Bridging the Gap Between Underspecification Formalisms: Hole Semantics as Dominance Constraints

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1 Bidging the Gap Between Undespecification Fomalisms: Hole Semantics as Dominance Constaints Alexande Kolle Joachim Niehen Stefan Thate Saaland Univesity, Saabücken, Gemany Abstact We define a back-and-foth tanslation between Hole Semantics and dominance constaints, two fomalisms used in undespecified semantics. Thee ae fundamental diffeences between the two, but we show that they disappea on pactically useful desciptions. Ou encoding bidges a gap between two undespecification fomalisms, and speeds up the pocessing of Hole Semantics. 1 Intoduction In the past few yeas thee has been consideable activity in the development of fomalisms fo undespecified semantics (Alshawi and Couch, 1992; Reyle, 1993; Bos, 1996; Copestake et al., 1999; Egg et al., 2001). These appoaches all aim at contolling the combinatoial explosion of eadings of sentences with multiple ambiguities. The common idea is to delay the enumeation of all eadings fo as long as possible. Instead, they wok with a compact undespecified epesentation fo as long as possible, only enumeating eadings fom this epesentation by need. At fist glance, many of these fomalisms seem to be vey simila to each othe. Now the question aises how deep this similaity is ae all undespecification fomalisms basically the same? This pape answes this question fo Hole Semantics and nomal dominance constaints, two logical fomalisms used in scope undespecification, by defining a back-and-foth tanslation between the two. Due to fundamental diffeences in the way the two fomalisms intepet undespecified desciptions, this encoding is only coect in a nonstandad sense. Howeve, we identify a class of chain-connected undespecified epesentations fo which these diffeences disappea, and the encoding becomes coect. We conjectue that all linguistically useful desciptions ae chainconnected. To suppot this claim, we pove that all desciptions geneated by a nontivial gamma we define ae indeed chain-connected. Ou esults ae inteesting because it is the fist time in the liteatue that two pactically elevant undespecification fomalisms ae fomally elated to each othe. In addition, the satisfiability poblems of Hole Semantics and nomal dominance constaints coincide on thei chain-connected fagments. This means that satisfiability of Hole Semantics, which is NP-complete in geneal (Althaus et al., 2003), becomes polynomial in pactice, and can be checked using the efficient algoithms available fo nomal dominance constaints (Ek et al., 2002). Enumeation of eadings becomes much moe efficient accodingly. 2 Some Intuitions The similaity of Hole Semantics and dominance constaints is illustated in Fig. 1. The pictues gaphically epesent the undespecified epesentations of all five eadings of the sentence Evey eseache of a company saw a sample in Hole Semantics (Bos, 1996) and as a dominance constaint (Egg et al., 2001). The undespecified epesentations specify the mateial that evey eading is made up of and constaints on the way in which they can be combined in obviously simila ways. Howeve, the intepetations of these undespecified epesentations diffe. In Hole Semantics, the intepetation is given by means of pluggings, whee holes (the h i ) and labels (l k ) ae identified. In contast, dominance constaints ae intepeted by embedding desciptions into tees that may contain moe mateial. This diffeence comes

2 h 0 l 1 : u comp u h 1 l 2 : w h 2 es w h 3 l 3 : x sample x h 4 l 4 : of w u l 5 : see x w u comp u of w w x sample es w x see u x w Figue 1: Gaphical epesentations of the Hole Semantics USR (left) and the nomal dominance constaint (ight) fo the sentence Evey eseache of a company saw a sample. out especially clealy in a desciption like in Fig. 2. It has no plugging in Hole Semantics, as two diffeent things would have to be plugged into one hole, but it is satisfiable as a dominance constaint. It is this fundamental diffeence that esticts ou esult in 5, and that we avoid by using chainconnected desciptions. a f b Figue 2: A desciption on which Hole Semantics and dominance constaints disagee. 3 Dominance Constaints Dominance constaints ae a geneal famewok fo the patial desciption of tees. They have been used in vaious aeas of computational linguistics (Roges and Vijay-Shanke, 1994; Gadent and Webbe, 1998). Fo undespecified semantics, we conside semantic epesentations like higheode fomulas as tees. Dominance constaints can be extended to CLLS (Egg et al., 2001), which adds paallelism constaints to model ellipsis and binding constaints to account fo vaiable binding without using vaiable names. We do not use these extensions hee, fo simplicity, but all esults emain tue if we allow binding constaints. 3.1 Syntax and Semantics We assume a signatue Σ of function symbols anged ove by f g, each of which is equipped with an aity a f 0, and an infinite set Î Ö of vaiables anged ove by X Y Z. A dominance constaint ϕ is a conjunction of dominance, inequality, and labeling liteals of the following fom: ϕ :: X Y X Y X: f X 1 X n ϕ ϕ whee a f n. Dominance constaints ae intepeted ove finite constucto tees, and thei vaiables denote nodes of a tee. We define an unlabeled tee to be a finite diected acyclic gaph V E, whee V is the set of nodes and E V V the set of edges. The indegee of each node is at most 1. Each tee has exactly one node (the oot) with indegee 0. Nodes with outdegee 0 ae called the leaves of the tee. A finite constucto tee τ is a tiple T L V L E consisting of an unlabeled tee T V E, a node labeling L V :V Σ, and an edge labeling L E : E Æ, s. t. fo each node u ¾V thee is an edge u v ¾ E with L E u v k iff 1 k a L V u. Now we ae eady to define tee stuctues, the models of dominance constaints: Definition 1 (Tee Stuctue). The tee stuctue M τ of a constucto tee τ V E L V L E is a fist-ode stuctue with domain V intepeting dominance and labeling. Let u v v 1 v n ¾ V. The dominance elationship u τ v holds iff thee is a path fom u to v in E and the labeling elationship u: f τ v 1 v n holds iff u is labeled by the n-ay symbol f and has the childen v 1 v n in this ode; that is, L V u f, a f n, u v 1 u v n E, and L E u v i i fo all 1 i n. Let ϕ be a dominance constaint and Î Ö ϕ be the set of vaiables of ϕ. A pai of a tee stuctue M τ and a vaiable assignment α : Î Ö ϕ V τ satisfies ϕ iff it satisfies each liteal in the obvious way. We say that M τ α is a solution of ϕ in this case; ϕ is satisfiable if it has a solution. Entailment ϕ ϕ holds between two constaints iff evey solution of ϕ is also a solution of ϕ. We often epesent dominance constaints as (diected) constaint gaphs; fo example, the gaph in Fig. 2 stands f g a b fo the constaint X: f Y Y Z Y Z Z:a Z :b. This constaint is satisfied e.g. by the tee stuctue displayed hee. Note the added g.

3 3.2 Solving Dominance Constaints The satisfiability poblem of dominance constaints (i.e. deciding whethe a constaint has a solution) is NP-complete (Ek et al., 2002). Howeve, Althaus et al. (2003) show that satisfiability becomes polynomial if the constaint ϕ is nomal, i.e. satisfies the following vey natual conditions: (N1) Evey vaiable occus in a labeling constaint. (N2) Evey vaiable occus at most once on the ight-hand side and at most once on the lefthand side of a labeling constaint. Vaiables that don t occu on a left-hand side ae called holes; vaiables that don t occu on a ighthand side ae called oots. (N3) If X Y occus in ϕ, X is a hole and Y is a oot. (N4) If X and Y ae diffeent vaiables that ae not holes, thee is a constaint X Y in ϕ. The gaph of a nomal constaint (e.g. the one in in Fig. 1) consists of solid tee fagments (N1, N2) that ae connected by dominance edges (N3); these fagments may not ovelap in a solution (N4). Because evey satisfiable dominance constaint ϕ has an infinite numbe of solutions, algoithms typically enumeate its solved foms instead (Ek et al., 2002). A solved fom is a constaint that diffes fom ϕ only in its dominance liteals. Its gaph must be a tee, and the eachability elation on the gaph must include the eachability in the gaph of ϕ. Evey solved fom of ϕ has a solution, and evey solution of ϕ satisfies one of its solved foms; so we can see solved foms as epesenting classes of solutions that only diffe in ielevant details (e.g. unnecessay exta mateial). Anothe way to avoid infinite solutions sets is to conside constuctive solutions only. A solution M α of a constaint ϕ is constuctive if evey node inm is denoted by a vaiable in Î Ö ϕ on the left-hand side of a labeling constaint. Intuitively, this means that the solution consists only of the mateial mentioned in the labeling constaints. Not all solutions ae constuctive; fo example, Fig. 2 is a solved fom but has no constuctive solutions. The poblem of deciding whethe a nomal dominance constaint does have constuctive solutions is again NP-complete (Althaus et al., 2003). 4 Hole Semantics Hole Semantics (Bos, 1996) is a famewok that defines undespecified epesentations ove abitay object languages. We use the vesion of (Bos, 2002) because it epais some sevee flaws in the oiginal definition of admissible pluggings. Hole Semantics configues fomulas of an object language (such as FOL o DRT) with holes, into which othe fomulas can be plugged. Fomally, a fomula with n holes is a complex function symbol of aity n as above. The equivalent of a dominance constaint is an undespecified epesentations (USR). An USR U consists of a finite set L U of labeled fomulas l:f h 1 h n, whee l is a label and F is an object-language fomula with holes h 1 h n, and a finite set C U of constaints. Constaints ae of the fom l h, whee l is a label and h a hole; this constaint means that h outscopes l. Like fo dominance constaints, thee is a natual way of witing USRs as gaphs (Fig. 1). An USR U is called pope if it has the following popeties: (P1) U has a unique top element, fom which all othe nodes in the gaph can be eached. (P2) The gaph of U is acyclic. (P3) Evey label and evey hole except fo the top hole occus exactly once in L U. 1 Fo example, the USR shown in Fig. 1 is pope; its top element is h 0. The solutions of undespecified epesentations ae called admissible pluggings. A plugging is a bijection fom the holes to the labels of an USR. Intuitively, we plug evey hole with a fomula (named by its label), and a plugging is admissible if it espects the constaints on the ode of holes and labels. Definition 2 (P-domination). Let k, k be holes o labels of some undespecified epesentation U, and P a plugging on U. Then k P-dominates k iff one of the following conditions holds: 1. k : F ¾ L U and k occus in F, o 2. P k k if k is a hole, o 3. Thee is a hole o label k such that k P- dominates k and k P-dominates k. 1 The estiction on hole occuences is missing in (Bos, 2002), but is necessay to ule out counteintuitive USRs.

4 Definition 3 (Admissible Plugging). A plugging P is admissible fo a pope USR U iff k k ¾ C U implies that k P-dominates k. 5 Hole Semantics as Dominance Constaints Now we have the fomal machiney to make the intuitive similaity between Hole Semantics and dominance constaints descibed in Section 2 pecise. We shall define encodings fom Hole Semantics to nomal dominance constaints and back, and show that this peseves models in a cetain sense. To keep things simple, the esults in this sections will only speak about compact nomal dominance constaints. A dominance constaint is compact if no vaiable occus in two diffeent labeling constaints. A vey nice popety (which we need below) of compact nomal constaints is that evey vaiable is eithe a oot o a hole. Howeve, any nomal constaint can be made compact by an opeation called compactification, which compesses conjunctions of labeling constaints into single labeling constaints with moe complex labels. So the encodings and esults ae moe moe geneally coect fo abitay nomal dominance constaints (with acyclic gaphs). Fom Hole Semantics to Dominance Constaints. Assume U L U C U is a pope USR. To obtain a compact dominance constaint ϕ U that encodes the same infomation, we fist encode evey labeled fomula l:f h 1 h n as the labeling constaint l:f h 1 h n. We encode evey constaint l h in C U as a dominance constaint h l except if h is the unique top hole and does not occu as a hole in a labeled fomula. Finally, we add a constaint l l fo evey label l. This encoding maps labels and holes to vaiables; labels end up as oots, and holes become holes. This means ϕ U satisfies axiom (N3). (N2) follows fom (P3). (N1) and (N4) ae obvious fom the constuction. Hence ϕ U is nomal. Fom Dominance Constaints to Hole Semantics. Assume ϕ is a compact nomal dominance constaint whose gaph is acyclic. To obtain a pope USR U ϕ encoding the same infomation, we fist split the vaiables Î Ö ϕ into holes and labels: oots become labels, and holes become holes. Then we encode evey labeling constaint X: f X 1 X n as the labeled fomula X: f X 1 X n, and we encode evey dominance constaint X Y as the constaint Y X. Finally, we add a top hole h 0 and a constaint l h 0 fo evey label l in U ϕ. U ϕ is a well-defined USR because of (N3). (P1) is obvious: h 0 is the unique top element. The gaph is acyclic because the gaph of ϕ is acyclic, so (P2) holds. (P3) holds because evey label names at least one fomula by constuction, and no moe than one by (N2). This back-and-foth encoding has the following popety: Theoem 4. Compact nomal dominance constaints ϕ with acyclic gaphs and pope USRs U can be encoded into each othe, in such a way that the pluggings of U and the constuctive solutions of ϕ coespond. Poof. We only show that the solutions of an USR U and its encoding ϕ U coespond; the othe diection is analogous. Assume fist that we have a plugging P of U. We build a tee which satisfies ϕ U constuctively and has one node fo evey label l of U. The node label of this node is the fomula that l addesses. Stating at the top element, we wok ou way down the USR; wheneve we find a hole h, we continue at the label P h. Convesely, assume we have a constuctive solutionm of ϕ. Evey node inm is denoted by a vaiable. Because holes have no labeling constaints, evey hole h must denote the same node as a oot P h. Futhe, evey oot that is not the oot of the entie tee denotes the child of anothe oot, i.e. denotes the same node as a hole. We obtain an admissible plugging by mapping each hole h to the label P h in the USR, and mapping the new top hole h 0 to the label denoting the oot of the tee. 6 Fom Solved Foms to Constuctive Solutions Theoem 4 establishes a vey stong connection between Hole Semantics and nomal dominance constaints. But it is not quite what we want: Nomal dominance constaints ae almost

5 always consideed with espect to abitay solutions (o solved foms), and not constuctive solutions. Constaints such as Fig. 2 ae solved foms, but have no constuctive solutions. The efficient algoithms available fo nomal constaints check fo solved foms, and aen t necessaily coect fo constuctive satisfiability. In this section, we establish that fo nomal dominance constaints which ae chain-connected and leaf-labeled (to be defined below), satisfiability and constuctive satisfiability ae equivalent; i.e. such a constaint has a constuctive solution if only it is satisfiable. The poof poceeds in thee steps: Fist we show that all solved foms of a nomal constaint ae simple iff the constaint banches constuctively. Then we show that if a constaint is chain-connected, it banches constuctively. Finally, evey simple solved fom of a leaf-labeled constaint has a constuctive solution. 6.1 Constuctive Banching We call a solved fom simple if its gaph has no node with two outgoing dominance edges (i.e. Fig. 2 is not simple). This means that we can decide fo any two vaiables how they will be situated in a solution of the solved fom. They can eithe dominate each othe in eithe diection, o they can be disjoint. But if they ae disjoint, we also know the lowest node that dominates them both, and this banching point is necessaily also denoted by a vaiable on the left-hand side of a labeling constaint. This motivates the following definition. We locally allow disjunctions of constaints and use an auxiliay constaint, the disjointness constaint X Y ato, wheeo is a set of vaiables. It is satisfied if X and Y denote disjoint nodes whose banching point is denoted by a membe ofo. Definition 5. A nomal dominance constaint ϕ banches constuctively if fo any two vaiables X Y ¾ Î Ö ϕ, ϕ X Y Y X X Y at L ϕ whee L ϕ is the set of vaiables that occu on the left-hand side of a labeling constaint in ϕ. Lemma 6. Let ϕ be a nomal dominance constaint. ϕ banches constuctively iff all solved foms of ϕ ae simple. Poof. Assume fist that all solved foms of ϕ ae simple; let ϕ 1 ϕ n be the set of all solved foms. Now because they ae simple solved foms, each ϕ i entails the ight-hand side of Def. 5. But ϕ entails the disjunction of all of its solved foms, and hence banches constuctively. Convesely, assume that ϕ has a non-simple solved fom ϕ. Then ϕ must contain a vaiable X with two outgoing dominance edges (to Y and Z). But this means that ϕ has a solution in which Y and Z ae diffeent childen of X, and hence thei lowest common ancesto is not in L ϕ. 6.2 Chain-Connectedness Constuctive banching is a semantic popety that can t conveniently be poved fo a gamma. We shall now elate it to a moe easily checkable popety called chain-connectedness. We will fist define chains, then chain-connectedness, and then pove the elation of the two concepts. Definition 7 (Fagments). A fagment in ϕ is a nonempty subset F Î Ö ϕ that is connected by labeling constaints in ϕ. We call the fagment maximal if it has no pope supeset that is also a fagment of ϕ. Exactly one vaiable in evey fagment is a oot; we witer F fo this oot. Definition 8 (Chains). Let ϕ be a nomal dominance constaint, and let F 1 F n (n 1) be disjoint fagments of ϕ.c F 1 F n is called a chain of ϕ iff thee is a disjoint patitiono U F 1 F n with the following popeties: 1. O is nonempty. 2. Fo each 1 i n, eithe (a) F i ¾O and F i 1 ¾U, and thee is a hole X i of F i s.t. ϕ X i R F i 1 ; o (b) F i ¾U and F i 1 ¾O, and thee is a hole X i 1 l of F i 1 s.t. ϕ X i 1 l R F i 3. Fo 1 i n s.t. F i ¾O, the holes X i l and X i ae diffeent. O is called the set of uppe fagments of the chain, andu is the set of lowe fagments. We call all the X i l and X i connecting holes ofc, and all othe holes in any of its fagments open holes. A schematic pictue of a chain is shown in Fig. 3. Note that although the definition of a chain involves the athe abstact condition that dominance between to vaiables is entailed by the con-

6 F 1 F 3 X 1 X 3 l X 3 F 2 F 4 X 5 l Figue 3: A schematic pictue of a chain. staint, this condition can often be established syntactically fo example in Fig. 3 by the explicit dominance edges. Chains wee oiginally intoduced by Kolle et al. (2000) because they have vey useful stuctual popeties. A paticulaly useful one is the following. Lemma 9 (Stuctual Popeties of Chains). Let ϕ be a nomal dominance constaint, and letc be a chain that contains all vaiables of ϕ. LetV O be the set of all vaiables in uppe fagments of C that ae not holes. Then if X Y ae vaiables in diffeent fagments of C, the following stuctual popety holds: ϕ X Y Y X X Y atv O Using this lemma, it is easy to show that wheneve a constaint is chain-connected, it also banches constuctively. Definition 10. Two vaiables X Y of a nomal dominance constaint ϕ ae chain-connected in ϕ iff thee is a chainc in ϕ that contains both X and Y. A constaint is chain-connected iff evey pai of vaiables is chain-connected. Poposition 11. Evey chain-connected dominance constaint ϕ banches constuctively. Poof. Let X Y be two abitay vaiables in ϕ. If X and Y belong to the same fagment, thee is obviously a connecting chain containing just this fagment. Othewise, constuctive banching fo X and Y follows fom Lemma 9. Fo the last step of the poof, we define that a nomal dominance constaint is leaf-labeled if evey vaiable occus on the left-hand side of a labeling o dominance liteal. Such constaints have the following popety: Lemma 12. Evey simple solved fom of a leaflabeled constaint has a constuctive solution. F 5 Putting it all togethe, we obtain the intended esult: Theoem 13. Evey solved fom of a chainconnected, leaf-labeled nomal dominance constaint has a constuctive solution. We can tansfe the notions of chainconnectedness and leaf-labeledness to USRs eithe by a diect definition o by defining that U is chain-connected o leaf-labeled iff ϕ U is. Then we can state the following theoem: Coollay 14 (Pocessing of Hole Semantics). The poblem whethe a chain-connected, leaflabeled pope USR has a plugging is polynomial. Poof. Simply check the coesponding dominance constaint fo satisfiability. Althaus et al. (2003) give a quadatic satisfiability algoithm; Thiel (2002) impoves this to linea. 7 Connectedness in a Gamma Finally, we claim that chain-connectedness and leaf-labeledness ae vey weak assumptions to make about a nomal dominance constaint, and conjectue that all linguistically useful constaints satisfy them. We define a nontivial gamma fo a fagment of English and show that it only geneates dominance constaints with these popeties. The agument we use to establish chainconnectedness (the less obvious popety) is faily geneal, and should be applicable to othe gamma fagments as well. The gamma we use is a vaiant of the one pesented in (Egg et al., 2001). Its syntax-semantics inteface poduces dominance constaints descibing fomulas of highe-ode logic; the stands fo functional application, and abstaction and vaiables ae witten as lam x and va x. We use dominance constaints because this gives us the logical tools we need in the poof; but by Theoem 4, we can tanslate all esults back into pope USRs, and those USRs will also be chainconnected. 7.1 The Gamma The syntactic component of the gamma consists of the following phase stuctue ules.

7 ν:s NP VP ν:vp IV b1 Xν Xν Xν b2 X ν X ν ν:np Det N b8 Xν lam x Xν ν:vp TV NP b3 Xν Xν Xν ν:n N b9 X ν X ν va x X ν ν:vp RV NP VP b4 Xν Xν Xν X ν ν:n N RC b10 lam x Xν Xν va x Xν va x ν:vp SV S ν:np PN b5 b7 Xν Xν Xν Xν lam x ν:rc RP i S ν:α W b11 va x Xν Figue 4: The syntax-semantics inteface lam x X ν X ν va x Xν X i b13 W Xν whee W α ¾ Lex (a1) S NP VP (a2) VP IV (a3) VP TV NP (a4) VP RV NP VP (a5) VP SV S (a7) NP PN (a8) NP Det N (a9) N N (a10) N N RC (a11) RC RP S (a13) α W if W α ¾ Lex Most categoy labels ae self-explanatoy, pehaps except fo SV, which efes to vebs taking sentence complements such as say, and RV, which efes to (object) aising vebs such as expect. The lexicon is defined by a elation Lex elating wods and lexical categoies. Rule (a13) expands lexical categoies to wods of the categoy. 7.2 The Syntax-Semantics Inteface Evey node ν in a syntax tee contibutes a constaint ϕ ν ; the vaiable Xν is intuitively the oot of this contibution. We assume that the syntax povides fo a coindexation of elative ponouns and thei coesponding taces, and associate each NP with index i with a coesponding vaiable X i. The vaiables ae elated by the ules in Fig. 4. Each syntactic poduction ule coesponds to one semantic constuction ule, which defines the semantic contibution of a syntactic node. A constuction ule of the fom ν:p Q R ϕ ν means that the node ν in the syntax tee is labeled with P, and its two daughte nodes ν and ν ae labeled with Q and R, espectively. The semantic contibution of ν is the constaint ϕ ν, with fesh instances of lam x and va x whee necessay. The complete constaint of a syntax tee with oot ν is the conjunction of the ϕ ν fo all nodes ν dominated by ν, and inequalities that ae needed to make the constaint nomal Connectedness of Constaints The poof that all constaints geneated by this gamma ae connected poceeds by stuctual induction ove pase tees. The semantic contibutions of leaves ae tivially chains, and hence connected. What we show in the est of this section is that if t is any subtee of the syntax tee, and all the semantics of all immediate subtees of t ae connected, then so is the semantics of t. We ignoe the globally intoduced inequality constaints because they have no effect on chain-connectedness. The cental popety of the constuction ules that we exploit is the following: Poposition 15. Let ϕ 0 ϕ n be chainconnected constaints such that 1. Va ϕ i Va ϕ j /0, fo 1 i j n, 2. Va ϕ 0 Va ϕ i X i, fo 1 i n, whee X 1 X n ae open holes in all connecting chains in ϕ 0. Then the constaint ϕ 0 ϕ n is chain-connected. 2 The oiginal gamma accounts fo scope island constaints by means of additional dominance liteals. We ignoe them hee, as they do not affect chain-connectedness.

8 This can be poved by induction. The base case n 0 is tivial, and fo the induction step we combine a connecting chain within ϕ 0 ϕ n 1 fom an abitay X to X n with a connecting chain within ϕ n fom X n to an abitay Y. Chains ae combined by taking all the fagments of the two smalle chains togethe. The assumption that the X i ae open holes in the connecting chains is needed fo the poblematic case in which the fagment containing X n is an uppe fagment in both chains. All constaints intoduced by a semantics constuction ule othe than (b11) ae of this fom: ϕ 0 coesponds to the constaint intoduced by the ule, and ϕ 1 ϕ n to the constaints associated with the daughte nodes. Hence, all constaints geneated using only these ules ae chainconnected. Fo the case of (b11), obseve that the elative ponoun is coindexed with its tace. This means that the vaiable Xν occus in the same fagment as Xν, so (b11) also satisfies the geneal scheme. An easie stuctual induction shows that the constaints ae also leaf-labeled. Hence: Coollay 16. All constaints geneated by the gamma ae chain-connected and leaf-labeled. 8 Conclusion We have established the equivalence of Hole Semantics and nomal acyclic dominance constaints with constuctive solutions. They ae equivalent to nomal acyclic dominance constaints with standad solutions if the constaints ae chainconnected and leaf-labeled. All constaints geneated by ou gamma have these popeties; we conjectue this is tue moe geneally. This bidges a gap between the two undespecification fomalisms, which means that we can now combine the simplicity of hole semantics with the efficient algoithms, poweful metatheoy, and extensibility of dominance constaints. A fist pactically useful esult is a polynomial satisfiability algoithm fo chain-connected, leaf-labeled USRs. Convesely, chain-connected dominance constaints inheit some of Hole Semantics esoucesensitivity: Additional mateial need neve be added to satisfy the constaint; but to model e.g. eintepetation (Kolle et al., 2000), this is still possible. This esouce-sensitivity was the cucial diffeence between the two fomalisms. In the futue, it will be inteesting to see how ou esults extend to othe esouce-sensitive undespecification fomalisms fo example, to MRS (Copestake et al., 1999), whose naive encoding into dominance constaints is less obviously nomal, and which adds a moe poweful outscopes elation. Refeences H. Alshawi and R. Couch Monotonic semantic intepetation. In Poc. 30th ACL, pages E. Althaus, D. Duchie, A. Kolle, K. Mehlhon, J. Niehen, and S. Thiel An efficient gaph algoithm fo dominance constaints. Jounal of Algoithms. In pess. Johan Bos Pedicate logic unplugged. In Poc. 10th Amstedam Colloquium, pages J. Bos Undespecification and esolution in discouse semantics. Ph.D. thesis, Saaland Univesity. A. Copestake, D. Flickinge, and I. Sag Minimal Recusion Semantics. An Intoduction. Unpublished manuscipt. M. Egg, A. Kolle, and J. Niehen The constaint language fo lambda stuctues. Jounal of Logic, Language, and Infomation, 10: K. Ek, A. Kolle, and J. Niehen Pocessing undespecified semantic epesentations in the constaint language fo lambda stuctues. Reseach in Language and Computation, 1(1). In Pess. Claie Gadent and Bonnie Webbe Descibing discouse semantics. In Poceedings of the 4th TAG+ Wokshop, Philadelphia. A. Kolle, J. Niehen, and K. Stiegnitz Relaxing undespecified semantic epesentations fo eintepetation. Gammas, 3(2-3). Uwe Reyle Dealing with ambiguities by undespecification: constuction, epesentation, and deduction. Jounal of Semantics, 10: J. Roges and K. Vijay-Shanke Obtaining tees fom thei desciptions: An application to teeadjoining gammas. Computational Intelligence, 10: Sven Thiel A linea time algoithm fo the configuation poblem of dominance gaphs. Submitted.