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<Paper uid="W04-0911">
  <Title>Lexical-Semantic Interpretation of Language Input in Mathematical Dialogs</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 Linguistic data
</SectionTitle>
    <Paragraph position="0"> In this section, we present an overview of the language phenomena prominent in the collected dialogs to indicate the overall complexity of input understanding in our setting.5 Interleaved natural language and formulas The following examples illustrate how the mathematical language, often semi-formal, is interleaved with the natural language informally verbalizing proof steps.</Paragraph>
    <Paragraph position="1"> A auch K(B) [Aalso K(B)] A\B ist 2 von C[(A\B) [... is 2 of ...] (da ja A\B=;) [(because A\B=;)] B enthaelt kein x2A [B contains no x2A] The mixture affects the way parsing needs to be conducted: mathematical content has to be identified before it is interpreted within the utterance. In particular, mathematical objects (or parts thereof) may lie within the scope of quantifiers or negation expressed in natural language (as in the last example above).</Paragraph>
    <Paragraph position="2"> Imprecise or informal naming Domain relations and concepts are described informally using imprecise and/or ambiguous expressions.</Paragraph>
    <Paragraph position="3">  In the above examples, contain and be in can express domain relations of (strict) subset or element, while be outside of and be different are informal descriptions of the empty intersection of sets.</Paragraph>
    <Paragraph position="4"> To handle imprecision and informality, we have designed an ontological knowledge base that includes domain-specific interpretations of conceptual relations that have corresponding formal counterparts in the domain of naive set theory.</Paragraph>
    <Paragraph position="5"> The dialogs were typed in German.</Paragraph>
    <Paragraph position="6"> 5As the tutor was also free in wording his turns, we include observations from both student and tutor language behavior. Metonymy Metonymic expressions are used to refer to structural sub-parts of formulas, resulting in predicate structures acceptable informally, yet incompatible in terms of selection restrictions.</Paragraph>
    <Paragraph position="7"> Dann gilt fuer die linke Seite, wenn C [(A \ B) = (A [ C) \(B [C), der Begriff A \ B dann ja schon dadrin und ist somit auch Element davon [Then for the left hand side it is valid that..., the term A \ B is already there, and so an element of it] where the predicate be valid for, in this domain, normally takes an argument of sort CONSTANT, TERM or FORMULA, rather than LOCATION; de morgan regel 2 auf beide komplemente angewendet [de morgan rule 2 applied to both complements] where the predicate apply takes two arguments: one of sort RULE and the other of sort TERM or FOR-MULA, rather than OPERATION ON SETS.</Paragraph>
    <Paragraph position="8"> Informal descriptions of proof-step actions Wende zweimal die DeMorgan-Regel an [I'm applying DeMorgan rule twice] damit kann ich den oberen Ausdruck wie folgt schreiben:. . . [given this I can write the upper term as follows:. . . ] Sometimes, &amp;quot;actions&amp;quot; involving terms, formulae or parts thereof are verbalized before the appropriate formal operation is performed. The meaning of the &amp;quot;action verbs&amp;quot; is needed for the interpretation of the intended proof-step.</Paragraph>
    <Paragraph position="9"> Discourse deixis der obere Ausdruck [the above term] der letzte Satz [the last sentence] Folgerung aus dem Obigen [conclusion from the above] aus der regel in der zweiten Zeile [from the rule in the second line] This class of referring expressions includes also references to structural parts of terms and formulas such as &amp;quot;the left side&amp;quot; or &amp;quot;the inner parenthesis&amp;quot; which are incomplete specifications: the former refers to a part of a formula, the latter, metonymic, to an expression enclosed in parenthesis. Moreover, they require discourse referents for sub-parts of mathematical expressions to be available.</Paragraph>
    <Paragraph position="10"> Generic vs. specific reference Potenzmenge enthaelt alle Teilmengen, also auch (A\B) [A power set contains all subsets, hence also(A\B)] Generic and specific references can appear within one utterance as above, where &amp;quot;a power set&amp;quot; is a generic reference, whereas &amp;quot;A\B&amp;quot; is a specific reference to a subset of a specific instance of a power set introduced earlier.</Paragraph>
    <Paragraph position="11"> Co-reference6 Da, wenn Ai K(Bj) sein soll, Ai Element von K(Bj) sein muss. Und wenn Bk K(Al) sein soll, muss esk auch Element von K(Al) sein.</Paragraph>
    <Paragraph position="12"> [Because if it should be that Ai K(Bj), Ai must be an element of K(Bj). And if it should be that Bk K(Al), it must be an element of K(Al) as well.] DeMorgan-Regel-2 besagt: K(Ai \ Bj) = K(Ai) [ K(Bj) In diesem Fall: z.B. K(Ai) = dem Begriff</Paragraph>
    <Paragraph position="14"> the term K(Ak [ Bl) K(Bj) = the term K(C [D)] Co-reference phenomena specific to informal mathematical discourse involve (parts of) mathematical expressions within text. In particular, entities denoted with the same literals may not co-refer, as in the second utterance.</Paragraph>
    <Paragraph position="15"> In the next section, we present the input interpretation procedure up to the level of lexical-semantic interpretation. We concentrate on the interface between the linguistic meaning representation (obtained from the parser) and the representation of domain-knowledge (encoded in a domain ontology), which we realize through a domain-motivated semantic lexicon.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 Interpretation strategy
</SectionTitle>
    <Paragraph position="0"> The task of the input interpretation component is two-fold. Firstly, it is to construct a representation of the utterance's linguistic meaning. Secondly, it is to identify within the utterance, separate, and construct interpretations of: (i) parts which constitute meta-communication with the tutor (e.g., &amp;quot;Ich habe die Aufgabenstellung nicht verstanden.&amp;quot; [I don't understand what the task is.] that are not to be processed by the domain reasoner; and (ii) parts which convey domain knowledge that should be verified by a domain reasoner; for example, the entire utterance &amp;quot;K((A [ B)) ist laut deMorgan-1 K(A) \ K(B)&amp;quot; [... is, according to deMorgan-1,...] can be evaluated in the context of the proof being constructed; on the other hand, the reasoner's knowledge base does not contain appropriate representations to evaluate the appropriateness of the focusing particle &amp;quot;also&amp;quot; in &amp;quot;Wenn A = B, dann ist A auch K(B) und B K(A).&amp;quot; [If A = B, then A is also K(B) and B K(A).].</Paragraph>
    <Paragraph position="1"> Domain-specific interpretation(s) of the proofrelevant parts of the input are further processed by  sentation of the proof constructed by the student;8 (ii) check appropriateness of the interpretation(s) found by the input understanding module with the state of the proof constructed so far; (iii) given the current proof state, evaluate the utterance with respect to soundness, relevance, and completeness.</Paragraph>
    <Paragraph position="2"> The semantic analysis proceeds in 2 stages: (i) After standard pre-processing9, mathematical expressions are identified, analyzed, categorized, and substituted with default lexicon entries encoded in the grammar. The input is then syntactically parsed, and an formal abstract representation of its meaning is constructed compositionally along with the parse; (ii) The obtained meaning representation is subsequently merged with discourse context and interpreted by consulting a semantic lexicon of the domain and a domain-specific ontology.</Paragraph>
    <Paragraph position="3"> In the next sections, we first briefly summarize the syntactic and semantic parsing part of the input understanding process10 and show the format of meaning encoding constructed at this stage (Sect. 4.1). Then, we show the lexical-semantic interface to the domain ontology (Sect. 4.2).</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
4.1 Linguistic Meaning
</SectionTitle>
      <Paragraph position="0"> By linguistic meaning (LM), we understand the dependency-based deep semantics in the sense of the Prague School sentence meaning as employed in the Functional Generative Description (FGD) (Sgall et al., 1986; Kruijff, 2001). It represents the literal meaning of the utterance rather than a domain-specific interpretation.11 In FGD, the central frame unit of a sentence/clause is the head verb which specifies the tectogrammatical relations (TRs) of  kenization, (spelling correction and) morphological analysis, part-of-speech tagging.</Paragraph>
      <Paragraph position="1"> 10We are concentrating on syntactically well-formed utterances. In this paper, we are not discussing ways of combining deep and shallow processing techniques for handling malformed input.</Paragraph>
      <Paragraph position="2"> 11LM is conceptually related to logical form, however, differs in coverage: while it does operate on the level of deep semantic roles, such aspects of meaning as the scope of quantifiers or interpretation of plurals, synonymy, or ambiguity are not resolved.</Paragraph>
      <Paragraph position="3"> its dependents (participants). Further distinction is drawn into inner participants, such as Actor, Patient, Addressee, and free modifications, such as Location, Means, Direction. Using TRs rather than surface grammatical roles provides a generalized view of the correlations between the conceptual content of an utterance and its linguistic realization. At the pre-processing stage, mathematical expressions embedded within input are identified, verified as to syntactic validity, categorized, and substituted with default lexical entries encoded in the parser grammar for mathematical expression categories. For example, the expression K((A [ B) \ (C [D)) = (K(A[B)\K(C [D)) given its top node operator, =, is of type formula, its &amp;quot;left side&amp;quot; is the expression K((A [ B) \ (C [ D)), the list of bracketed sub-expressions includes: A[B, C[D, (A [ B) \ (C [ D), etc.</Paragraph>
      <Paragraph position="4"> Next, the pre-processed input is parsed with a lexically-based syntactic/semantic parser built</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
on Multi-Modal Combinatory Categorial Gram-
</SectionTitle>
      <Paragraph position="0"> mar (Baldridge, 2002; Baldridge and Kruijff, 2003).</Paragraph>
      <Paragraph position="1"> The task of the deep parser is to produce an FGD-based linguistic meaning representation of syntactically well-formed sentences and fragments. The linguistic meaning is represented in the formalism of Hybrid Logic Dependency Semantics. Details on the semantic construction in this formalism can be found in (Baldridge and Kruijff, 2002).</Paragraph>
      <Paragraph position="2"> To derive our set of TRs we generalize and simplify the collection of Praguian tectogrammatical relations from (HajiVcov'a et al., 2000). One reason for simplification is to distinguish which relations are to be understood metaphorically given the domain-specific sub-language. The most commonly occurring relations in our context (aside from the roles of Actor and Patient) are Cause, Condition, and Result-Conclusion (which coincide with the rhetorical relations in the argumentative structure of the proof):  the meaning contain and in this frame takes dependents in the relations Actor and Patient, shown schematically in Fig. 2 (FORMULA represents the default lexical entry for the identified mathematical expressions categorized as formulas). The linguistic meaning of this utterance returned by the parser obtains the following representation: @h1(contain ^ &lt;ACT&gt;(f1 ^ FORMULA:B) ^ &lt;PAT&gt;(f2 ^ FORMULA: x 2 A) where h1 is the state where the proposition contain is true, and the nominals f1 and f2 represent dependents of the head contain, in the relations Actor and Patient, respectively.</Paragraph>
      <Paragraph position="3"> More details on our approach to parsing interleaved natural and symbolic expressions can be found in (Wolska and Kruijff-Korbayov'a, 2004a) and more information on investigation into tectogrammatical relations that build up linguistic meaning of informal mathematical text can be found in (Wolska and Kruijff-Korbayov'a, 2004b).</Paragraph>
    </Section>
    <Section position="3" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
4.2 Conceptual Semantics
</SectionTitle>
      <Paragraph position="0"> At the final stage of input understanding, the linguistic meaning representations obtained from the parser are interpreted with respect to the given domain. We encode information on the domain-specific concepts and relations in a domain ontology that reflects the knowledge base of the domainreasoner, and which is augmented to allow resolution of ambiguities introduced by natural language (Horacek and Wolska, 2004). We interface to the domain ontology through an upper-level ontology of concepts at the lexical-semantics level.</Paragraph>
      <Paragraph position="1"> Domain specializations of conceptual relations are encoded in the domain ontology, while a semantic lexicon assigns conceptually-oriented semantics in terms of linguistic meaning frames and provides a link to the domain interpretation(s) through the domain ontology. Lexical semantics in combination with the knowledge encoded in the ontology allows us to identify those parts of utterances that have an interpretation in the given domain. Moreover, productive rules for treatment of metonymic expressions are encoded through instantiation of type compatible concepts. If more than one lexical-semantic interpretation is plausible, no disambiguation is performed. Alternative conceptual representations are further interpreted using the domain ontology, and passed on to the Proof Manager for evaluation. Below we explain some of the entries the semantic lexicon encodes: Containment The Containment relation specializes into the domain relations of (strict) SUB-SET and ELEMENT. Linguistically, it can be realized, among others, with the verb &amp;quot;enthalten&amp;quot; (&amp;quot;contain&amp;quot;). The tectogrammatical frame of &amp;quot;enthalten&amp;quot; involves the roles of Actor (ACT) and Patient (PAT):</Paragraph>
      <Paragraph position="3"> Location The Location relation, realized linguistically by the prepositional phrase introduced by &amp;quot;in&amp;quot;, involves the tectogrammatical relations HasProperty-Location (LOC) and the Actor of the predicate &amp;quot;sein&amp;quot;. We consider Location in our domain as synonymous with Containment. Another realization of this relation, dual to the above, occurs with the adverbial phrase &amp;quot;ausserhalb von ...(liegen)&amp;quot; (&amp;quot;lie outside of&amp;quot;) and is defined as negation of Containment:</Paragraph>
      <Paragraph position="5"> Common property A general notion of &amp;quot;common property&amp;quot; we define as follows:</Paragraph>
      <Paragraph position="7"> Property is a meta-object that can be instantiated with any relational predicate, for example as in &amp;quot;(A und B)&lt;ACT&gt; haben (gemeinsame Elemente)&lt;PAT&gt;&amp;quot; (&amp;quot;A and B have common elements&amp;quot;): common(ELEMENT, ACTplural(A:SET;B:SET)) ELEMENT(p1 ;A)^ ELEMENT(p1 , B) Difference The Difference relation, realized linguistically by the predicates &amp;quot;verschieden (sein)&amp;quot; (&amp;quot;be different&amp;quot;; for COLLECTION or STRUCTURED OBJECTS) and &amp;quot;disjunkt (sein)&amp;quot; (&amp;quot;be disjoint&amp;quot;; for objects of type COLLEC-TION) involves a plural Actor (e.g. coordinated noun phrases) and a HasProperty TRs. Depending on the type of the entity in the Actor relation, the interpretations are:</Paragraph>
      <Paragraph position="9"> Mereological relations Here, we encode part-of relations between domain objects. These concern both physical surface and ontological properties of objects. Commonly occurring part-of relations in our domain are:  Using these definitions and polysemy rules such as polysemous(Object, Property), we can obtain interpretation of utterances such as &amp;quot;Dann gilt f&amp;quot;ur die linke Seite, . . . &amp;quot; (&amp;quot;Then for the left side it holds that . . . &amp;quot;) where the predicate &amp;quot;gilt&amp;quot; normally takes two arguments of types STRUCTURED OBJECTterm;formula, rather than an argument of type Property.</Paragraph>
      <Paragraph position="10"> For example, the previously mentioned predicate contain (Fig. 2) represents the semantic relation of Containment which, in the domain of naive set theory, is ambiguous between the domain relations EL-EMENT, SUBSET, and PROPER SUBSET. The alternative specializations are encoded in the domain ontology, while the semantic lexicon provides the conceptual structure of the head predicate. At the domain interpretation stage, the semantic lexicon is consulted to translate the tectogrammatical frame of the predicate into a semantic relation represented in the domain ontology. For the predicate contain, from the semantic lexicon, we obtain: contain(ACTtype:FORMULA, PATtype:FORMULA) (SUBFORMULAP AT , embeddingACT ) ['a Patient of type FORMULA is a subformula embedded within a FORMULA in the Actor relation with respect to the head contain']</Paragraph>
      <Paragraph position="12"> ['the Containment relation involves a predicate contain and its Actor and Patient dependents, where the Actor and Patient are the container and containee parameters respectively'] Translation rules that consult the domain ontology expand the conceptual structure representation into alternative domain-specific interpretations preserving argument structure. As it is in the capacity of neither sentence-level nor discourse-level analysis to evaluate the appropriateness of the alternative interpretations in the proof context, this task is delegated to the Proof Manager.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
5 Example analysis
</SectionTitle>
    <Paragraph position="0"> In this section, we illustrate the mechanics of the approach on the following example: A enthaelt keinesfalls Elemente, die auch in B sind. [A contains no elements that are also in B] The analysis proceeds as follows.</Paragraph>
    <Paragraph position="1"> The mathematical expression tagger first identifies the expressions A and B. If there was no prior discourse entity for &amp;quot;A&amp;quot; and &amp;quot;B&amp;quot; to verify their types, they are ambiguous between constant, term, and formula12. The expressions are substituted with generic entries FORMULA, TERM, CONST represented in the parser grammar. The sentence is assigned alternative readings: &amp;quot;CONST contains no elements that are also in CONST&amp;quot;, &amp;quot;CONST contains no elements that are also in TERM&amp;quot;, &amp;quot;CONST contains no elements that are also in FORMULA&amp;quot;, etc. Here, we continue only with &amp;quot;CONST contains no elements that are also in CONST&amp;quot;; the other readings would be discarded at later stages of processing because of sortal incompatibilities.</Paragraph>
    <Paragraph position="2"> The linguistic meaning of the utterance obtained from the parser is represented by the following formula13: null</Paragraph>
    <Paragraph position="4"> ['(set) A contains no elements that are in (set) B'] Next, the semantic lexicon is consulted to translate the linguistic meaning representation into a conceptual structure. The relevant lexical semantic entries are Containment and Location (see Sect. 4.2).</Paragraph>
    <Paragraph position="5"> The transformation is presented schematically below: null contain(ACTOBJECT:A, PATOBJECT:element)</Paragraph>
    <Paragraph position="7"> Finally, in the domain ontology, we find that the conceptual relation of Containment, in naive set theory, specializes into the domain relations of ELE-MENT, SUBSET, STRICT SUBSET. Using the linguistic meaning, the semantic lexicon, and the domain ontology, we obtain all the combinations of interpretations, including the target one paraphrased below: 'it is not the case that there exist elements e, such that e 2 A and e 2 B', Using translation rules the final interpretations are translated into first-order logic formulas and passed on for evaluation to the Proof Manager.</Paragraph>
  </Section>
  <Section position="7" start_page="0" end_page="0" type="metho">
    <SectionTitle>
6 Related work
</SectionTitle>
    <Paragraph position="0"> Language understanding in dialog systems, be it with speech or text interface, is commonly performed using shallow syntactic analysis combined 12In prior discourse, there may have been an assignment A := , where is a formula, in which case, A would be known from discourse context to be of type FORMULA (similarly for term assignment); by CONST we mean a set or element variable such as A, x denoting a set A or an element x respectively.</Paragraph>
    <Paragraph position="1"> 13Irrelevant parts of the meaning representation are omitted; glosses of the formula are provided.</Paragraph>
    <Paragraph position="2"> with keyword spotting. Tutorial systems also successfully employ statistical methods which compare student responses to a model built from preconstructed gold-standard answers (Graesser et al., 2000). This is impossible for our dialogs, due to the presence of symbolic mathematical expressions and because of such aspects of discourse meaning as causal relations, modality, negation, or scope of quantifiers which are of crucial importance in our setting, but of which shallow techniques remain oblivious (or handle them in a rudimentary way).</Paragraph>
    <Paragraph position="3"> When precise understanding is needed, tutorial systems use closed-questions to elicit short answers of little syntactic variation (Glass, 2001) or restricted format of input is allowed. However, this conflicts with the preference for flexible dialog do achieve active learning (Moore, 1993).</Paragraph>
    <Paragraph position="4"> With regard to interpreting mathematical texts, (Zinn, 1999) and (Baur, 1999) present DRT analyzes of course-book proofs. The language in our dialogs is more informal: natural language and symbolic mathematical expressions are mixed more freely, there is a higher degree and more variety of verbalization, and mathematical objects are not properly introduced. Both above approaches rely on typesetting information that identifies mathematical symbols, formulas, and proof steps, whereas our input does not contain any such information.</Paragraph>
    <Paragraph position="5"> Forcing the user to delimit formulas would not guarantee a clean separation of the natural language and the non-linguistic content, while might reduce the flexibility of the system by making the interface harder to use.</Paragraph>
  </Section>
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