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<Paper uid="C92-1020">
  <Title>On the Satisfiability of Complex Constraints</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2 An example
</SectionTitle>
    <Paragraph position="0"> From an intuitive point of view, our approach attempts to explore the fact that complex constraints arising from combining descriptions of linguistic objects, tend to take the form of a large conjunction of smaller constraints.</Paragraph>
    <Paragraph position="1"> To illnstrate this approach we will use the following example from \[MK91\] originated by conjoining the agreement features of the lexi-AcrEs DE COLING-92. NANTES, 23-28 AOr3T 1992 1 0 g PROC. OF COLING-92, NANTES. AUG. 23-28, 1992 eal information for tim German words die and</Paragraph>
    <Paragraph position="3"> By looking at the top equality conjuncts in as it will be slmwn in the next section, is equivalent to the satisfiability of (1). However we should point out that the above process, wtfieh takes at most quadratic time since, by using adequate data structures, each rewrite can be done in linear time and the number of rewrites is limited by the size of the fornmla, has succeed in drastically reducing the size of the original formula.</Paragraph>
    <Paragraph position="4">  3 A rewriting system (1) we can conclude that any model of (1) is In this section we iutroduee a rewriting system, subsumed by which is essentially an adaptation to feature</Paragraph>
    <Paragraph position="6"> Using this information and the standard axioms of Feature Logics we can rewrite (1) as follows  \[(f case) = nora V (f ease) = ace\] A \[\[false A (f num)= sg\] V (f n~,n) : pl\] ^ true A true A \[\[(f .... ) = sg A (f .... ) 7 k Ben\] V \[(f,,ur.) = pl A (f e~se)C/ ,tall\] (a)  which, by using the standard rules of Propositional Calculus, can be simplified to \[(f case) =noru V (lease) = ace\] A</Paragraph>
    <Paragraph position="8"> Again, froin the atomic top conjunct in (4) we can refine (2) to obtain</Paragraph>
    <Paragraph position="10"> after which (4) reduces to \[(: case) = sort, V (f ease) ..... \] ^ (f ease) # ant (6) Since 11o atomic top conjunct remains we would now have to resort to the exponentiM algorithm to cheek the satisfiability of (6), which, structures of the one presented in \[DMV91a\] for constraints Oil first order terms.</Paragraph>
    <Paragraph position="11"> The purpose of this rewriting system is to fornlalize the rewriting process illustrated in thc previous section. The rules of the rewriting system, whidt are justified by theorems of the logic underlying features structures, have been dcsigncd with the aim of bringing out conjunctions to the top while avoiding more than a linear growth of the size of constraints.</Paragraph>
    <Paragraph position="12"> Tc~ formalize the rewriting system we will use Feature Logics \[Smo89\] and its notation. Thus we will use a and b to denote atoms, f to denote a feature name, x and y to denote variables and and t to denote either an atom or a variable.</Paragraph>
    <Paragraph position="13"> We start by recalling the notion of solved feature clause of \[Smo89\] which is the feature logic w~rsion of the standard definition of solved form \[lier30, Mah88\].</Paragraph>
    <Paragraph position="14"> A set of formulae C is said to be a solved feature clause if it satisfies the following conditions: null  1. every constraint in 6' has one of the following forms: fx-s, fx \[, x=s, x~s 2. ifx:-s is in C then x occurs exactly once in C 3. if fx-s and fx-t are in C then s = t 4. iffx I is in C, then C contains no constraint fx-s 5. if x~s is in C, then x C/ s.</Paragraph>
    <Paragraph position="15">  Smolka a\]so shows that any satisfiable set C of, possibly negated, atomic formulae can be reduced to solved form by using the following simplification rules, which again are the feature logic version of llerbrand's rules for solving sets of equations of first order terms: AcrEs DE COL1NG-92, NANTES, 23-28 Aotrr 1992 1 0 9 PROC. OF COLING-92. NANTES, AUG. 23-28, 1992  1. {x--s) LI C ---* {x &amp;quot;-s} U \[s/x\]C if * ~k .... I x occurs in C 2, {a--'=}uc-, {='----a}UC 3. {fx--s, fx--t} UC ---* {fx--s,s--f} UC 4. {s-s}uC~C 5. {Sa T}uC ~C 7. {atb}uC-*Cifutb.</Paragraph>
    <Paragraph position="16">  We call a solved feature clause positive iff it includes only constraints of the form fx-s and x-s.</Paragraph>
    <Paragraph position="17"> We can now make precise the notion of partial model used in the previous section as a positive solved feature clause .PS4.</Paragraph>
    <Paragraph position="18"> Note that the form required for A~ is essentially the one produced by an unification algorithm for feature structures.</Paragraph>
    <Paragraph position="19"> Given a set of constraints C we say that a feature clause C ~ is a minimal model of C if every model ofC r is a model of C and no proper subset of C ~ satisfies this condition.</Paragraph>
    <Paragraph position="20"> From Theorem 5.6 of \[Smo89\] we can conclude that for any C there is a finite uumber of minimal models of C.</Paragraph>
    <Paragraph position="21"> &amp;quot;\]2he aim of the CLG rewriting system is to produce from a set of constraints Co a partial model .PS4 and a smaller set of constraints C such that any minimal model of Co can be obtained by conjoining (i.e. &amp;quot;unifying&amp;quot;) a minimal model of C with .~ and moreover for any minimal model of C the reunion Ad U C is satisfiable. We start by defining a set of rewriting rules  We now define a rewriting system for pairs (34,17) by first closing C under ---*s~4 and then using the following rules</Paragraph>
    <Paragraph position="23"> with the convention that after each application of one of the rewrite rules the new partial model is reduced to solved tbrm and the resulting set of constraints is closed nmler ~M.</Paragraph>
    <Paragraph position="24"> As in \[DMV91a\] the rewrite process could be extended to use negated atomic constraints in C.</Paragraph>
    <Paragraph position="25"> We will now sketch the proof of the claims made above about the rcwriting system.</Paragraph>
    <Paragraph position="26"> We will first argue that if given an initial set of constraints Co we apply the rewriting system to (~,Co) to obtain {34,C} then Co ( ....... precisely the conjunct of all the constraints in Co) is equivalent to 3.4 UC. As a matter of fact this follows from the fact that each rewrite rule is associated with a similar meta-theorem of l:irst Order Logic and/or the axioms of Feature Logits. null As for the other property of the rewriting system, namely that the minimal models of Co are obtained by conjoining NI with those of C, we will only sketch the argument of the proof which follows from the fact that if some minimal model C' of C was inconsistent witl~ .M then, after reducing C to disjunctive form, at least one of the disjunctions s|muld subsume C ~ and thus should be inconsistent with A4 while admiring itself a model. Now, since every atomic formula occurring ill the disjunction already occurcd in C it is possible to derive a contradiction with the hypothesis that C was closed under --~M.</Paragraph>
    <Paragraph position="27"> To see this we notice that each disjunct in the disjunctive form of C can be regarded as feature clause C'. Now if A4 U C&amp;quot; was unsatisfiable then some sequence of simplification rules should lead to a clash. Now, noticing that any atomic formula in C&amp;quot; umst already be present in C, it is easy to check that any sequence of simplification rules would involve only fornmlae from C&amp;quot; and a clash with a formula in f14 would thus be impossible.</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 On the factorization of
</SectionTitle>
    <Paragraph position="0"> feature constraints Although tile rewriting system introduced in tile previous section can be seen as a factorization of tile original constraint it is also important, in accordance with tile discussion in the introduction, to further factorizing tile con-Acres DE COLING-92, NANTES, 23-28 hOt~T 1992 1 1 0 PgOC. OF COLING-92, NawrEs, Auo. 23-28, 1992 straint produced before an exponential satisfaction algorithm is api)licd.</Paragraph>
    <Paragraph position="1"> Now given a conjunction C A C' of feature constraints it is easy to provc, using argumcnts similar to the ones used in thc previuus section, that a sufficient condition for tim conjunction to be satisliablc iff each of the conjunets is independently satisfiable, can be expressed ms follows: null  1. any variable x which occurs in C in an atolnic formula x--s or s--a: does not occur in C' and vlce-versa 2. for every variable x and feature f such that  fx occurs in C, /x does nnt occur in C' and vice-versa.</Paragraph>
    <Paragraph position="2"> Note that given a conjunctiw~ wet of constraints it is possible to l)artition it into miidmal disjoint sets satisfying the. conditions above in quadratic time.</Paragraph>
    <Paragraph position="3"> Using the above criteria results in achievillg the objective described in \[MK91\] of separating not only the treatment of constraints dealing with completely indel)endcnt linguistic descriptions but also of independent phenomena for the same the \[iaguistic descriptions.</Paragraph>
  </Section>
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