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Nominal rewriting
 Information and Computation
"... Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting ma ..."
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Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the objectlanguage (atoms) and of the metalanguage (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced firstorder character, since substitution of terms for variables is not captureavoiding. We show how good properties of firstorder rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem
The Complexity of Equivariant Unification
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004), volume 3142 of LNCS
"... Nominal logic is a firstorder theory of names and binding based on a primitive operation of swapping rather than substitution. Urban, Pitts, and Gabbay have developed a nominal unification algorithm that unifies terms up to nominal equality. However, because of nominal logic's equivariance pri ..."
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Cited by 32 (7 self)
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Nominal logic is a firstorder theory of names and binding based on a primitive operation of swapping rather than substitution. Urban, Pitts, and Gabbay have developed a nominal unification algorithm that unifies terms up to nominal equality. However, because of nominal logic's equivariance principle, atomic formulas can be provably equivalent without being provably equal as terms, so resolution using nominal unification is sound but incomplete. For complete resolution, a more general form of unification called equivariant unification, or "unification up to a permutation" is required. Similarly, for rewrite rules expressed in nominal logic, a more general form of matching called equivariant matching is necessary. In this paper, we study the complexity of the decision problem for equivariant unification and matching. We show that these problems are NPcomplete in general. However, when one of the terms is essentially firstorder, equivariant and nominal unification coincide. This shows that equivariant unification can be performed efficiently in many interesting common cases: for example, anypurely firstorder logic program or rewrite system can be run efficiently on nominal terms.
Captureavoiding Substitution as a Nominal Algebra
 Formal Aspects of Computing
, 2008
"... Abstract. Substitution is fundamental to computer science, underlying for example quantifiers in predicate logic and betareduction in the lambdacalculus. So is substitution something we define on syntax on a casebycase basis, or can we turn the idea of ‘substitution ’ into a mathematical objec ..."
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Abstract. Substitution is fundamental to computer science, underlying for example quantifiers in predicate logic and betareduction in the lambdacalculus. So is substitution something we define on syntax on a casebycase basis, or can we turn the idea of ‘substitution ’ into a mathematical object? We exploit the new framework of Nominal Algebra to axiomatise substitution. We prove our axioms sound and complete with respect to a canonical model; this turns out to be quite hard, involving subtle use of results of rewriting and algebra. 1
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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Cited by 14 (9 self)
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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.
Implementing nominal unification
 In 3rd Int. Workshop on Term Graph Rewriting (TERMGRAPH’06), Vienna, Electronic
"... Nominal matching and unification underly the dynamics of nominal rewriting. Urban, Pitts and Gabbay gave a nominal unification algorithm which finds the most general solution to a nominal matching or unification problem, if one exists. Later the algorithm was extended by Fernández and Gabbay to deal ..."
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Cited by 9 (2 self)
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Nominal matching and unification underly the dynamics of nominal rewriting. Urban, Pitts and Gabbay gave a nominal unification algorithm which finds the most general solution to a nominal matching or unification problem, if one exists. Later the algorithm was extended by Fernández and Gabbay to deal with name generation and locality. In this paper we describe first a direct implementation of the nominal unification algorithm, including the extensions, in Maude. This implementation is not efficient (it is exponential in time), but we will show that we can obtain a feasible implementation by using termgraphs.
Principal Types for Nominal Theories
"... Abstract. We define rank 1 polymorphic types for nominal rewrite rules and equations. Typing environments type atoms, variables, and function symbols, and since we follow a Currystyle approach there is no need to fully annotate terms with types. Our system has principal types, and we give rule and ..."
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Abstract. We define rank 1 polymorphic types for nominal rewrite rules and equations. Typing environments type atoms, variables, and function symbols, and since we follow a Currystyle approach there is no need to fully annotate terms with types. Our system has principal types, and we give rule and axiom formats to guarantee preservation of types under both rewriting and equality reasoning. This is nontrivial because substitution does not avoid capture, so a substituted symbol can—if we are not careful—appear in inconsistent typing contexts.
Contents Preface v
, 2006
"... Rewriting (your) calculus............................................ 1 ..."
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GDP Festschrift ENTCS, to appear Abstract Nominal Equational Logic
"... This paper studies the notion of “freshness ” that often occurs in the metatheory of computer science languages involving various kinds of names. Nominal Equational Logic is an extension of ordinary equational logic with assertions about the freshness of names. It is shown to be both sound and comp ..."
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This paper studies the notion of “freshness ” that often occurs in the metatheory of computer science languages involving various kinds of names. Nominal Equational Logic is an extension of ordinary equational logic with assertions about the freshness of names. It is shown to be both sound and complete for the support interpretation of freshness and equality provided by the GabbayPitts nominal sets model of names, binding and αconversion.