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<Paper uid="P99-1038">
  <Title>Two Accounts of Scope Availability and Semantic Underspecification</Title>
  <Section position="5" start_page="294" end_page="296" type="metho">
    <SectionTitle>
3 Alternative Account of
Availability
</SectionTitle>
    <Paragraph position="0"> The NP-hardness result of the previous section arises from the assumption that the availability of scopings is determined by the well formedness of the associated logical forms. Park (1995) has proposed an alternative theory of scope availability which states that available scopes are accounted for by relative scopes of arguments around relations, whereby quantifiers may not move across NP boundaries. For example, consider the sentence Every representative of a company saw most samples, containing two relations, saw and of. Around saw, every (representative of a company) can outscope most (samples), or vice versa, and around of, every (representative) can outscope a (company), or vice versa. Park generalises this observation to the claim that for any n-ary relation in a sentence, there are n! possible orderings of quantified arguments around that relation. Other quantitiers in the sentence should not &amp;quot;intercalate&amp;quot; between those which are single arguments to a relation. So in the example sentence there are four possible scopes, because there are 2! = 2 scopings around saw and 2! = 2 scopings around of. What is not possible is a reading where a outscopes most which outscopes every; although this can be represented by a well formed sentence of logic (with no unbound variables), it is not available to a speaker of English.</Paragraph>
    <Paragraph position="1"> By using this theory as the basis of underspecification, we can say: * underspecification is to be captured by allowing different possible relative scope assignments around the predicates, and * partial scopes between arbitrary quantitiers in the sentence will be translated into the equivalent scoping of quantifiers around their predicates.</Paragraph>
    <Paragraph position="2"> The chosen representation will be based upon a sentence's quantifiers and relations (for example, verbs and prepositions).</Paragraph>
    <Paragraph position="3"> Quantifiers and the relations which determine their relative scope are represented by a set of elements under a strict partial order, where the ordering represents the relative scopes. A strict order will be taken to be transitive, antisymmetric and irreflexive. However, because the interaction between the predicates in the sentence has implications for possible scopings, it is also necessary to consider the relationships between the ordered sets.</Paragraph>
    <Paragraph position="4"> Consider again the sentence Every man loves a woman. The quantifiers and relation in this sentence can be represented by a set of elements {every, a, love}. A strict partial order, ~-, is defined over the set which states that the relation love must be outscoped by both quantifiers: ({every, a, love}, (every ~- love, a ~- love)) The partial order states that both quantifiers outscope the verb, but says nothing about their scopes relative to each other. This represents a completely underspecified meaning. An unambiguous reading of the sentence is represented when ~- defines a total order on the set. So if the relation every ~- a were added, the reading:</Paragraph>
    <Paragraph position="6"> every ~- a ~- love would be represented. Alternatively, adding a ~- every to the underspecified form would represent the reading:</Paragraph>
    <Paragraph position="8"> The introduction of a further relation which does not lead to a well formed sentence (such as love ~- every) is shown by the irreflexivity of ~- being violated.</Paragraph>
    <Paragraph position="9"> While using a single set of elements correctly accounts for the possible scopes of quantifiers in the sentences discussed so far, relative clauses and prepositional attachment to NPs are more complex. Consider the sentence Every representative of a company saw most samples. The presence of two binary relations, of and saw, implies that there should be 2!.2! -- 4 readings. Continuing with the system developed so far, these possibilities could be represented by a pair of strictly partially ordered sets: ({every, most, see},(everyNsee, most Nsee)) ({every, a, of}, (every ~' of, a ~' of)) where the four possible ways of completing the strict orders on the sets correspond to the four available readings. To represent relative scope between arbitrary quantifiers in the sentence, a further transitive relation, .&gt;, is defined. Say that if (S, ~-) is a strictly partially ordered set in the structure where x, y E S and x ~- y then x .&gt; y. So for example, consider the pair of strictly partially ordered sets: ({every, most, see},(every~most~see)) ({every, a, of}, (a ~' every ~-' of)) which would represent the reading (in a format similar to generalised quantifiers): a(y, every(x, rep.of(x, y), most(z, sample(z), see(x, z)))) The orders on the sets state that every .:&gt; most see and a .&gt; every .:&gt; of, and from the transitivity of .&gt; it can be inferred (correctly) that a .:&gt; most. Similarly, given the ambiguous sentence and the partial scope requirement that a should outscope most, the required partial scope can be obtained by adding the relations a ~-~ every and every ~- most.</Paragraph>
    <Paragraph position="10"> The transitivity of .&gt; is not enough to capture all the available scope information. Suppose it were required that most should outscope a. There are two readings of the sentence which satisfy this partial scope, those being: most(z, sample(z), every(x, a(y, co(y), rep.of (x, y)), see(x, z))) and most(z, sample(z), a(y, co(y), every(x, rep.oI (x, y), see(x, z)))).</Paragraph>
    <Paragraph position="11"> These readings are precisely those for which the object of see outscopes its subject; the partial scope is captured by the pair: ({every, most, see}, (most ~- every ~- see)) ({every, a, of}, (every ~-' of, a ~-' of)) where there is no additional information about the relative scope of every and a. However, the transitivity of -&gt; alone does not capture the fact that most .:&gt; a follows from most .:&gt; every.</Paragraph>
    <Paragraph position="12"> We remedy this by defining a domination relation. In the current case, say that every dominates a, which means that a is nested within the QNP whose head quantifier is every. Then because quantifiers may not &amp;quot;intercalate&amp;quot; across NP boundaries, anything that outscopes every also outscopes anything that every dominates (here, a); if most outscopes one it must outscope both. We capture this behaviour by putting the sets into a tree structure, where each of the nodes is one of the strictly ordered sets representing the scopes around a relation. For any node, N, each of the daughter nodes has (exactly) one element in common with N, otherwise, any element appears only once in the structure. So, consider again the sentence Every representative of a company saw most samples. The scope information of the underspecifled form is represented by the tree: ({every, most, see}, (every see, most see)) / ({every, a, of},(every ~-' of, a ~' of)) Now, say that an element X dominates another  element Y (denoted as X ~-~ Y) if X and Y are (distinct) elements in a set at some node, and X is also in the parent node. Also, ~-+ is transitive and irreflexive. So in the example given: every ~-+ a and every ~ of, but every ~-+ every.</Paragraph>
    <Paragraph position="13">  We can now extend the definition of -&gt; by saying that:  if (P,~-) is a node in the tree, and x, y E P and x ~- y, then x.&gt;y and x.&gt;z where z is any term that y dominates.</Paragraph>
    <Paragraph position="14"> Also, .&gt; is transitive and irreflexive.</Paragraph>
    <Paragraph position="15"> This captures the scoping behaviour for nested quantifiers. So from the ambiguous representation of scopes: ({every, most, see}, (most every see)) I ({every, a, of}, (every of, a of)) where most ~-- every and every ~ a, it is possible to infer correctly that most .&gt; a, whatever the relation is between every and a.</Paragraph>
  </Section>
  <Section position="6" start_page="296" end_page="297" type="metho">
    <SectionTitle>
4 Formal Definition of Scope
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="296" end_page="297" type="sub_section">
      <SectionTitle>
Representations
</SectionTitle>
      <Paragraph position="0"> We now provide a formal description of the structures described in section 3. The definition is divided into two parts. First a scope structure is defined, which is a tree structure whose nodes are sets under a strict order and describes the correct possible scopings of quantiffed arguments around their relations. Next, a scope representation is defined, which is the pair of a scope structure and an outscoping relation, * &gt;, which is defined over all the elements in the structure.</Paragraph>
      <Paragraph position="1"> The analysis presented here differs from that of the previous section in that the nodes in the scope ~ structures are sets under a strict total order, rather than under a partial order.</Paragraph>
      <Paragraph position="2"> The structures therefore represent unambiguous readings of the sentence. Underspecification will then be captured in the constraint language, rather than in the underlying structures, as discussed in section 5.</Paragraph>
      <Paragraph position="3"> A scope structure is a finite tree, where each node of the tree is a finite, non-empty set of elements, P, taken from a set (9 = {a,/~,-),,...} under a strict total order. For any node, each daughter node is also a strictly ordered set, such that each daughter set di has exactly one element in common with P, a different element for each of the di. An element can only appear once in the tree, unless it is the common node between a mother and a daughter. So: is a correct scope structure, because no element appears twice except c~ and 8, which appear in mother/daughter pairs (the ordering relations have been omitted for clarity).</Paragraph>
      <Paragraph position="4"> A scope structure is defined as a triple (P, ~, :D), where P is a set of elements, ~- is a strict total order over P and 7:) is the set of daughters. We say that an element occurs in a scope structure if it is a member of the set at any node in the scope structure. If (9 is a (countable) set of elements, then scope structures can be recursively defined as:  * If S = (Ps, &gt;-s, {}), where Ps is a finite, non-empty subset of (9 and &gt;-s is a strict total order on Ps, then S is a scope structure, where: 1. if x E Ps, then x occurs in S, * If R and S are scope structures such that</Paragraph>
      <Paragraph position="6"> where no element occurs in both R and S, and there is some element a such that a E Pn, then if T = (PT, N'T,~T), where</Paragraph>
      <Paragraph position="8"> a strict total order on PT then T is a scope structure, where:  1. If some element x occurs in either R or S then x occurs in T 2. If some element x occurs in R and x a, then a dominates x in T 3. If x and y occur in R and x dominates y in R then x dominates y in T 4. If x and y occur in S and x dominates y in S then x dominates y in T If S is a scope structure, then a node in S is defined as: * If S is a scope structure such that S -(Ps, &gt;-s, T~S), then: - (Ps, &gt;'-s) is a node in S - if di E :Ds, then any node in di is a node in S.</Paragraph>
      <Paragraph position="9">  Having defined scope structures, we now define a scope representation, which is a pair iS, &amp;quot;&gt;s), where S is a scope structure and &amp;quot;&gt;s is a relation between pairs of elements which occur in S. &amp;quot;&gt;s represents outscoping between any  pair of elements in the structure, rather than just between elements at a common node. If S is a scope structure such that S = (Ps,~-s,7)s), then (S, &gt;s) is a scope representation, where &amp;quot;&gt;s is the minimum relation such that:  * If (P, ~-p) is a node in S and x, y E P and x N-p y, then x &amp;quot;&gt;s Y.</Paragraph>
      <Paragraph position="10"> * If (P, ~-p) is a node in S and x, y E P and x ~-p y, then ifz is an element which occurs in S and y dominates z in S then x &amp;quot;&gt;s z. * &amp;quot;&gt;s is transitive.</Paragraph>
      <Paragraph position="11">  If (S, &amp;quot;&gt;s) is a well formed scope representation, then &amp;quot;&gt;s is a strict partial order over the set of elements which occur in S.</Paragraph>
    </Section>
  </Section>
  <Section position="7" start_page="297" end_page="298" type="metho">
    <SectionTitle>
5 Constraints for Scope
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="297" end_page="298" type="sub_section">
      <SectionTitle>
Underspecification
</SectionTitle>
      <Paragraph position="0"> We now consider a constraint language for representing the available scopes in a sentence. The structure of the sentence can be defined in terms of common arguments to a relation (which is represented by membership of a common set in the scope structure) and the domination relation. The constraint language is: C/, C/ ::= x o y Common set membership  where x, y are members of a (countable) set of constants, COAl = {x, y, z, . . . }.</Paragraph>
      <Paragraph position="1"> It is intended that these constraints be defined over terms in an underspecified semantic representation, such as QLF or UDRT, with a function mapping grammatical objects in the representation onto members of CON. Representing the quantifiers and relations in the sentence is sufficient for our current needs. Constraints of the form x o y (where o is symmetric) state either that x and y represent common arguments to a relation, or that x and y represent a relation and a quantifier which quantifies over it. Constraints of the form x ~-4 y indicate that x is the head quantifier of a complex NP, in which y, another grammatical object (either a quantifier or a relation), is nested.</Paragraph>
      <Paragraph position="2"> So for example, consider again the sentence Every representative of a company saw most samples, and assume that terms in the underspecified representation representing the the grammatical objects every, exists, most, rep.of and see map onto the elements e, a, m, o and s respectively, where {e, a, m, o, s} C CON. Then the constraint representing the fully underspecified meaning is:  Note that the symmetry of o is stated explicitly in the constraint. The (underspecified) constraint is generated either from the grammar or directly from the underspecified structure, so the inference rules for determining the availability of a partial scope only generate constraints of the form X t&gt; Y. These rules are discussed further in section 6. Underspecification is now captured within the constraint language; note the parallels between the constraints of the form X t&gt; Y in this example and the partial orders used in section 3.</Paragraph>
      <Paragraph position="3"> The satisfiability of the constraints is given in terms of the scope representations defined in section 4. A scope representation, (S, &amp;quot;&gt;s), satisfies a constraint of the form X o Y if (P, &gt;-p) is a node in S such that X', Y' E Ps, X' # Y', where some assignment function maps X and Y onto X' and Y'. Similarly, constraints of the form X ~-+ Y are satisfied if X' dominates Y' in S, and constraints of the form X D Y are satisfied if X' &amp;quot;&gt;s Y'. So the above constraint is satisfied by a set of scope structures of the form: ({every, most, see}, &gt;-) / ({every, a, of}, ~-') where the assignment function maps the constants e,a,m,o and s onto the elements every, a, most, of and see respectively, and where every ~- see, most ~- see, every ~-' of and a ~-' of.</Paragraph>
      <Paragraph position="4"> We can now define the semantics for the constraint language. An assignment function, I\[-~/, maps constants of the constraint language onto  elements which occur in S and wffs of the constraint language onto one of the pair of values {t,f}. I is a pair ((I),~4}, where (I) is a scope representation, such that (I) = (S, &amp;quot;&gt;s}, and .4 is a function mapping constants of the constraint language onto the set of elements which occur in S. The denotation of the constraints is then given by:  iff there is at least one I such that IC/~ / = t for all constraints C/ where C/ E A.</Paragraph>
      <Paragraph position="5"> The satisfiability of a constraint set represents the existence of a reading of the sentence which respects the partial scoping.</Paragraph>
    </Section>
  </Section>
  <Section position="8" start_page="298" end_page="299" type="metho">
    <SectionTitle>
6 Availability of Partial Scopes
</SectionTitle>
    <Paragraph position="0"> We now turn to the question of determining whether a partial scoping is available. In section 3 it was stated that scope availability is accounted for by the relative scope of quantitiers around their predicates. It turns out (although we do not prove it here) that for any partial scoping, there is a necessary and sufficient set of scopings of quantifiers around their relations that gives the partial scoping. For example, we showed that for the sentence Every representative of a company saw most samples, the readings where most outscopes a are exactly those where the subject of see outscopes its object. Therefore, from the constraint most C&gt; a, it should be possible to infer most E&gt; every. The aim of the constraint solver is to determine what scopings of quantifiers about their relations are required to obtain the required partial scoping, and therefore to state whether the partial scope is available.</Paragraph>
    <Paragraph position="1"> A set of rules is defined on the constraints, so that additional scope information may be inferred. The introduction of further scope constraints does not affect scope information already present (monotonicity). The rules are given in figure 3, where F represents any conjunction of literals and the associativity and commutativity of A are assumed. The inference rules S1, $2 and $3 operate by recursively reducing the (arbitrary) outscoping constraint X~&gt;Z to XI&gt;YAYE&gt;Y~, where Y and Y~ represent arguments to a common relation, and Y' either dominates or is equal to Z. Repeated application of these constraints gives the set of scopes of quantifiers around their relations for the initial partial scoping. The rules Trans and Dora then generate the remaining possible scope constraints. If a scope is unavailable, then completing the transitive closure of D across the structure yields a constraint of the form X ~&gt; X.</Paragraph>
    <Paragraph position="2"> We then say that: * A constraint set is in normal \]orm iff applying the rules S1, $2, $3, Trans and Dom does not yield any new constraints.</Paragraph>
    <Paragraph position="3"> If F is a constraint set in normal form then: * F represents an available scoping iff it does not contain a constraint of the form X ~&gt; X.</Paragraph>
    <Paragraph position="4"> * F represents a complete scoping iff it represents an available scoping, and for every constraint of the form X o Y there is either a constraint X D Y or a constraint Y D X.</Paragraph>
    <Paragraph position="5"> The condition for a scoping to be available follows from the irreflexivity of -&gt;. The condition for a scoping to be complete states that if two elements are arguments to a relation, or are a relation and one of its arguments, then they must have scope relative to each other. This corresponds to considering sets under a total order, rather than under a partial order.</Paragraph>
    <Paragraph position="6"> Complexity Issues Let F be a constraint representing an available scoping of a sentence, and let X~&gt;Y be a constraint representing a partial scope between two terms in that sentence.</Paragraph>
    <Paragraph position="7"> Then the worst case of applying the inference rules to F A X ~&gt; Y to saturation turns out to be equivalent to completing the transitive closure of i&gt;, which is known to be soluble in better than O(n 3) time (Cormen et al., 1990), where n is the number of elements in the structure.</Paragraph>
    <Paragraph position="8">  Application of rules $1, $2 and $3 to completion can be completed in linear time; if X i&gt; Y is a constraint between two arbitrary quantitiers X and Y where X fi Y, then exactly one of the rules S1, $2 or $3 applies (lack of space prevents us proving this here). If X o Y, then none of these three rules applies. Application of S1, $2 or $3 adds at most two new constraints, of which at most one is a scope constraint XC&gt;Y ~ where X fi Y~. At most n - 1 such constraints are generated.</Paragraph>
    <Paragraph position="9"> Application of the rules S1, $2 and $3 reduces an arbitrary partial scope into relative scopes of arguments around their relations. If a scoping is unavailable, this is represented by the irreflexivity of C&gt; being violated. Testing for this requires that the transitive closure of C&gt; be completed; this is known to be soluble in better than cubic time. We conclude that testing for the availability of a partial scope in this framework can be achieved in better than cubic time in the worst case.</Paragraph>
  </Section>
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