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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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