring operations). Modularity results can now be obtained directly at the level of combining monads without recourse to the syntax at all. As an application and demonstration of the usefulness of this approach, two modularity results for the disjoint union of two term rewriting systems are proven, the modularity of confluence (Toyama's theorem) and the modularity of strong normalization for a particular class of term rewriting systems (noncollapsing term rewriting systems). The proofs in the categorical setting provide a mild generalisation of these results.
Formal Software Development: From Foundations to Tools
"... This exposé gives an overview of the author’s contributions to the area of formal software development. These range from foundational issues dealing with abstract models of computation to practical engineering issues concerned with tool integration and user interface design. We can distinguish three ..."
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This exposé gives an overview of the author’s contributions to the area of formal software development. These range from foundational issues dealing with abstract models of computation to practical engineering issues concerned with tool integration and user interface design. We can distinguish three lines of work: Firstly, there is foundational work, centred around categorical models of rewriting. A new semantics for rewriting is developed, which abstracts over the concrete term structure while still being able to express key concepts such as variable, layer and substitution. It is based on the concept of a monad, which is wellknown in category theory to model algebraic theories. We generalise this treatment to term rewriting systems, infinitary terms, term graphs, and other forms of rewriting. The semantics finds applications in functional programming, where monads are used to model computational features such as state, exceptions and I/O, and modularity proofs, where
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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. 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. O...