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2011): Nominal terms and nominal logics: from foundations to metamathematics
 In: Handbook of Philosophical Logic
"... ABSTRACT: Nominal techniques concern the study of names using mathematical semantics. Whereas in much previous work names in abstract syntax were studied, here we will study them in metamathematics. More specifically, we survey the application of nominal techniques to languages for unification, rew ..."
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ABSTRACT: Nominal techniques concern the study of names using mathematical semantics. Whereas in much previous work names in abstract syntax were studied, here we will study them in metamathematics. More specifically, we survey the application of nominal techniques to languages for unification, rewriting, algebra, and firstorder logic. What characterises the languages of this chapter is that they are firstorder in character, and yet they can specify and reason on names. In the languages we develop, it will be fairly straightforward to give firstorder ‘nominal ’ axiomatisations of namerelated things like alphaequivalence, captureavoiding substitution, beta and etaequivalence, firstorder logic with its quantifiers—and as we shall see, also arithmetic. The formal axiomatisations we arrive at will closely resemble ‘natural behaviour’; the specifications we see typically written out in normal mathematical usage. This is possible because of a novel namecarrying semantics in nominal sets, through which our languages will have namepermutations and termformers that can bind as primitive builtin features.
Investigations into Algebra and Topology over Nominal Sets
, 2011
"... The last decade has seen a surge of interest in nominal sets and their applications to formal methods for programming languages. This thesis studies two subjects: algebra and duality in the nominal setting. In the first part, we study universal algebra over nominal sets. At the heart of our approach ..."
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The last decade has seen a surge of interest in nominal sets and their applications to formal methods for programming languages. This thesis studies two subjects: algebra and duality in the nominal setting. In the first part, we study universal algebra over nominal sets. At the heart of our approach lies the existence of an adjunction of descent type between nominal sets and a category of manysorted sets. Hence nominal sets are a full reflective subcategory of a manysorted variety. This is presented in Chapter 2. Chapter 3 introduces functors over manysorted varieties that can be presented by operations and equations. These are precisely the functors that preserve sifted colimits. They play a central role in Chapter 4, which shows how one can systematically transfer results of universal algebra from a manysorted variety to nominal sets. However, the equational logic obtained is more expressive than the nominal equational logic of Clouston and Pitts, respectively, the nominal algebra of Gabbay and Mathijssen. A uniform fragment of our logic with the same expressivity
Instances of computational effects: an algebraic perspective
"... Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different ins ..."
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Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different instances of a particular computational effect. We use our framework to give a general account of several notions of computation that had previously been analyzed in terms of monads on presheaf categories: the analysis of local store by Plotkin and Power; the analysis of restriction by Pitts; and the analysis of the pi calculus by Stark. I.
Nominal semantics for predicate logic: algebras, substitution, quantifiers
"... and limits ..."
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Unity in nominal equational reasoning: the algebra of
"... equality on nominal sets ..."
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