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<?xml version="1.0" standalone="yes"?> <Paper uid="C00-1067"> <Title>On Underspecified Processing of Dynamic Semantics</Title> <Section position="3" start_page="0" end_page="461" type="metho"> <SectionTitle> 2 Tree Descriptions </SectionTitle> <Paragraph position="0"> In this section, we define the Constraint Language tbr DPL structures, CL(DPL), a language of tree descriptions which conserw~tively extends donfinance constraints (Marcus et al., 1983; Rainbow et al., 1995; Keller et al., 2000) by variable binding constraints. CL(DPL) is a close relative of the Constraint Language for Lamb(la Structures (CLLS), 1)resented in (Egg et al., 1998). It; is interl)reted over DPL structures - trees extended by a variable 1)inding function which can be used to encode tbrmulas of dynamic (or static) predicate logic. We will define DPL structures in two steps and then the language to talk al)out them.</Paragraph> <Section position="1" start_page="460" end_page="460" type="sub_section"> <SectionTitle> 2.1 Tree Struetures </SectionTitle> <Paragraph position="0"> For the definitions below, we assulne a signature Y\] = {(~12, varl0,Vl~,~ll,Al.z, manll , likel2,... } of node labels, each of which is equipped with a fixed arity n _> 0. The labels A,_~,_V,... are the.</Paragraph> <Paragraph position="1"> tirst-order commctives. Node lal)els are. ranged over by f, g, a, b, and the arity of a lal)el f is denoted by ar(./'); i.e. if J'l,~ C E then ar(f) = n.</Paragraph> <Paragraph position="2"> Let N l)e the set; of natural numbers ~. > 1.</Paragraph> <Paragraph position="3"> As usual, we write N* tbr the set of words over N, C for the elnl)ty word, and 7r~-' for the concatellatioll of two words 7r, 7r t C N*. A wor(t 7t is a prefiz of 7c' (written rc _< re') if there is a word 7r u such that 7rTr tt = 7r t.</Paragraph> <Paragraph position="4"> A node of a tree is the word rr E N* which addresses the node. The empty word e C N* is called the root node. A tree domain A is a nonempty, t)refixed-closed subset of N* which is closed under the left-sil)ling relation.</Paragraph> <Paragraph position="5"> Definition 2.1 A tree strllctilre iS a t'a\])l(: (A, c,) consisting of a finite, tree dmn, ain A and a total labeling t'unction co deg : A -+ E s'ach th, al, for all rr 6 A and i G N: c A 1 < i < We say that the nodes r 6 rcl,..., 7c~ m'e in the labeling ,'elationsh/ip 7c: J' On,..., 7r,~) ill! a (Tr) = J' and tbr each 1 < i < n, ~-i = ~-i. Similarly, we say that a node ~c properly dom, inatcs a node 7c' and write 7r<\]+rc ' iffrr is a proper prefix of 7c'. We take Ir and It' to be disjoint (~r J_ ~r') if ttley are different and neither node dominates the other. So any two nodes in a tree structure are in one of the four relations = (equality), <1 +, ~>+ (the inverse of <1+), or _L. We shall ~lso t)e interested the coml)inations of these l&quot;elatiolls by set operators: intersection, coml)lementation , union, and inversion. For instance, the dominance relation <~* is detined as the union of node equality and proper dominance = U<1 +.</Paragraph> <Paragraph position="6"> Finally, we detine the ternary non-intervention relation ~(Tr<l*Td<Frc&quot;) to hold ifl' it is not the</Paragraph> </Section> <Section position="2" start_page="460" end_page="461" type="sub_section"> <SectionTitle> 2.2 DPL structures </SectionTitle> <Paragraph position="0"> Now we extend tree structures by variable binding and obtain DPL structures. To this end, we I)artition E into three sets: conneetive.s Econ = {V_,A,_%...}, predicate symbols Epred = {man, likes,...}, and tcr'm symbols Ere,.,. = {var, peter, mother_of,...} which substone, the variable symbol var and fimction sym1)ols. null Definition 2.2 A DPL structure is a triple (A,c&A) con.~isting of a trcc structure. (A,c,) and a partial varial)le t)inding flmction A : A A which sati.sfies for all % ~r t ~ A: 1. the. < r.co,, u .fo,.</Paragraph> <Paragraph position="1"> all 7ci C A; 2. < the,,,, fo,.</Paragraph> <Paragraph position="2"> all 7ci ~ A; I)PL structures can be used to re, present ibrmulas of first-order predicate logic. For instance, the DPL structure in Fig. 1 represents the (unique) meaning of (1). So far, however, variables bound by a quantifier do not need to be in any special position in a DPL structure; in particular, not in its scope. To entbrce seeping as in static predicate logic, we could simpy add the condition ~c'<~*~r in condition 3 of Definition 2.2. We will define an appropriate counterpart \]'or DPL in Section 3 (properness).</Paragraph> <Paragraph position="3"> Modeling variable binding with an explicit binding flmction instead of variable nmnes was first proposed in (Egg et al., 1998). There, binding flmctions heJp to avoid a capturing problem in the context of scope underspecitication which t)ecomes most ~q)l)arent in the presence of ellipsis. Her(; the 1)inding flmction mainly gives us a different t)erspective on variable binding which</Paragraph> <Paragraph position="5"> is useflfl for defining properness of DPL structures. null</Paragraph> </Section> <Section position="3" start_page="461" end_page="461" type="sub_section"> <SectionTitle> 2.3 The Constraint Language CL(DPL) </SectionTitle> <Paragraph position="0"> The syntax of CL(DPL) is defined in Figure 2. It provides constraints tbr all the relations discussed above. There are labeling constraints X:.f(X~,...,Xr~), expressive combinations XRY of dominance constraints 'with set operators (Dtu:hier and Niehren, 200(}; Cornell, 1994), non-intervention constraints ~( X <1* Y <~* Z), and binding constraints a(X)=Z.</Paragraph> <Paragraph position="1"> CL(DPL) is interpreted over DPL structures.</Paragraph> <Paragraph position="2"> A variable assignment into a DPL structure 54 is a total flmction fi'om the set of variables of a constraint to the domain of 54. A pair (54, oz) of a DPL structure 54 and a variable assignment (t into 54 satisfies a constraint qo ifl' it satisfies all of its atomic constraints; that is, if the relation with the same sylnbol holds of the nodes assigned to their arguments. We also call the pair (54, oz) a solution and Ad a model of ~o. Only some of the atonfic constraints in CL(DPL) are used in mlderspecified descriptions in t)articular, labeling, dominance, and binding constraints; the other constraints are helpful in processing the others. These three types of constraints can be transparently displayed in constraint graphs. For instance, the constraint graph ill Fig. 3 represents a constraint describing the readings of example (1) including the scope ambiguity. The nodes of the graph stand for variables in tile constraint; labels and solid edges represent labeling COl> straints, dotted edges, donlinance constraints, and dashed arrows, binding constraints, hi addition, the constraint graph represents an inequality constraint X-~=Y between each two variables whose nodes carry a label. A constraint with the latter property is called overlapfree. The intuition is that the solid-edge tree fragments in the constraint graph must never overlap properly in a solution.</Paragraph> </Section> </Section> <Section position="4" start_page="461" end_page="462" type="metho"> <SectionTitle> 3 Dynamic Semantics in CL(DPL) </SectionTitle> <Paragraph position="0"> The semantics of DPL is built in a way that allows quantifiers to bind only variables in certain positions: inside their scopes and, if it is an existential quantifier, from the left-hand sides of conjunctions and implications into the right-hand sides. In CL(DPL), we model this as a purely syntactic restriction on the accessibility of binders which we define as a structural prop-erty of DPL structures. DPL structures which have this property will be called proper.</Paragraph> <Paragraph position="1"> A useflll auxiliary concept tbr the definition is that of an infimum of two nodes with respect to the dominance relation q*, which constitutes a lower senfilattice because of the underlying treeness of DPL structures. Furthermore, we will use the standard DPL notions of internally v@&quot; = {A,~} and ex- dynamic COlUlectives ~con ternalh d, static connectives ~con'~stat = {2, V, __=::k, _V}. The semantics definition of DPL gives these two groups special relevance tbr variable binding.</Paragraph> <Paragraph position="2"> Now we can define pTvper \])PL structures as tbllows.</Paragraph> <Paragraph position="3"> Definition 3.1 A DPL structure 54 is called proper '~f for each, node ~r of 54 on which ~ is defined, one of th, e following cases holds true wh, ere p, is the i'nfimum of rc and A(TC).</Paragraph> <Paragraph position="4"> 1. ff = ;~(~), or 2. )@r) is labeled 'with ~_, ttl<l*.~(Tr), p,2q*% I t is labeled with art internally dynamic connective, and no node between ttl and A(Tc), inclusively, is labeled with an externally static connective.</Paragraph> <Paragraph position="5"> Intuitively, the first branch of the definitiou corresponds to usual binding of variables inside tile scope of a quanfifer. In the second branch, the positions of the variable and the (existential) quantifier in the DPL structure are dis.joint, and the quantifier is dominated by tile left child of the infimum. Then the infimum must be labeled with an internally dynamic connective, and there must be no externally static connective between this node and the quantifier. This restriction is what we are going to exploit</Paragraph> <Paragraph position="7"> to capture the influence 055 scope. There is 51o such restriction for the lmth 1)etwcen the intimmn and the w~riM)le.</Paragraph> <Paragraph position="8"> Sohll;ions of & constraint thnt violate the (lynmnie accessibility conditions are now excluded silnply by restricting the, class of ~utmissible solutions to i)roper ones. As exl)ected from the linguistic intuition, only one sohd;ion of the nmning exmnple (1) is proper: the one where &quot;a woman&quot; is assigned wide scope (Fig. 1). Tit(; other solution is not prot)er because the path Kern the infimum ((lellol;cd by Z0 in Fig. 3) to the antecedent contains ;~ mfiversal qu~mtiticl&quot;.</Paragraph> <Paragraph position="10"> iff the CI~(I)I'I~)~xioms (l)y,,\])to (l)yn4)in Fig. 4 ~re, wflid over it. The rule (Dynl) threes universM qmmtifiers to bind only variM)les in their scopes, and the rules (l)yn2) to (Dyn4) enforce properness of binding when a wtrial)le is not in the scope of its binder.</Paragraph> </Section> <Section position="5" start_page="462" end_page="464" type="metho"> <SectionTitle> 4 Underspecified Reasoning </SectionTitle> <Paragraph position="0"> We next present a procedure for mMorspecitied reasoning with dynmnic semmltics. Th(' goal is to narrow an mMerst)ecified (les(;rit)tion such that improi)er DPL-structure, s are removed flom the solution set. Narrowing should apply as soon as possible, so unde, rspeciliedness (:~m be</Paragraph> <Section position="1" start_page="462" end_page="462" type="sub_section"> <SectionTitle> 4.1 Inference Procedure </SectionTitle> <Paragraph position="0"> This infi;rence procedure ,s'atuvatt,.s a, constrMnt ttccording to the rules in Figures 4 and 5; that is, whenever a constraint conttdns the lefl;-hmM side of a rule,, it adds il;s right-hand side, until no new conjuncts ca, n 1)e ndded. Fig. 4 contains simply the prot)erness axioms from the, l)revious sections, turned into deterministic proof rules. The rules in Pig. 5 are t)ropagation rules t'ronl Algorithm DO in (Duchier and Niehren, 2000), plus new rules for non-intervention constrainl;s. Algorithm DO contains some ~Mditional rules, in I)ari;iculm' distribution rules that perform case distinctions, because DO is a complete solver tbr dominance constraints with set operators, wlfich improves on (Duchier and Oardent, 1999; Keller et M., 1998). We have omitted the (listril)ution rules here l)e,(;ause we do 'not wmlt to perform case distinctions; l)y ndding 1;\]1(;115 ~tg~l, ill~ WC COll\]d ellll511era, I;e all proper solutions, ~:ls Schiehlen (1997) does tbr UDI1T.</Paragraph> <Paragraph position="1"> The new rules (NonI1) ~md (NonI2) Mlow to derive dominan(:e infbrmation from non-intervention constraints. As we will see, the most interesting rule in Fig. 4 is (l)yn2), which derives explicit non-intervention int'ormalion fi'om the structurM t)roperLies of dynamic 1)inding. Note that while the rules in Fig. 5 are sound over ;my DPL strucl;ure, those in Fig.</Paragraph> <Paragraph position="2"> 4 are only serum eve5&quot; proper DPL structures.</Paragraph> <Paragraph position="3"> This is intended: Application of a prot)erness rule is s'upposcd to exclude (improper) solutions.</Paragraph> </Section> <Section position="2" start_page="462" end_page="464" type="sub_section"> <SectionTitle> 4.2 Examples </SectionTitle> <Paragraph position="0"> The inii;rence rules go a long w:ty towards makink tile eft'cot of dynamic seminal;its on scope e, xt)lieit. Let us consider |;15(; running example in Figure 3 to see how this works; we show how to derive Y3<I*X, which specifies the relative quantifier scope.</Paragraph> <Paragraph position="1"> First of all, we need to make the information</Paragraph> <Paragraph position="3"> Z2<1*Za explicit by application of (Lab.Dom) and (Inter). In this instance, (Inter) is used as a rule of weakening.</Paragraph> <Paragraph position="5"> Now we can apply the rule (Dyn2) to the variable binding constraint A(Za) = Y (drawn in boldface in the graph) and the V labeling constraint to derive a non-intervention constraint.</Paragraph> <Paragraph position="7"> All that is left to do is to make the positive dominance intbrmation contained in the new non-intervention constraint explicit. As the constraint also contains Zo<1*X, we can apply (NonI1) on the new non-intervention constraint and derive X~<FY.</Paragraph> <Paragraph position="8"> We can now combiue all of our constraints tbr X and Y with the intersection rule and obtain Y<1*X, which basically determines the order of the two quantifiers: (Inter) X~<*Y A X-~ PS Y ~ Y<*X By exploiting the fact that the constraint is overlap-ti'ee (i.e. contains an inequality coststraint for each two labeled variables), we (:an even derive Y3<I*X by repeated application of the rules (Child.down), (Lab.Disj), (NegDisj), and (NegDom). This means that we have flflly disambiguated the scope ambiguity by saturntion with deterministic inference rules. Now let us consider a more complicated example. Fig. 6 is the underspecified description of the semantics of (2) Every visitor of a company saw one of its departments.</Paragraph> <Paragraph position="9"> The constraint graph has five solntions, three of which are proper. Unfortunately, the constraint language is not expressive enough to describe these three solutions ill a single constraint: Both X and Z can be either above or below Y, even in a proper solution, but if X is below Y, Z must be too, and ifX is above Y, Z must be anywhere below X (but; may be above * IX V ,, Y &quot;&quot;&quot;-- q ,,,Z company .&quot; &quot; x &quot;. .&quot; i &quot;. ,&quot; &quot; var &quot; researcher i &quot;. / of~----~ depar.tmen-gh'~ ; , &quot;~&quot;. var \~! ..&quot; ' Var ~ var'~ .&quot;&quot;&quot; var ~&quot;.&quot; * i ..&quot; &quot; &quot;, &quot;. L&quot; i . &quot; / var'~ vat ~ .......</Paragraph> <Paragraph position="10"> YI). In other words, this constraint is an exampie where the inference procedure is not strong enough to narrow the description. In this case, we must still resort to pertbrming nondeterministic case distinctions; at worst, the rules will apply on solved forms of CL(1)PL) constraints. constraints over these set variables; examples for set constraints are V C V' and V = V~ U V.2. The new non-intervention constraint This inferen('e procedure fits nicely with all imph;mentation of (lominance constraints t)ased on constraint programming (Marriott and Stuckey, 1.998; Koller and Niehren, 2000) with tinite set constraints (Miiller, 1999). Constraint programlning is a technology for solving combinatoric puzzles eificiently. The main idea is to replace &quot;generate and test&quot; by &quot;propagate and distrilmt(f'. Constraint prot~agation t)eribrms deterministic inferences which prune the search space, whereas distribution tmrfonns (nondeterrain|st|c) case distinctions.</Paragraph> <Paragraph position="11"> Duchier and Niehren (2000) show how to implenmnt a (lominance constraint solver by encoding donfinance constraints as finite set constraints and disjunctive propagators. This solver does not handle non-intervention constrain|s, lint we show here that they can tm added very naturally. The (Dyn) rules still have to be implemented as saturation rules.</Paragraph> <Paragraph position="12"> The idea of this implementation is to encode a solution (Ad, ~) of a donfinance constraints by introducing for each variable X in the constraint and each relation symbol R C {<1 +, t> +, =, J_ } a finite set variable R(X). This w~riable is supposed to denote the set of all variables denoting nodes that are in tile relation R to ~(X):</Paragraph> <Paragraph position="14"> Dominance constr~fints can now be stated as The bull|in t)rot)agation tbr set constraints automatically implenmnts the rules (NonI1) and (NonI2). For instance, assume that X<1*Y t)elongs to ~; then there will 1)e a set constraint Y C/ <1 +(X), so set constraint propagation will derive Y ~ ~_(Z) U t>+(Z). This is the |mined|at(; encoding of Y_L U t>+Z, which is equiwdent to Y~<1* Z.</Paragraph> </Section> </Section> class="xml-element"></Paper>