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<Paper uid="C02-1141">
  <Title>A complete integrated NLG system using AI and NLU tools</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 GePhoX
</SectionTitle>
    <Paragraph position="0"> PhoX is an extensible proof assistant based on higherorderlogic, whichhasbeendeveloppedto help mathematicians building proofs and teaching mathematics. Like other proof assistants, PhoX works interactively. The user (a mathematician) first gives the theorem to be proven (a goal). PhoX returns a list of subgoals which should be easier to prove than the initial goal.</Paragraph>
    <Paragraph position="1"> The user enters a command to guide PhoX in choosing or achieving a subgoal. The proof is thus computed top-down from goals to evidences. The user's commands form a Proof script. PhoX's output is a list of successive goals equivalent to a Proof tree.</Paragraph>
    <Paragraph position="2"> Both the Proof script and PhoX's output are difficult to read (even for a mathematician), as the reader can see for himself in Table 1 and Table 2. Hence, the need of an NLG system in order to obtain an easy-to-read version of the proof.</Paragraph>
    <Paragraph position="3"> GePhoX is given as input both the Proof script and the Proof tree. It is one of the Here is the goal:</Paragraph>
    <Paragraph position="5"> main original proposals in our generator (similar generators, such as PROVERB (Huang and Fiedler, 1997), take as input only the Proof tree). It makes it possible for GePhoX to start from an incomplete proof and produce texts during the interactive session. These texts help the mathematician user: before entering a new command in the Proof script, he can read a text reminding himself what he has been doing so far. The Proof script is also useful for identifying the reasoning strategies that have been used (reasoning by contradiction or induction), while it is very hard (if not impossible) to retrieve this information from a Proof tree with its numerous deduction steps.</Paragraph>
    <Paragraph position="6"> Another originality of GePhoX is that it takes into account the knowledge of the user who can be either a mathematician using PhoX or a person more or less novice in mathematics.</Paragraph>
    <Paragraph position="7"> For the same proof, GePhoX can generate several texts according to the user model.</Paragraph>
    <Paragraph position="8"> 3 Using a descrition logic (DL) The knowledge representation system Kl-One (Branchman et al., 1979) was the first DL. It was created to formalize semantic networks and frames (Minsky, 1974). It introduces the notions of TBoxes and ABoxes respectively for terminological and assertional knowledge. Kl-One has been widely used in the NLG community to formalize the domain model. On the other hand, this is not the case for the more recent DLs. Nevertheless, they present at least two advantages compared to Kl-One : 1) for a large variety of DLs, sound and complete algorithms have been developped for main inference problems such as subsumption, concepts satisfiability and consistency (Donini et al., 1996); 2) the relations between instances and classes are well defined for all the constructors, and their mathematical and computational properties have been studied in detail (Horrocks et al., 2000). So we believe that DLs are appropriate for the content determination task as shown in 3.3. Let us first briefly present DLs.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.1 A brief introduction to DL
</SectionTitle>
      <Paragraph position="0"> The three fundamental notions of DLs are individuals (representing objects in the domain), concepts (describing sets of individuals), and roles (representing binary relations between individuals or concepts). A DL is characterized by a set of constructors that allow us to build complex concepts/roles from atomic ones.</Paragraph>
      <Paragraph position="1"> The set of constructors which seem useful for GePhoX and their syntax are shown in Table 3; examples of concepts and roles with their semantic are shown underneath Table 3.</Paragraph>
      <Paragraph position="2">  conjonction C [?] D disjonction (U) C [?] D complement (C) rightangleneC univ. quant. [?]R.C exist. quant. (E) [?]R.C numeral restrictions (N) &gt;n R.C [?]n R.C collection of individuals (O) {a1,...,an} atomic role P roles conjonction (R) Q[?]R inverse role R[?]1 role composition Q * R Table 3: Syntax of standard constructors Examples of concepts with their semantic Theorem, Variable, {H1}, [?]CHOOSE.User</Paragraph>
      <Paragraph position="4"> The choice of constructors is domain dependent. Constructors other than those used in GePhoX (e.g. temporal extension) can be used for other domains (e.g. domains with non trivial temporal information), without altering the mathematical and computational properties.</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.2 Domain and user models in DL
</SectionTitle>
      <Paragraph position="0"> The Domain model is the set of concepts and roles necessary to express the input of the generator. More formally, let TD be a TBox, such that each input I can be described by means of an ABox AD corresponding to TD. The knowledge base SD = (TD,AD) is called knowledge base for the domain and noted dkb. The User model is a knowledge base SU = (TU,AU) such that TU and AU are respectivly subsets of TD and AD. SU is noted ukb. Table 4 shows a part of the dkb for GePhoX.</Paragraph>
    </Section>
    <Section position="3" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.3 Content determination tasks
</SectionTitle>
      <Paragraph position="0"> The content determination module performs four tasks, as shown in Figure 2.</Paragraph>
      <Paragraph position="1"> Translation: The input of the generator (assertional information) is first translated into concepts of the TBox. For that purpose, a correspondence is established between the elements of the input and concepts and roles in the dkb. The O constructor is used to keep information about the individuals occurring in the input. For example, command 2 in Table 1 with individual H is translated into the concept</Paragraph>
      <Paragraph position="3"> {H}, and commands 8 to 11 are translated into</Paragraph>
      <Paragraph position="5"> Selection: The selection task consists of choosing the relevant concepts among those constructed in the translation phase with regard to the ukb. For example, if C0 is an unknown concept for the user, a conceptC must be looked up in the ukb such as C approximates1 C0.</Paragraph>
      <Paragraph position="6">  Verification: At this point, the coherence of all the concepts of the selection is verified. For example, if the user tries to reason by induction on a real number, GePhoX tells him that it is not possible.</Paragraph>
      <Paragraph position="7"> Instanciation: With the information about individuals, which have been kept in the translation phase (with the use of the O constructor), the instanciation task is straightforward. Table 5 shows some instanciated concepts for the Euclidian division.</Paragraph>
      <Paragraph position="8"> As is well known, designing knowledge bases (dkb and ukb) and translating the input of the generator into concepts and roles of the DL is a difficult task which has to be fulfilled for every generator. However, with a DL, the selection, verification and instanciation tasks are domain independent: algorithms and their implementation are reusable. Moreover, when using a DL for the content determination task, the &amp;quot;message&amp;quot; is a first order logic formula (a standard representation shared by a large commu1Given two TBoxes T and T prime with T [?] T prime and a concept C [?] T [?] T prime, Cprime [?] T prime approximates C if C minimally subsumes Cprime or Cprime minimally subsumes C.</Paragraph>
      <Paragraph position="10"> nity) which takes into account the user knowledge and whose coherence has been checked.</Paragraph>
    </Section>
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