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<?xml version="1.0" standalone="yes"?> <Paper uid="C00-1076"> <Title>Extending a Formal and Computational Model of Rhetorical Structure Theory with Intentional Structures h la Grosz and Sidner</Title> <Section position="2" start_page="523" end_page="523" type="metho"> <SectionTitle> 2 The limits of Moser and Moore's </SectionTitle> <Paragraph position="0"> approach In a recent proposal, Moser and Moore (1996) argued that the primary intentions in a GST representation can be derived fi'om the nuclei ot'the corresponding RST representation. Although their proposal is consistent with the cases in which each textual span is characterized by an explicit nucleus that encodes the primary intention of that span (as in the case of text (I)), it seems that an adequate account of the correspondence between GST and RST is somewhat more complicated. For example, in tile case of text (3) below, whose RST analysis is shown in ligure 2, we cannot apply Moser and Moore's approach because we can associate tile primary intention of discourse segment \[a2, B2\] neither to trait A2 nor to trait B2. also wanted to have dinner with Suzanne. '~2\] \[He went crazy, c2 \] In Grosz and Sidner's terms, we can say that the primary intention ot' segment \[A2, B~\] is (Intend writer (Believe reader &quot;John wanted to do two things that were incompatible&quot;)). But in order to recognize this relation, we need to recognize that the two desires given in units A~ and B2 are incompatible, which is captured by the CONTRAST relation that holds between the two units. In other words, the intention associated with segment \[A2, B2\] is a function both el' its nuclei, A 2 and B2, and of the rhetorical relation of CONTRAST that holds between them.</Paragraph> <Paragraph position="1"> In this paper, we generalize this obserwttion by making use o1&quot; the compositionality criterion proposed in (Marcu, 1996), which stipulates that it'a rhetorical relation holds between two textual spans, a si,nilar relation also holds between two salient constructs of those spans. 2 Similarly, we will assume that the primary intention of a discourse segment is not given by the nucleus of the corresponding relation but rather that it depends on the corresponding relation and the salient constructs associated with that segment.</Paragraph> </Section> <Section position="3" start_page="523" end_page="527" type="metho"> <SectionTitle> 3 Melding text structures and intentions </SectionTitle> <Paragraph position="0"/> <Section position="1" start_page="523" end_page="525" type="sub_section"> <SectionTitle> 3.1 Formulation of the problem </SectionTitle> <Paragraph position="0"> Formally, the problem that we want to solve is the following. Given a sequence of textual units U = tq, u2,..., UN, a set 1U~ of rhetorical relations that hold among these units, and a set o1' intentional judgments IH that pertain to the same units, find all legal discourse structures (trees) of U, and determine the dominance, satisl'action-precedence relations, and primary intentions of each span of these trees.</Paragraph> <Paragraph position="1"> Following (Marcu, 1996), we use tile predicates posiHo,z(ui, j) and vl, eId'd(,za,,,e, s, ,z) with the fol2Seclion 3 discusses in detail how the salient construcls are delermined. null lowing semantics: tim predicate posilion(ui, j) is tree for a textual unit ul in sequence U if and only if ul is the j-th element in the sequence; the predicate rhei_vel(namc, ui, uj) is true for textual units ul and uj witb respect to rhetorical relation name, if and only it' the detinition provided by RST for rhetorical relation name applies to textual units ui, in most cases a satellite, and uj, a nucleus. In order to enable discourse problems to be characterized by rhetorical judgments that hold between large textual spans as well, we use predicate rh.cl_rel_ext(namc, s~, s~, *~, n~). This predicate is trl, e for textual spans \[ss, .%\] and \[,,.,, n0\] with respect to rhetorical relation name if and only if the detinition of rhetorical relation name applies for tim textual span that ranges over units ss--se, ill most cases a satellite, alld textual spans that ranges over units n.~-nc, a nucleus. 3 From a rhetorical perspective, text (I) is described at the minimal unit level by the relations given in (2) and (4) below.</Paragraph> <Paragraph position="3"> The intentional judgments 1~1 are given by the following functions and predicates: * The predicate dom(l~, lq, 1~, h-,) is true whenever tbe DSP of discourse segment/span \[I1, hl\] dominates ttle DSI' of discourse segment \[l~, h:~\]. A dominance relation is well-formed if segment \[/~, h~\] is a proper subsegment of segment \[ll, h,t\]. i.e., l, </~ < h., < h, A (h C/ z~ v h~ # h~).</Paragraph> <Paragraph position="4"> * The predicate salpvec(ll, Ih, lu, h..,) is true whenever an intentional satisfactiol>precedence relation holds between the DSI's of segments Ill, hi\] and \[/2, h2\]. A satisfaction-precedence relation is well-formed if tile segments do not overlap.</Paragraph> <Paragraph position="5"> * Tile oracle function .fl(r, aq,..., ;%) takes as at: guments a rhetorical relation r and a set of texttufl units, and returns tbe primary intention that pertains to that relation and those units. For example, in the case of segment \[A2, Be\] in text (3), the oracle function .l) (CONTRAST, A2, B2) is assullted to returu a Iirst-order object wltose meaning can be glossed as &quot;inform the reader that John wanted to do two things that were incompatible&quot;. And the oracle function .1) (EWDI ~;NcE, B1) associated with segntent \[A1,1)~\] in text (1) is assuntcd to return a \[irst-oMer object whose nteaning can be glossed as &quot;increase the reader's belief that the pressure to smoke in junior high is greater than it will be any other time of one's life&quot;.</Paragraph> <Paragraph position="6"> Without restricting the generality of the problem, discourse structures are assented to be binary trees. In our formalization, each node era discourse structure is characlerized by l()tu&quot; features: the status (nucleus or satellite), tim O'lJe (the rhetorical relations tlmt hold between 3'Fhe s ~llld e subscripls COlTCgpond Io .~tm'ling ~.lll(I ending posilions. the text spans that that node spans over), the l)romotion set (the set of units that constitute the most &quot;salient&quot; (irapertain) part of the text that is spanned by that node), and tile i)rima O, intelltion. By convention, for each leaf node, the type is LEAF, the promotion set is tile textual unit to which it corresponds, and tbe primary intention is that of inJbmting the content of that unit. For exampie, a representation of the tree in ligure 1.a that makes explicit the features el' all spans that play an active role in the final representation is given in \[igure 3. In general, the salient units are computed using the comlmsitionality criterion proposed in (Marcu, 1996), i.e, they are given by the union of the salient units of the immediate subordinated nuclei. Similarly, the primary intentions are a function of tbe rhetorical relation (type) and salient units of each span.</Paragraph> <Paragraph position="7"> The status, type, promotion set, and primary intention that are associated with each node in a discourse trec provide suflieient information for a full description of an instance of a tree structure. Given the linear nature of text and the fact that we cannot predict in advance where the boundaries between various segments will be d,'awn, we should provide a lnethodology that permits one to enumerate all possible ways in which a tree could bc built on the lop of a linear sequence of elementary discourse units. The solution we use relies on tile same intuition that constitutes tile foundation of chart parsing: just as a chart parser is capable of consklering all possible ways in which different words in a sentence could be chlstered into higher-order grammatical units, so our formalization is capable of considering all the possible ways in which different segments coukl be joined into discourse trees.</Paragraph> <Paragraph position="8"> l,et spa,tLj, or simply \[i,j\], denote a text span thai includes all tile elementary discourse unils between position i and j. Then, if we consider a sequence of discourse units .u~, It2:... ~'lt~t, there are n ways in which spans o1' length one could be built, spa'~Zl,l, st)(tLt2,2, * * * , 'sl)(t'/tn,n; it - \] ways in which spans of length two could be built, * spa~zl,2: Sl)~Ut..&3~... , spaltn-l,n; 11 -- 2 ways in which spans of length three could be built, and one 6&quot;\])(t?l.l ; h Sl)(tll.2,4~ . . . ~ .5\]}altn-2,n; . . . ; way in which a span of length n coukl be built, spa771,n.</Paragraph> <Paragraph position="9"> Since it is impossible to determine a priori the sl)ans that will be used to make up a discourse tree, we will associate with each span that could possibly become part of a tree a status, a type, promotion, and primary intention relation and let discourse and intentional constraints determine the valid discourse trees. In other words, we want to C/tetermine from the set of ha- (,,.- 1)-t- (n-2) +...+ 1 = n(n4- 1)/2 potentM spans that pertain to a sequence of n discourse units, the subset that adheres to some constraints of rhetorical and intentional well-formedness. For example, for text 1, there are d + 3 -t- 2 + \[ = l0 potential spans, i.e., S\])(17tl ,1 ~ 8\])aTt2,2: S1)fl713,3, sl)a?).,1,4, 8P(t?tl,2~ $1)(/N.2,3, sPa~l:l,4: 8Payt.1,3~ s'\])a?12,,t, and 8p(I.711,.I, but</Paragraph> <Paragraph position="11"> characterize every node that does not have a NONE status. The nunlbers associated with each node denote the limits of the text span that that node characterizes.</Paragraph> <Paragraph position="12"> only seven of them play an act?ve role in the representation given in figure l.a, i.e., 8\])(l~.1,1, SP(llZ2,2, $1)(t1~.3,3, 8\])(t1~'4,4, St)Ctl~l ,2, spa~.3,4, alld .5'\])a IZ 1,4-To formalize the constraints that pertain both to RST and GST, we thus assume that each potential span \[1, hi is characterized by the following predicates: * S(I, h, s lalus) provides the status of span El, h\], i.e., the text span that contains units / to h; staZus can take one of the values NUCLEUS, SATELLITE, or NUNS. according to the role played by that span in the tinal discot,rse tree. For example, for the tree depicted in tigure 3, some of the relations that hold are: ,5'(1, 2, NUCLEUS),,5'(3, 4, SATELLITE), ,5&quot;(1,3, NONE).</Paragraph> <Paragraph position="13"> * T(1, h, relation_ua.rn.e) provides the name of the rhetorical relation that holds between the text spans that are immediate subordinates o1' span El, h\] in the discourse tree. If the text span is not used in the construction of the final tree, the type assigned is NONE. For example, for the tree in ligure 3, some o1' the relations that hold are: T(I, J, LEAF), 5/'(1,2, JUSTW~CATION), T(3, 4, CONC~SSrON), T(1, 3, NONE).</Paragraph> <Paragraph position="14"> * P(I, h.,unit_name) provides one of the set of units that are salient for span El, h\]. The collection of units for which the predicate is true provides the promotion set of a span, i.e., all units that are salient for that span. If span \[1, h\] is not used in the tilml tree, by convention, the set of salient units is NONE. For example, for the tree in figure 3, some of the relations that hold are: P(1, 1., &), P(1, 2, lh), P(1,3, NONE), 1'(3, 4, D,).</Paragraph> <Paragraph position="15"> * Ill, h, intention) provides the primary intention of discourse span El, h\]. The term iulenlion is represented using the oracle ftmction J). For example, for the tree in figure 3, some of the relations that tloi(t arc: I(3, 4, f/(CONCESSION, Cj )), l(J,/1, .fI(P:VIDENCI~:, B\])), l(J, 3, NONE).</Paragraph> </Section> <Section position="2" start_page="525" end_page="527" type="sub_section"> <SectionTitle> 3.2 An integrated formalization of RST and GST </SectionTitle> <Paragraph position="0"> Using the ideas that we have discussed ill the previous section, we present now a first-order formalization of discourse structures that makes use both of RST- and GSTlike constraints. In this lbrmalization, wc assume a universe that consists of the set of natural numbers fi'om J tO N, where N represents the number of textual units in the text that is considered; the set of names thai were defined by Mann and Thompson for each rhetorical relation; the set of unit names that are associated with each textual unit; and four exlra constants: NUCLEUS, SATEL-LITE, NONE, and LI~2AF. The formalization is assumed lo provide unique name axioms for all these constants.</Paragraph> <Paragraph position="1"> The only funclion symbols that operate eve,&quot; the assumed domain are the mlditional + and - functions that are associated with the set of natural numbers and the oracle function J). The formalization uses the traditional predicate symbols that pertain to the set of natural numbers (<, <, >, >, =, C/) and eight other predicate symbols: ,5', T, P and I to account for the status, type, salienl units, and primary intention that are associated with every text span; vhel_vel to account for the rhetorical relalions that hold between different textual units; position to account for the index of the textual units in lhe text dmt one considers; dora to account for dominance relations; and satprec to account for satisfaction-precedence relations.</Paragraph> <Paragraph position="2"> Throughout the paper, we apply the convention that all unbound variables are universally quantified and that variables are represented in lower-case italics&quot; and constants in SMALL CAPITALS. We also make use of the two extra relations, vclevaul_uni~ and relevant_tel.</Paragraph> <Paragraph position="3"> For every text span span \[/, hi, relevant_unit(l, h, u) describes the set ot' textual units that are relevant for that text span, i.e., the units whose positions in the initial sequence are numbers in the interval \[l, hi. It is only these units that can be used to label the pro- null motion set associated with a tree that subsumes all units in the interval \[l, hi. For every text span \[1, h.\], vclevcm.Z_vcl(l, h, name) describes the set of rhetorical relations that are relevant to that text span, i.e., the set of rhetorical relations that span over text units in the interval \[1, h\] and the set of extended rhetorical relations that span over text spans that cover the whole interval \[/, h\] (see (Marcu, 1996) for the formal delinitions of these rehttions.) null For example, fin&quot; text (1), which is descrihed formally in (2) and (4), the following is the set of all rclc'~a~zl_rel and vclevctn~_unil, relations that hold with respect to text segment \[l,3\]: {vclcvanLvcl(l,3, JUSTWlCaTtON), 'rclcvanl_vcl(l, 3, EVII)ENCl0, relevcr, t_m~it(I, 3, &), ,.~z~v.,,t_~,,nit(l, a, B~), ,.d~,:.,,z_,,,,it(l, :~, q)}.</Paragraph> <Paragraph position="4"> The constraints that pertain to the discourse trees that we formalize can be partitioned into constraints related to the domain of objects over which each predicate ranges, constraints related to the structure of the tree, and constraints that relate the slrucltlral COlnponenl with the intentional component. The axioms that pertain to the domains over which predicates ,5, P, and 7' range and the constraints related to the structure of the live are the same as those given by Marcu (1996). For lhe sake of completeness, in this paper we only enumerate then\] informally. In contrast, the axioms that pertain to intentions and the relation between structure and intentions are discussed in detail.</Paragraph> <Paragraph position="5"> Constraints that concern the objects over which the predicates that describe every segment \[1, hi of a text structure range (Mareu, 1996, pp. 1072-1073).</Paragraph> <Paragraph position="6"> ,, For every siren \[/, h\], the set or objects over which predicate ,5' ranges is the set {NUC1A,~US, SNI'ELIJTI,\], NONE).</Paragraph> <Paragraph position="7"> * The status of any discourse segment is unique.</Paragraph> <Paragraph position="8"> * For every segment \[l, h\], the set of objects over which predicate 7' ranges is the set of rhetorical relations that are relevant to that span.</Paragraph> <Paragraph position="9"> * At most one rhetorical rdation can connect two adjacent discourse spans * The primary intention of a discourse segment is ei null ther NONE or is a function of the salient units that pertain to that segment and of the rhetorical relation that holds between the immediate subordinated segments.</Paragraph> <Paragraph position="10"> Since we want to stay within the boundaries of Iirst-order logic, we express this (see formula (5) below) by means of a disjunction of at most N sulfformulas, which correspond to the cases in which the span has I, 2 .... , or N salient traits. 4 4Formula (5) reflects no preference concerning lhe order in which rhetorical relalions and intentions should be computed (Asher and Lascarides, 1998). It only asserts a consh'ailll on the two.</Paragraph> <Paragraph position="12"> (~'r, a:,, a:2 .... , :,:N)\[S\]~(/, h, r) A r y:- NONEA a;1 7~ a:~ A a:l # a::~ A ... A :cl C/ ~;NA * ~;2 -7 k a'3 A ... A :C# ~ :;';NA ,~:N--I :~ XNA P(/, h, ,:, ) A e(t, h, ~) A... A PU, h,, ,;,)A (V~)(P(t, h, y) -+ (:/= ~, v... v y = ,;,)> inl.c,,.lio,tu, --- fz(r, :c,, a;u,. *. , ,;,)\]} * The primary intention of any discourse segment is unique.</Paragraph> <Paragraph position="13"> (6) \[(i < 1,. < N) A (1 5_ t < 1,.)\] \[(1(~,/,, i, ) A J(I, h, <)) -- .i, = <4 * For every segmeut \[l, hi, the set of objects over which predicate P ranges is the set of units that make up that segment Constraints that concern /lie strnctmm of the discourse trees * The status, type, and promotion set that are associated with a discourse segment reflect the COmlmsition ality criterion. That is, whenever a rhetorical relation holds between two spans, either a simihu&quot; relation holds between Ihe mosl salicnl units of those spans or an extended rhetorical relation holds between those spans. * Discourse segments do not overlap.</Paragraph> <Paragraph position="14"> * A discourse segment with status NONE does not participate in the tree at all.</Paragraph> <Paragraph position="15"> * There exists a discourse segment, the root, that sirens over the entire text.</Paragraph> <Paragraph position="16"> ~,S'(1, N, nonl') A ~P(\], N, NONF,)A (7) ~&quot;(1, N, NONIi) A -71(1, N, NONE) * The dominance relations described by Grosz and Sidner hold Between the DSP of a discoorse segment and the DSP of'its most immediate subordinated satellite. This constraint is consistent with Moser and Moore's (1996) discussion of RST and GST. In fact, this is not surprising if we examine the definitions of dominance relation given by Grosz and Sidner and satellite given by Mann and Thompson: a discourse segment purpose D,5't1/2 dominates a discourse segment purpose D,5'1&quot;1 if I),5'P\] contributes to the satisfaction el' the I),5'11/2. But this is exactly the role that satellites play in P, ST: they do not express what is most essential for the writer's purpose, but rather, provide supporting informalion that contributes to the understanding of the nucleus. The relationship between Grosz and Sidner's dominance relations and Mann and Thompson's distinction between nuclei and satellites is formalized by axioms (8) and (9).</Paragraph> <Paragraph position="18"> s(/+, ha, SATI+LLm,:))\] dom(ll, hq, 12, h2)} \[(+ < h, < N) ,X (+ _< h _< h,) /, (l _< h+ < N)A (9) (1 ~ 1.9 .~ 11.2) A do?l+(l,, lt, l, 12, //.2)\] &quot;--+ \[-~,5'(h, hi, NON.:) A S(6, h_~, SATJILUTE)\] Axiom (8) specities that if segment \[12, h.2\] is the imme null diate satellite el'segment \[lt, lq\], then there exists a dominance relation between the DSP of segment \[/1,/q\] and the DSP of segment \[12, h2\]. Hence, axiom (8) explicates the relationship between the structure of discourse and intentional dominance. In contrast, axiom (9) explicates the relationship between intentional dominance and discourse structure. That is, if we know that the intention associated with span \[lj, 1,1\] dominates the intention associated with span \[12, h,2\], then both those spans play an active role in the representation and, moreover, the segment \[12,11,2\] plays a SATELLITE role.</Paragraph> <Paragraph position="19"> * The satisfaction-precedence rdations described by Grosz and Sidner are parataetie relations that hold between arlfitrarily large textual spans. Nevertheless, as we have seen in the examples discussed in this paper, the fact that a paratactic relation holds between spans does not imply that there exists a satisfaction-precedence relation at the intentional level between those spans. Therefore, for satisfaction-precedence relations, we will have only OnE axiom, that shown in (I0), below.</Paragraph> <Paragraph position="20"> \[(t 5 hJ ~ N) A (1 ~ 11 ~ hl) A (\] <&quot; h,2 ~ N)A</Paragraph> <Paragraph position="22"> This specifiES that the spans that are arguments of a satisfaction-precedence relation have a NUCLEUS status in the linal representation.</Paragraph> </Section> </Section> <Section position="4" start_page="527" end_page="527" type="metho"> <SectionTitle> 4 A computational view of the </SectionTitle> <Paragraph position="0"> axiomatization Given the formulation discussed abovE, tinding the discourse trees and the primary intentions lkw a text such as that given in (1) amounts to finding a model for a first-order theory that consists of formulas (2), (4), and the axioms enumerated in section 3.</Paragraph> <Paragraph position="1"> There are a number of ways in which one can proceed with an implementation: for cxalnple, a smtightforward choice is one that applies constraint-satisl'action techniques, an approach that extends that discussed in (Marcu, 1996). Given a sequence U of N textual units, one can take advantage of the structure of the domain and associate with each of the N(N-F 1)/2 possible text spans a status and a type variable whose domains consist in the set of objects over which the corresponding predicates ,5 + and T, range. For each of the N(N + 1)/2 possible text spans \[l, h.\], one can also associate h, - l + \] promolion variables. These are boolean variables that specify whether units l, 1 + \],... , h belong to the promotion set of span \[/, hi. For each of the N(N + 1)/2 possible text spans \[l, hi, one can also associate h - 1 + 2 intentional variables: one of these wtriables has as domain the set of rhetorical relations that are relevant for the span \[1, hi. The rest of the h -/+ 1 wwiables are boolean and specify whether unit l, l-t- \] .... , or h are arguments of the oracle function f~ that intentionally characterizes that span.</Paragraph> <Paragraph position="2"> Hence, each text of N units yields a constraint-satisfaction prohlem with N(N + I)(2N + \]3)/6 variables (NCN q- \])(2N -}- 13)/(J = 2NCN q- \])/~ -}-</Paragraph> <Paragraph position="4"> The constl+aints associated with these wtriables arc a one-to-onE mapping o1' the axioms in section 3. Finding the set of RS-trees and the intentions that are associated with a given discourse reduces then to/inding all the solutions for a traditional constraint-satisfaction problem.</Paragraph> </Section> class="xml-element"></Paper>