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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.
Dependent Types for Nominal Terms with Atom Substitutions
"... Abstract Nominal terms are an extended firstorder language for specifying and verifying properties of syntax with binding. Founded upon the semantics of nominal sets, the success of nominal terms with regard to systems of equational reasoning is already well established. This work first extends th ..."
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Abstract Nominal terms are an extended firstorder language for specifying and verifying properties of syntax with binding. Founded upon the semantics of nominal sets, the success of nominal terms with regard to systems of equational reasoning is already well established. This work first extends the untyped language of nominal terms with a notion of noncapturing atom substitution for objectlevel names and then proposes a dependent type system for this extended language. Both these contributions are intended to serve as a prelude to a future nominal logical framework based upon nominal equational reasoning and thus an extended example is given to demonstrate that this system is capable of encoding various other formal systems of interest.
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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