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<Paper uid="P90-1022">
  <Title>EXPRESSING DISJUNCTIVE AND NEGATIVE FEATURE CONSTRAINTS WITH CLASSICAL FIRST-ORDER LOGIC.</Title>
  <Section position="3" start_page="174" end_page="175" type="metho">
    <SectionTitle>
AXIOMATIZING ATTRIBUTE-VALUE
STRUCTURES
</SectionTitle>
    <Paragraph position="0"> This section shows how attribute-value structures can be axiomatized using first-order quantifier-free formulae with equality. In the next section we see that equality and inequality constraints on the values of the attributes can also be expressed as such formulae, so systems of these constraints can be solved using standard techniques such as the Congruence Closure algorithm \[15\], \[5\].</Paragraph>
    <Paragraph position="1"> The elements of the attribute-value structure, both constant and complex, together with an additional element +- constitute the domain of individuals of the intended interpretation. The attributes are unary partial functions over this domain (i.e. mappings from elements to elements) which are always undefined on constant elements. We capture this partiality by the standard technique of adding an additional element 3_ to the domain to serve as the value 'undefined'. Thus a(x) = 3_ if x does not have an attribute a, otherwise a(x) is the value of x's attribute a.</Paragraph>
    <Paragraph position="2"> We proceed by specifying the conditions an interpretation must satisfy to be an attribute-value structure. Modelling attributes with functions automatically requires attributes to be single-valued, as required.</Paragraph>
    <Paragraph position="3"> Axiom schema (A1)describes the properties of constants. It expresses the requirement that constants have no attributes.</Paragraph>
    <Paragraph position="4"> Axiom schema (A2) requires that distinct constant symbols denote distinct elements in any satisfying model. Without (A2) it would be possible for two distinct constant elements, say singular and plural, to denote the same individual. 2 Axiom schema (A3) and (A4) state the properties of the &amp;quot;undefined value&amp;quot; 3-. It has no attributes, and it is distinct from all of the constants (and from all other elements as well this will be enforced by the translation of equality constraints).</Paragraph>
    <Paragraph position="5"> This completes the axiomatization. This axiomatization is finite iff the sets of attribute symbols and constant symbols are finite: in the intended computational and linguistic applications this is always the case. The claim is that any interpretation which satisfies all of these The close relationship between quantifier-free formulae and attribute-value constraints was first noted in Kaplan and Bresnan \[10\].</Paragraph>
    <Paragraph position="6">  Such a schema is required because we are concerned with satisfiability rather than validity (as in e.g. logic programming).</Paragraph>
    <Paragraph position="7"> axioms is an attribute-value structure; i.e. (A1) - null (A4) constitute a definition of attribute-value structures.</Paragraph>
    <Paragraph position="8"> Example (continued): The interpretation corresponding to the attribute-value structure (1) has as its domain the set D = { el ..... e6,  singular, third, 3-}. The attributes denote functions from D to D. For example, agr denotes the function whose value is 3_ except on e2 and es, where its values are e4 and e6 respectively. It is straight-forward to check that all the axioms hold in the three attribute-value structures given above.</Paragraph>
    <Paragraph position="9"> In fact, any model for these axioms can be regarded as a (possibly infinite and disconnected) attribute-value feature structure, where the model's individuals are the elements or nodes, the unary functions define how attributes take their values, the constant symbols denote constant elements, and _L is a sink state.</Paragraph>
  </Section>
  <Section position="4" start_page="175" end_page="175" type="metho">
    <SectionTitle>
EXPRESSING CONSTRAINTS AS
QUANTIFIER-FREE FORMULAE.
</SectionTitle>
    <Paragraph position="0"> Various notations are currently used to express attribute-value constraints: the constraint requiring that the value of attribute a of (the entity denoted by) x is (the entity denoted by) y is written as (x a&gt; = y in PATR-II \[19\], as (x a) = y in LFG \[10\], and as x(a) = y in \[9\], for example.</Paragraph>
    <Paragraph position="1"> At the risk of further confusion we use another notation here, and write the constraint requiring that the value of attribute a of x is y as a(x) = y.</Paragraph>
    <Paragraph position="2"> This notation emphasises the fact that attributes are modelled by functions, and simplifies the definition of '-'.</Paragraph>
    <Paragraph position="3"> Clearly for an attribute-value structure to satisfy the constraint u = v then u and v must denote the same element, i.e. u = v. However this is not a sufficient condition: num(x) = num(y) is not satisfied if num(x) or num(y) is I. Thus it is necessary that the arguments of '=' denote identical elements distinct from the denotation of_L.</Paragraph>
    <Paragraph position="4"> Even though there are infinitely many instances of the schema in (A5) (since there are infinitely many terms) this is not problematic, since u = v can be regarded as an abbreviation for U=VAU~:/.</Paragraph>
    <Paragraph position="5"> Thus equality constraints on attribute-value structures can be expressed with quantifier-free formulae with equality. We use classically interpreted boolean connectives to express conjunctive, disjunctive and negative feature constraints.</Paragraph>
    <Paragraph position="6"> Example (continued): Suppose each variable xi denotes the corresponding e i, 1 &lt;_i &lt;_11, of(l) and (2). Then subj(xl) ~ x2,</Paragraph>
    <Paragraph position="8"> hence x8 ~Xll is true. Thus &amp;quot; ~&amp;quot; means &amp;quot;not identical to&amp;quot; or &amp;quot;not unified with&amp;quot;, rather than &amp;quot;not unifiable with&amp;quot;.</Paragraph>
    <Paragraph position="9"> Further, since agr(xl ) = J-, agr( x l ) = agr(x l ) is false, even though agr(xl) = agr(xl) is true. Thus t = t is not a theorem because of the possibility that t = J_.</Paragraph>
  </Section>
  <Section position="5" start_page="175" end_page="177" type="metho">
    <SectionTitle>
SATISFACTION AND UNIFICATION
</SectionTitle>
    <Paragraph position="0"> Given any two formulae ~ and q0, the set of models that satisfy both ~) and q0 is exactly the set of models that satisfy ~ ^ q). That is, the conjunction operation can be used to describe the intersection of two sets of models each of which is described by a constraint formula, even if these satisfying models do not form principal filters \[11\] \[9\]. Since conjunction is idempotent, associative and commutative, the satisfiability of a conjunction of constraint formulae is independent of the order in which the conjuncts are presented, irrespective of whether they contain negation. Thus the evaluation (i.e.</Paragraph>
    <Paragraph position="1"> simplification) of constraints containing negation can be freely interleaved with other constraints.</Paragraph>
    <Paragraph position="2"> Unification identifies or merges exactly the elements that the axiomatization implies are equal. The unification of two complex elements e and e' causes the unification of elements a(e) and a(e') for all attributes a that are defined on both e and e'. The constraint x = x' implies a(x) : a(x') in exactly the same circumstances; i.e. when a(x) and a(x') are both distinct from 3-.</Paragraph>
    <Paragraph position="3"> Unification fails either when two different constant elements are to be unified, or when a complex element and a constant element are unified (i.e. constant-constant clashes and constantcomplex clashes). The constraint x : x' is unsatisfiable under exactly the same circumstances, x -~ x' is unsatisfiable when x and x' are also required to satisfy x = c and x' = c' for distinct constants c, c', since c ~ c' by axiom schema (A2). x = x&amp;quot; is also unsatisfiable when x and x' are required to satisfy a(x) : t and x' ~ c'  for any attribute a, term t and constant c', since a(c') = _t_ by axiom schema (A3).</Paragraph>
    <Paragraph position="4"> Since unification is a technique for determining the satisfiability of conjunctions of atomic equality constraints, the result of a unification operation is exactly the set of atomic consequences of the corresponding constraints.</Paragraph>
    <Paragraph position="5"> Since unification fails precisely when the corresponding constraints are unsatisfiable, failure of unification occurs exactly when the corresponding constraints are equivalent to False.</Paragraph>
    <Paragraph position="6"> Example (continued): The sets of satisfying models for the formulae (1&amp;quot;) and (2') are precisely the principal filters generated by (1) and (2) above.</Paragraph>
    <Paragraph position="8"> Because the principal filter generated by the unification of el and e7 is the intersection of the principal filters generated by (1) and (2), it is also the set of satisfying models for the conjunction of (1') and (2') with the formula</Paragraph>
    <Paragraph position="10"> The satisfiability of a formula like (3') can be shown using standard techniques such as the Congruence Closure Algorithm \[15\], \[5\]. In fact, using the substitutivity and transitivity of equality, (3') can be simplified to (3&amp;quot;). It is easy to check that (3) is a satisfying model for both (3&amp;quot;) and the axioms for attribute-value structures.</Paragraph>
    <Paragraph position="11"> The treatment of negative and disjunctive constraints is straightforward. Since negatiou is interpreted classically, the set of satisfying models do not ahvays form a filter (i.e. they are not always upward closed \[16\]). Nevertheless, the quantifier-free language itself is capable of characterizing exactly the set of feature structures that satisfy any boolean combination of constraints, so the failure of upward closure is not a fatal flaw of this approach.</Paragraph>
    <Paragraph position="12"> At a methodological level, I claim that after the mathematical consequences of two different interpretations of feature structure constraints have been investigated, such as the classical and intuitionistic interpretations of negation in feature structure constraints \[14\], it is primarily a linguistic question as to which is better suited to the description of natural language. I have been unable to find any linguistic analyses which can yield a set of constraints whose satisfiablity varies under the classical and intuitionistic interpretations, so the choice between classical and intuitionistic negation may be moot.</Paragraph>
    <Paragraph position="13"> For reasons of space the following example (based on Pereira's example 116\] demonstrating a purported problem arising from the failure of upward closure with classical negation) exhibits only negative constraints.</Paragraph>
    <Paragraph position="14"> Example: The conjunction of the formulae</Paragraph>
    <Paragraph position="16"> can be simplified by substitution and transitivity of equality and boolean equivalences to (4') agr(x) = y A number(y) ~- singular A pers(y) ~ 3rd.</Paragraph>
    <Paragraph position="17"> This formula is satisfied by the structure (4) when x denotes e and y denotes f. Note the failure of upward closure, e.g. (5) does not satisfy (4'), even though (4) subsumes (5).</Paragraph>
    <Paragraph position="19"> However, if (4') is conjoined with pers(agr(x) ) ~- 3rd the resulting formula (6)/s unsatisfiable since it is equivalent to (6'), and</Paragraph>
    <Paragraph position="21"/>
  </Section>
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