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5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	4	<TITLE>Nominal Unification</TITLE>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	5	<BASE HREF="http://www4.in.tum.de/~urbanc/Unification">
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5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	25	<H2>Nominal Unification [<a HREF="http://www4.in.tum.de/~urbanc/Unification/nomu-tcs.ps">ps</a>]</H2>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	26
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	27	<A HREF="http://www4.in.tum.de/~urbanc/">Christian Urban</A>,
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	28	<A HREF="http://www.cl.cam.ac.uk/~amp12/">Andrew Pitts</A>,
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	29	Murdoch Gabbay
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	30	<p>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	31	We present a generalisation of first-order unification to the
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	32	practically important case of equations between terms involving
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	33	<i>binding operations</i>. A substitution of terms for variables
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	34	solves such an equation if it makes the equated terms
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	35	<i>alpha-equivalent</i>, i.e. equal up to renaming bound names.
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	36	For the applications we have in mind, we must consider the simple,
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	37	textual form of substitution in which names occurring in terms may
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	38	be captured within the scope of binders upon substitution. We are
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	39	able to take a "nominal" approach to binding in which bound
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	40	entities are explicitly named (rather than using nameless,
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	41	de Bruijn-style representations) and yet get a version of this
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	42	form of substitution that respects alpha-equivalence and
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	43	possesses good algorithmic properties. We achieve this by adapting
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	44	an existing idea and introducing a key new idea. The existing
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	45	idea is terms involving explicit substitutions of names for names,
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	46	except that here we only use <i>explicit permutations</i>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	47
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	48	(bijective substitutions). The key new idea is that the
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	49	unification algorithm should solve not only equational problems,
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	50	but also problems about the <i>freshness</i> of names for
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	51	terms. There is a simple generalisation of the classical
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	52	first-order unification algorithm to this setting which retains
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	53	the latter's pleasant properties: unification problems involving
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	54	alpha-equivalence and freshness are decidable; and solvable
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	55	problems possess most general solutions.
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	56	<BR><BR>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	57
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	58	In an <a HREF="http://www4.in.tum.de/~urbanc/Unification/app-nomu.ps">appendix</a> we
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	59	discuss some issues about the relationship between nominal unification and higher-order
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	60	pattern unification.
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	61
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	62	<BR><BR><BR><BR>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	63	All results in the paper have been verified in
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	64	<A HREF="http://www.cl.cam.ac.uk/Research/HVG/Isabelle/">Isabelle</A> (2004 version).
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	65	Below are the theory files.
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	66
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	67	<ul>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	68	<li><A HREF="Unification/Swap.thy">Swap.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	69	<br /> amongst other facts proves that swapping is a bijection (on atoms)
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	70	<li><A HREF="Unification/Atoms.thy">Atoms.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	71	<br /> facts about atoms occurring in swappings
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	72	<li><A HREF="Unification/Terms.thy">Terms.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	73	<br /> defines terms, occurs check and the notion of subterms
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	74	<li><A HREF="Unification/Disagreement.thy">Disagreement.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	75	<br /> proves various facts about disagreement sets
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	76	<li><A HREF="Unification/Fresh.thy">Fresh.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	77	<br /> defines the freshness relation and shows facts about its behaviour under swapping
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	78	<li><A HREF="Unification/PreEqu.thy">PreEqu.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	79	<br /> defines the relation capturing the notion of alpha-equivalence (on open terms)
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	80	and proves a long lemma by mutual induction over the depth of terms which
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	81	is needed to show that the relation is an equivalence relation
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	82	<li><A HREF="Unification/Equ.thy">Equ.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	83	<br /> proves various facts about the equivalence relations
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	84	<li><A HREF="Unification/Substs.thy">Substs.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	85	<br /> defines substitutions and composition of substitutions, and establishes
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	86	some facts of substitution and our equivalence relation
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	87	<li><A HREF="Unification/Mgu.thy">Mgu.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	88	<br /> defines the notion of unification problems and reduction rules over sets
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	89	of such problems; proves that every reduction leading to the empty set
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	90	produces an mgu
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	91	<li><A HREF="Unification/Termination.thy">Termination.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	92	<br /> shows that every reduction reduces a (well-founded) measure, thus
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	93	proves that every reduction sequence must terminate
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	94	<li><A HREF="Unification/Unification.thy">Unification.thy</A>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	95	<br /> proves that all solvable problems reduce only to the empty set
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	96	(the "ok" configuration which provides an mgu) and all unsolvable
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	97	problems reduce to a "fail" configuration
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	98	(for which no unifier exists)
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	99	</uL><BR>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	100	The old Isabelle-2002 files can be downloaded
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	101	<A HREF="http://www4.in.tum.de/~urbanc/Unification/nomu-2002.tgz">here</A>.
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	102	<BR><BR>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	103	A "quick and dirty" implementation of nominal unification in
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	104	<A HREF="http://caml.inria.fr/">Ocaml</A> can be
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	105	downloaded
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	106	<A HREF="http://www4.in.tum.de/~urbanc/Unification/unification.ml">elsewhere</A>.
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	107
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	108
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	109	<BR><BR><BR><BR>
5c816239deaa added Unification subdirectory Christian Urban <urbanc@in.tum.de> parents: diff changeset	110
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author	Christian Urban <christian.urban@kcl.ac.uk>
	Fri, 23 Feb 2024 11:59:17 +0000
changeset 645	552a6b834d43
parent 107	5c816239deaa
permissions	-rw-r--r--