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<TITLE>Nominal Unification</TITLE>
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<BASE HREF="http://www4.in.tum.de/~urbanc/Unification">
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<H2>Nominal Unification [<a HREF="http://www4.in.tum.de/~urbanc/Unification/nomu-tcs.ps">ps</a>]</H2>
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<A HREF="http://www4.in.tum.de/~urbanc/">Christian Urban</A>,
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<A HREF="http://www.cl.cam.ac.uk/~amp12/">Andrew Pitts</A>,
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Murdoch Gabbay
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<p>
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We present a generalisation of first-order unification to the
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practically important case of equations between terms involving
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<i>binding operations</i>. A substitution of terms for variables
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solves such an equation if it makes the equated terms
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<i>alpha-equivalent</i>, i.e. equal up to renaming bound names.
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For the applications we have in mind, we must consider the simple,
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textual form of substitution in which names occurring in terms may
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be captured within the scope of binders upon substitution. We are
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able to take a "nominal" approach to binding in which bound
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entities are explicitly named (rather than using nameless,
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de Bruijn-style representations) and yet get a version of this
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form of substitution that respects alpha-equivalence and
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possesses good algorithmic properties. We achieve this by adapting
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an existing idea and introducing a key new idea. The existing
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idea is terms involving explicit substitutions of names for names,
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except that here we only use <i>explicit permutations</i>
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(bijective substitutions). The key new idea is that the
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unification algorithm should solve not only equational problems,
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but also problems about the <i>freshness</i> of names for
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terms. There is a simple generalisation of the classical
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first-order unification algorithm to this setting which retains
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the latter's pleasant properties: unification problems involving
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alpha-equivalence and freshness are decidable; and solvable
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problems possess most general solutions.
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<BR><BR>
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In an <a HREF="http://www4.in.tum.de/~urbanc/Unification/app-nomu.ps">appendix</a> we
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discuss some issues about the relationship between nominal unification and higher-order
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pattern unification.
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<BR><BR><BR><BR>
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All results in the paper have been verified in
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<A HREF="http://www.cl.cam.ac.uk/Research/HVG/Isabelle/">Isabelle</A> (2004 version).
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Below are the theory files.
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<ul>
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<li><A HREF="Unification/Swap.thy">Swap.thy</A>
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<br /> amongst other facts proves that swapping is a bijection (on atoms)
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<li><A HREF="Unification/Atoms.thy">Atoms.thy</A>
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<br /> facts about atoms occurring in swappings
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<li><A HREF="Unification/Terms.thy">Terms.thy</A>
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<br /> defines terms, occurs check and the notion of subterms
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<li><A HREF="Unification/Disagreement.thy">Disagreement.thy</A>
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<br /> proves various facts about disagreement sets
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<li><A HREF="Unification/Fresh.thy">Fresh.thy</A>
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<br /> defines the freshness relation and shows facts about its behaviour under swapping
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<li><A HREF="Unification/PreEqu.thy">PreEqu.thy</A>
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<br /> defines the relation capturing the notion of alpha-equivalence (on open terms)
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and proves a long lemma by mutual induction over the depth of terms which
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is needed to show that the relation is an equivalence relation
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<li><A HREF="Unification/Equ.thy">Equ.thy</A>
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<br /> proves various facts about the equivalence relations
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<li><A HREF="Unification/Substs.thy">Substs.thy</A>
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<br /> defines substitutions and composition of substitutions, and establishes
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some facts of substitution and our equivalence relation
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<li><A HREF="Unification/Mgu.thy">Mgu.thy</A>
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<br /> defines the notion of unification problems and reduction rules over sets
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of such problems; proves that every reduction leading to the empty set
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produces an mgu
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<li><A HREF="Unification/Termination.thy">Termination.thy</A>
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<br /> shows that every reduction reduces a (well-founded) measure, thus
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proves that every reduction sequence must terminate
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<li><A HREF="Unification/Unification.thy">Unification.thy</A>
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<br /> proves that all solvable problems reduce only to the empty set
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(the "ok" configuration which provides an mgu) and all unsolvable
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problems reduce to a "fail" configuration
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(for which no unifier exists)
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</uL><BR>
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The old Isabelle-2002 files can be downloaded
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<A HREF="http://www4.in.tum.de/~urbanc/Unification/nomu-2002.tgz">here</A>.
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<BR><BR>
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A "quick and dirty" implementation of nominal unification in
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<A HREF="http://caml.inria.fr/">Ocaml</A> can be
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downloaded
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<A HREF="http://www4.in.tum.de/~urbanc/Unification/unification.ml">elsewhere</A>.
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<BR><BR><BR><BR>
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</TABLE>
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<P><!-- Created: Tue Mar 4 00:23:25 GMT 1997 -->
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