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Nominal Unification [chapter]

Christian Urban, Andrew Pitts, Murdoch Gabbay
2003 Lecture Notes in Computer Science  
We are able to take a "nominal" approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijn-style representations) and yet get a version of this form of substitution  ...  See http://www.cl.cam.ac.uk/users/cu200/Unification for the Isabelle proof scripts.  ...  A machine-checked proof of all results using the theorem prover Isabelle can be found at http://www.cl.cam.ac.uk/users/cu200/Unification.  ... 
doi:10.1007/978-3-540-45220-1_41 fatcat:lsqsitcvyzdoxlhmwesba5tbdq

Nominal unification

Christian Urban, Andrew M. Pitts, Murdoch J. Gabbay
2004 Theoretical Computer Science  
We are able to take a "nominal" approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijn-style representations) and yet get a version of this form of substitution  ...  See http://www.cl.cam.ac.uk/users/cu200/Unification for the Isabelle proof scripts.  ...  A machine-checked proof of all results using the theorem prover Isabelle can be found at http://www.cl.cam.ac.uk/users/cu200/Unification.  ... 
doi:10.1016/j.tcs.2004.06.016 fatcat:5yt6t3bd7nhchenstde5xlid7q

Nominal C-Unification [article]

Mauricio Ayala-Rincón, Washington de Carvalho-Segundo, Maribel Fernández, Daniele Nantes-Sobrinho
2017 arXiv   pre-print
We propose a sound and complete procedure for nominal unification with commutative operators, or nominal C-unification for short, which has been formalised in Coq.  ...  Nominal unification is an extension of first-order unification that takes into account the \alpha-equivalence relation generated by binding operators, following the nominal approach.  ...  The nominal C-unification problem is to decide, for a given P, if U C (P) is non empty; that is, whether P has nominal C-unifiers.  ... 
arXiv:1709.05384v1 fatcat:fi4d4l6mobc3pc4x4hjij2gnhy

Unifying Nominal Unification

Christophe Calvès, Marc Herbstritt
2013 International Conference on Rewriting Techniques and Applications  
Nominal unification is proven to be quadratic in time and space.  ...  This isomorphism is of prime importance to prove intricate properties on both sides and a step further to the real complexity of nominal unification.  ...  Urban, Pitts and Gabbay [13] showed that nominal unification, i.e. unification up to α-equivalence, is decidable.  ... 
doi:10.4230/lipics.rta.2013.143 dblp:conf/rta/Calves13 fatcat:dksjrtjbpfh2fgxiz22ijz443y

Implementing Nominal Unification

Christophe Calvès, Maribel Fernández
2007 Electronical Notes in Theoretical Computer Science  
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.  ...  In this paper we describe two implementations of nominal unification (we have actually implemented the extended nominal unification algorithm of [4] ).  ... 
doi:10.1016/j.entcs.2006.09.027 fatcat:oq3nhp3x2ffklifcjjv3nchs34

Nominal Unification Revisited

Christian Urban
2010 Electronic Proceedings in Theoretical Computer Science  
The purpose of this paper is to simplify a clunky proof from the original paper on nominal unification and to give an overview over some results about nominal unification.  ...  Nominal unification calculates substitutions that make terms involving binders equal modulo alpha-equivalence.  ...  Unification problems solved by nominal unification occur frequently in practice.  ... 
doi:10.4204/eptcs.42.1 fatcat:xzpsv26e75fc3f6wvzdip3lcoy

An Efficient Nominal Unification Algorithm

Jordi Levy, Mateu Villaret, Marc Herbstritt
2010 International Conference on Rewriting Techniques and Applications  
Nominal Unification is an extension of first-order unification where terms can contain binders and unification is performed modulo α-equivalence.  ...  First, we linearly-reduce nominal unification problems to a sequence of freshness and equalities between atoms, modulo a permutation, using ideas as Paterson and Wegman for first-order unification.  ...  The Nominal unification problem a 3 .a 2 .a 1 .f (c 1 , c 2 ) ?  ... 
doi:10.4230/lipics.rta.2010.209 dblp:conf/rta/LevyV10 fatcat:ztxn77f7c5cjjdhicziok6u34u

Nominal Unification from a Higher-Order Perspective

Jordi Levy, Mateu Villaret
2012 ACM Transactions on Computational Logic  
From this view, we show that nominal unification can be reduced to a particular fragment of higher-order unification problems: Higher-Order Pattern Unification.  ...  Despite these differences, nominal unification can be seen from a higher-order perspective.  ...  Nominal Unification is quadratic reducible to Higher-Order Pattern Unification. Nominal Unification can be decided in quadratic deterministic time.  ... 
doi:10.1145/2159531.2159532 fatcat:taumfjush5gz7c4on2yyy3aoxu

Nominal Unification with Atom and Context Variables

Manfred Schmidt-Schauß, David Sabel, Michael Wagner
2018 International Conference on Rewriting Techniques and Applications  
Our proposal is to extend nominal unification to solve these unification problems. We succeeded in constructing the nominal unification algorithm NomUnifyASD.  ...  We define ASD1-unification problems, which are higher-order equational unification problems employing variables for atoms, expressions and contexts, with additional distinct-variable constraints, and which  ...  Our proposal is to extend nominal unification to solve these unification problems.  ... 
doi:10.4230/lipics.fscd.2018.28 dblp:conf/rta/Schmidt-Schauss18 fatcat:dldzfeytmrhxhh2mr3cgeyoeem

Fixed-Point Constraints for Nominal Equational Unification

Mauricio Ayala-Rincón, Maribel Fernández, Daniele Nantes-Sobrinho, Michael Wagner
2018 International Conference on Rewriting Techniques and Applications  
This gives rise to a new notion of nominal unification, where solutions for unification problems are pairs of a fixed-point context and a substitution.  ...  are considered: for example, nominal unification remains finitary in the presence of commutativity, whereas it becomes infinitary when unifiers are expressed using freshness contexts.  ...  Nominal Unification via fixed-point constraints In this section we define the notion of nominal unification in terms of fixed-point constraints.  ... 
doi:10.4230/lipics.fscd.2018.7 dblp:conf/rta/Ayala-RinconFN18 fatcat:i2v4hes4srdvdoyvmxs7grvocq

Nominal Unification and Matching of Higher Order Expressions with Recursive Let [article]

Manfred Schmidt-Schauß and Temur Kutsia and Jordi Levy and Mateu Villaret and Yunus Kutz
2022 arXiv   pre-print
A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time.  ...  In addition we prove that there is a guessing strategy for nominal unification with letrec and atom-variable that is a trade-off between exponential growth and non-determinism.  ...  We omit failure rules, since these obviously follow from the nominal matching algorithm.  ... 
arXiv:2102.08146v4 fatcat:6m6lzbidl5fp5p27ut4csmafee

Completeness in PVS of a Nominal Unification Algorithm

Mauricio Ayala-Rincón, Maribel Fernández, Ana Cristina Rocha-Oliveira
2016 Electronical Notes in Theoretical Computer Science  
Functional nominal computation can be modelled through nominal rewriting, in which α-equivalence, nominal matching and nominal unification play an important role.  ...  In this work, we present a new specification of nominal unification in the language of PVS and a formalisation of its completeness.  ...  Definition 4. 2 ( 2 Nominal Unification Function) Let t, s be two nominal terms.  ... 
doi:10.1016/j.entcs.2016.06.005 fatcat:pgsndhnpbfd5nlpbjht7qo3fge

Nominal Syntax with Atom Substitutions: Matching, Unification, Rewriting [chapter]

Jesús Domínguez, Maribel Fernández
2019 Lecture Notes in Computer Science  
Nominal extensions have been developed to deal with syntax involving binding operators: nominal unification takes into account α-equivalence; however, it does not take into account non-capturing substitutions  ...  We consider an extension of nominal syntax with non-capturing substitutions and show that matching is decidable and finitary but unification is undecidable.  ...  Kumar and Norrish [19] also studied efficient forms of nominal unification. Cheney [9] proved that a more general version of nominal unification, called equivariant unification, is NP-complete.  ... 
doi:10.1007/978-3-030-25027-0_5 fatcat:fscghmsaurdvpghhxxd45ufegq

(Nominal) Unification by Recursive Descent with Triangular Substitutions [chapter]

Ramana Kumar, Michael Norrish
2010 Lecture Notes in Computer Science  
We mechanise termination and correctness for two unification algorithms, written in a recursive descent style.  ...  One computes unifiers for first order terms, the other for nominal terms (terms including α-equivalent binding structure).  ...  Nominal Unification Classical unification works over first-order terms.  ... 
doi:10.1007/978-3-642-14052-5_6 fatcat:dnc4a3cjcrhp3bjxxptdsfazte

Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques

Gilles Dowek, Murdoch J. Gabbay, Dominic P. Mulligan
2010 Logic Journal of the IGPL  
We investigate the relation between permissive nominal unification and higher-order pattern unification.  ...  We prove that expressivity is not lost moving to the permissive case and provide an injection of nominal terms unification problems and their solutions into permissive nominal terms problems and their  ...  Therefore, we relate permissive nominal terms and their unification to nominal unification, and to unification of higher-order patterns.  ... 
doi:10.1093/jigpal/jzq006 fatcat:z44woxiotzfnjnpn46pjwtsdqu
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