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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.
A Variant of HigherOrder AntiUnification
"... We present a rulebased Huet’s style antiunification algorithm for simplytyped lambdaterms, which computes a least general higherorder pattern generalization. For a pair of arbitrary terms of the same type, such a generalization always exists and is unique modulo αequivalence. The algorithm com ..."
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We present a rulebased Huet’s style antiunification algorithm for simplytyped lambdaterms, which computes a least general higherorder pattern generalization. For a pair of arbitrary terms of the same type, such a generalization always exists and is unique modulo αequivalence. The algorithm computes it in cubic time within linear space. It has been implemented and the code is freely available.
Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free
 Journal of Symbolic Logic
, 2012
"... By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissivenominal models, so the construction hinges on generating from an insta ..."
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By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissivenominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atomsabstraction. The results are of interest in their own right, but also, we factor the mathematics so as to maximise the chances that it could be used offtheshelf for other nominal reasoning systems too. Models with infinite support can be easier to work with, so it is useful to have a semiautomatic theorem to transfer results from classes of infinitelysupported nominal models to the more restricted class of models with finite support. In conclusion, we consider different permissivenominal syntaxes and nominal models and discuss how they relate to the results proved here.
Dependent Types for a Nominal Logical Framework
, 2012
"... We present a logical framework based on the nominal approach to representing syntax with binders. First we extend nominal terms, which have a builtin nameabstraction operator and a firstorder notion of substitution for variables, with a captureavoiding substitution operator for names. We then bu ..."
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We present a logical framework based on the nominal approach to representing syntax with binders. First we extend nominal terms, which have a builtin nameabstraction operator and a firstorder notion of substitution for variables, with a captureavoiding substitution operator for names. We then build a dependent type system for this extended syntax
Nominal Henkin Semantics: simplytyped
"... lambdacalculus models in nominal sets ..."
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connecting the logic of permutation models with the logic
"... From nominal sets binding to functions and λabstraction: ..."
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unknown title
, 2010
"... Twolevel nominal sets and semantic nominal terms: an extension of nominal set theory for handling metavariables ..."
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Twolevel nominal sets and semantic nominal terms: an extension of nominal set theory for handling metavariables