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+<TITLE>Nominal Unification</TITLE>
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+<H2>Nominal Unification [<a HREF="http://www4.in.tum.de/~urbanc/Unification/nomu-tcs.ps">ps</a>]</H2>
+
+<A HREF="http://www4.in.tum.de/~urbanc/">Christian Urban</A>,
+<A HREF="http://www.cl.cam.ac.uk/~amp12/">Andrew Pitts</A>,
+Murdoch Gabbay
+<p>
+ We present a generalisation of first-order unification to the
+ practically important case of equations between terms involving
+ <i>binding operations</i>. A substitution of terms for variables
+ solves such an equation if it makes the equated terms
+ <i>alpha-equivalent</i>, i.e. equal up to renaming bound names.
+ For the applications we have in mind, we must consider the simple,
+ textual form of substitution in which names occurring in terms may
+ be captured within the scope of binders upon substitution. 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 that respects alpha-equivalence and
+ possesses good algorithmic properties. We achieve this by adapting
+ an existing idea and introducing a key new idea. The existing
+ idea is terms involving explicit substitutions of names for names,
+ except that here we only use <i>explicit permutations</i>
+
+ (bijective substitutions). The key new idea is that the
+ unification algorithm should solve not only equational problems,
+ but also problems about the <i>freshness</i> of names for
+ terms. There is a simple generalisation of the classical
+ first-order unification algorithm to this setting which retains
+ the latter's pleasant properties: unification problems involving
+ alpha-equivalence and freshness are decidable; and solvable
+ problems possess most general solutions.
+<BR><BR>
+
+In an <a HREF="http://www4.in.tum.de/~urbanc/Unification/app-nomu.ps">appendix</a> we
+discuss some issues about the relationship between nominal unification and higher-order
+pattern unification.
+
+<BR><BR><BR><BR>
+All results in the paper have been verified in
+<A HREF="http://www.cl.cam.ac.uk/Research/HVG/Isabelle/">Isabelle</A> (2004 version).
+Below are the theory files.
+
+<ul>
+<li><A HREF="Unification/Swap.thy">Swap.thy</A>
+ <br /> amongst other facts proves that swapping is a bijection (on atoms)
+<li><A HREF="Unification/Atoms.thy">Atoms.thy</A>
+ <br /> facts about atoms occurring in swappings
+<li><A HREF="Unification/Terms.thy">Terms.thy</A>
+ <br /> defines terms, occurs check and the notion of subterms
+<li><A HREF="Unification/Disagreement.thy">Disagreement.thy</A>
+ <br /> proves various facts about disagreement sets
+<li><A HREF="Unification/Fresh.thy">Fresh.thy</A>
+ <br /> defines the freshness relation and shows facts about its behaviour under swapping
+<li><A HREF="Unification/PreEqu.thy">PreEqu.thy</A>
+ <br /> defines the relation capturing the notion of alpha-equivalence (on open terms)
+ and proves a long lemma by mutual induction over the depth of terms which
+ is needed to show that the relation is an equivalence relation
+<li><A HREF="Unification/Equ.thy">Equ.thy</A>
+ <br /> proves various facts about the equivalence relations
+<li><A HREF="Unification/Substs.thy">Substs.thy</A>
+ <br /> defines substitutions and composition of substitutions, and establishes
+ some facts of substitution and our equivalence relation
+<li><A HREF="Unification/Mgu.thy">Mgu.thy</A>
+ <br /> defines the notion of unification problems and reduction rules over sets
+ of such problems; proves that every reduction leading to the empty set
+ produces an mgu
+<li><A HREF="Unification/Termination.thy">Termination.thy</A>
+ <br /> shows that every reduction reduces a (well-founded) measure, thus
+ proves that every reduction sequence must terminate
+<li><A HREF="Unification/Unification.thy">Unification.thy</A>
+ <br /> proves that all solvable problems reduce only to the empty set
+ (the "ok" configuration which provides an mgu) and all unsolvable
+ problems reduce to a "fail" configuration
+ (for which no unifier exists)
+</uL><BR>
+The old Isabelle-2002 files can be downloaded
+<A HREF="http://www4.in.tum.de/~urbanc/Unification/nomu-2002.tgz">here</A>.
+<BR><BR>
+A "quick and dirty" implementation of nominal unification in
+<A HREF="http://caml.inria.fr/">Ocaml</A> can be
+downloaded
+<A HREF="http://www4.in.tum.de/~urbanc/Unification/unification.ml">elsewhere</A>.
+
+
+<BR><BR><BR><BR>
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