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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 principl ..."
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Cited by 21 (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.
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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Cited by 19 (7 self)
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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
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 6 (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.
ON THE MATHEMATICAL SYNTHESIS OF EQUATIONAL LOGICS
"... Birkhoff [1935] initiated the general study of algebraic structure. Importantly for our concerns here, his development was from (universal) algebra to (equational) logic. Birkhoff’s starting point was the informal conception of algebra based on familiar concrete examples. Abstracting from these, he ..."
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Birkhoff [1935] initiated the general study of algebraic structure. Importantly for our concerns here, his development was from (universal) algebra to (equational) logic. Birkhoff’s starting point was the informal conception of algebra based on familiar concrete examples. Abstracting from these, he introduced the concepts of signature and equational presentation,
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