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<Paper uid="A00-3003">
  <Title>Generating Text with a Theorem Prover</Title>
  <Section position="3" start_page="0" end_page="13" type="metho">
    <SectionTitle>
2 A Logical Semantics for
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="13" type="sub_section">
      <SectionTitle>
Statecharts
</SectionTitle>
      <Paragraph position="0"> The graphical language of statecharts as proposed by David Harel (Harel et al., 1987; Harel and Naamad, 1996), has been widely recognized as a important tool for analyzing complex reactive systems.</Paragraph>
      <Paragraph position="1"> It has been implemented in commercial applications like STATEMATE (Harel and Politi, 1998) 1A full description of this algorithmic translation of a statechart from its graphical formalism to the propositional logic input format used in this work is described in Garibay (2000).  and RHAPSODY from ilogix (I-Logix Inc., 2000) and has been adopted as a part of the Unified Modeling Language (UML Revision Task Force, 1999; Booch, 1999), an endeavor to standardize a language of blueprints for software.</Paragraph>
      <Paragraph position="2"> Statecharts (Fig. 3) are an extension of conventional finite state machines in which the states may have a hierarchical structure. A configuration is defined as a maximal set of non-conflicting states which are active at a given time. A transition connects states and is labeled with the set of events that trigger it, and a second set of events that are generated when the transition is taken. A step of the statechart relates the current configuration and the events that are active to the next configuration and the events that are generated. A configuration and the set of events that are active is referred to as a status.</Paragraph>
      <Paragraph position="3"> We capture a step of a statechart as a pair of propositional models, one for the current status and</Paragraph>
      <Paragraph position="5"> tion of the example statechart (Fig. 3).</Paragraph>
      <Paragraph position="6"> one for the next status. In practice, we incorporate this into a single model with two versions of each propositional variable: P for the truth value in the current status and Pn for the truth value in the next status 2. A full description of the algorithm for translating statecharts to sets of formulae can be found in Garibay (2000). For a example of this translation see Fig. 4.</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="13" end_page="14" type="metho">
    <SectionTitle>
3 The Minimum Clausal Theory of
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="13" end_page="14" type="sub_section">
      <SectionTitle>
the Statecharts
</SectionTitle>
      <Paragraph position="0"> At this point, we have a formula that entails the theory of the single step transition behavior of a Statechart. We can fulfill our requirement of generating a sound and complete report just by translating this formula into English. However, this approach presents a number of problems. For instance, the AND and OR connectives do not in general have the same meaning in English as they do in logic (Gazdar, 1979), furthermore, unlike in the logical formula the scope of the connectives in English is not, in general, well defined (Holt and Klein, 1999). To minimize the ambiguity, we need to take the formula to a form with minimal nesting of operators.</Paragraph>
      <Paragraph position="1"> Potentially a more significant problem is the fact that much of the theory (the formula plus all its logical consequences) is obtainable only via complicated inferences. Since the reader understands the translation of the formula at an intuitive level, making only limited inferences, a direct translation will fail to communicate the entire theory. Hence, we would like to take the formula to a form that is closed, in some sense, under logical consequences.</Paragraph>
      <Paragraph position="2"> We address both issues by using what we refer to as minimal (fully) resolved conjunctive normal form (MRCNF). A formula is in a MRCNF if and only if  it is in conjunctive normal form (CNF) and is closed under resolution, absorption and tautology (Fitting, 1990; Rogers and Vijay-Shanker, 1994). The closure under resolution is effectively a finite approximation of closure under consequence, that is, every clause that is a logical consequence of the theory entailed by the formula is a direct consequence of some clause in the MRCNF. The other two operations guarantee minimality in size by removing clauses that are trivially true (tautology), and those that are proper super-sets of another (absorption).</Paragraph>
      <Paragraph position="3"> Hence, the translation will communicate not only the initial facts but also those inferred by resolution. Moreover, a formula in this form is just a conjunction of disjunctions--eliminating the scoping problem. If we interpret the disjunctions as implications, the translation into English will be just a sequence of implicative sentences that are to be interpreted conjunctively--a typical structure for such information in English.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="14" end_page="14" type="metho">
    <SectionTitle>
4 Organizing the Hyper-text
Report: The Question Tree
</SectionTitle>
    <Paragraph position="0"> A formula in MRCNF is organized in a way that resembles a sequence of implicative sentences. The problem now is the size of this sequence. Large to begin with, its size is increased by the transformation to CNF and closure under resolution. Hence, the translation of MRCNF directly into a sequence of statements would present an uninterpretable sequence of facts. If they are going to be understood by the reader there is a need for some kind of structure. The correct organization depends heavily on the reader's goals and expectations. However, beyond the assumption that the reader's generic goal is to obtain information about the transition behavior of the statechart under consideration, we do not make any assumptions about what the particular reader's goals may be. Instead we present the report as a hyper-text document and allow the reader to interactively refine their goal by following hyper-links. Effectively, the reader's queries focus the theory of the statechart in a particular aspect of its behavior 3.</Paragraph>
    <Paragraph position="1"> In this way, as in Reiter et al. (1992) and Levine et al. (1991), we use hyper-text as an implicit text planner, in the sense that we account for every possible model of the user/system interaction and let the actual reader decide which goal to pursue.</Paragraph>
    <Paragraph position="2"> We will call the reader's selections choices. Each choice the reader makes narrows the information we have to convey, limiting it to all and only the part that is logically consistent with that choice. We will say that the reader refines the theory by making the choice. At each point, the choices available to the reader are all the propositional variables that 3In a process that will be precisely described shortly.</Paragraph>
    <Paragraph position="3"> the theory is contingent upon. The reader effectively fixes the valuation of one of these variables to true or false. The system then adds the reader's choice to the theory and recalculates the MRCNF. If the newly obtained theory remains contingent upon some variables, the reader then will have available a new set of choices. If not, the reader will have reached a set of non-contingent facts (henceforth facts) which are consequences of all the previous choices.</Paragraph>
    <Paragraph position="4"> While this process makes the information more accessible by giving it a logical structure, it does nothing to reduce the size of the report. We resolve this by generating the document on demand. While the refinement process (the core computation for on-demand generation) can potentially be very expensive in terms of time, the fact that we are adding singleton clauses to an already minimum set of clausal consequences allows us to use a simplified form of the theorem prover with asymptotic time complexity linear in the number of clauses.</Paragraph>
    <Paragraph position="5"> We can visualize the process of the reader making choices as navigating a question tree, in which each branch is labeled with a choice and each node contains the theory of the Statechart as refined by the path of choices from the root to that node. In this tree, a reader's choice is equivalent to the question: &amp;quot;What are the circumstances/situations if X is true/false?.&amp;quot; The root is the full theory of the transition behavior of the Statechart. The children of a node are obtained by fixing the valuation of each of its contingent propositional variables in turn and recomputing the MRCNF. The leaves are non-contingent theories (those containing only facts) a Conceptually, the labels of each path from the root to a leaf together with each one of the facts in that leaf corresponds to all and only the valuations which are models of the original theory. Therefore, the question tree is sound and complete in the logical sense.</Paragraph>
  </Section>
  <Section position="6" start_page="14" end_page="15" type="metho">
    <SectionTitle>
5 Generating the Hyper-text Page
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="14" end_page="15" type="sub_section">
      <SectionTitle>
under Pragmatic Considerations:
Information Extraction Module
</SectionTitle>
      <Paragraph position="0"> This tree turns out to provide a useful framework to address pragmatic issues--those that arise principally from the structure of the report itself (Gazdar, 1979). By addressing these issues in the context of the question tree, rather than in its realization as a report, we abstract away from a great deal of subtle semantic detail that would otherwise obscure the analysis. Our approach consists of applying a series of transformations that resolve these issues while 4In general this structure is a directed acyclic graph which Reiter et al. call the question space (Reiter et al., 1995), but since we work with a tree that spans it, we prefer question tree.</Paragraph>
    </Section>
    <Section position="2" start_page="15" end_page="15" type="sub_section">
      <SectionTitle>
5.1 Promoting facts
</SectionTitle>
      <Paragraph position="0"> In the question tree, the facts are either reported at the end of a chain of choices or are encoded in the choices themselves. A sequence of these choices is analogous to a chain of nested implications in which the antecedents are the choices made by the user and the consequence is the theory as refined by the choices. This refinement continues until we obtain a non-contingent theory--one in which all variables have valuations. Thus, the chain of implications eventually leads to a set of facts as its final consequence. The pragmatic problem in this case relates to the amount of information to be provided (Grice's Maxim of Quantity (Grice, 1975)). This maxim states that speakers will make their contribution as informative as is required, but not more informative than that (Gazdar, 1979). Under this assumption, reporting a fact as a consequence of a sequence of choices explicitly denies that this fact is a consequence of any prefix of that sequence, in contrast to the logical semantics of implication. Such implicatures, while not consequences of the logical content, are valid inferences that people make on the basis of well established expectations about the communicative act.</Paragraph>
      <Paragraph position="1"> To avoid this false implicature, we present the facts to the reader as soon as they become available, that is, as soon as they become non-contingent in the theory. The transformation, in this case, moves the facts from the leaves to the interior nodes. This transformation does not change the set of models represented in the tree simply because the movement of facts does not eliminate any path of the tree. Hence, the transformation preserves soundness and completeness of the tree.</Paragraph>
      <Paragraph position="2"> In practice, the facts are just the singleton clauses of a theory, therefore we can realize this transformation by simply reporting singleton clauses as soon as they appear in the theory.</Paragraph>
    </Section>
    <Section position="3" start_page="15" end_page="15" type="sub_section">
      <SectionTitle>
5.2 Reporting facts only once
</SectionTitle>
      <Paragraph position="0"> On the other hand, facts in a theory are also facts in every consistent refinement of that theory. Hence, reporting all the facts at each node of the question tree leads us to report many of them repeatedly. In effect, every fact reported in a node will be reported in each of its children as well. This repetition of facts violates the &amp;quot;upper-bound&amp;quot; of Quantity--it reports more than is relevant. In this case Quantity requires us to report only information that is &amp;quot;new&amp;quot;.</Paragraph>
      <Paragraph position="1"> In general, what is new will depend not only on what is reported but on inferences the reader is likely to have made (McDonald, 1992). We have, however, already committed to being explicit; our assumption is that the reader makes essentially no inferences, that they know all and only what we have explicitly reported. Therefore, we can satisfy the upper bound of Quantity by reporting each fact exactly once on each branch--when it first becomes non-contingent.</Paragraph>
      <Paragraph position="2"> To do this, we simply keep a list of all facts that have been reported in the current branch; this is the extent of our model of the user.</Paragraph>
      <Paragraph position="3"> This transformation does not change the set of models represented in the tree, since it only eliminates repeated literals.</Paragraph>
    </Section>
    <Section position="4" start_page="15" end_page="15" type="sub_section">
      <SectionTitle>
5.3 Promoting single level implications
</SectionTitle>
      <Paragraph position="0"> One of the difficulties in using Quantity is to determine what information is &amp;quot;required&amp;quot;. At each node of the question tree we have a current theory to report. The issue, in essence, is what to report at that node and what to report at its descendents. On one hand, it seems clear that we are, at least, required to report the non-contingent facts at each node. On the other hand, we don't want to report the whole theory at the root.</Paragraph>
      <Paragraph position="1"> Our intuition is that the degree to which facts are relevant is inversely proportional to the difficulty of interpreting them. Under these circumstances, un-nested implications (i.e., binary disjunctions) are simple enough that the reader is likely to expect them to be reported. From the perspective of the question tree, this suggest, that in addition to the facts at a node, we should also report, as implications, the facts at its non-contingent children (those that are leaves). We refer to the choices leading to non-contingent theories as conclusive choices. These are reported as single-level implications (&amp;quot;If X then (some sequence of facts~&amp;quot;). This has the effect of promoting the leaves of the tree to their parent pages.</Paragraph>
      <Paragraph position="2"> Note that a choice that is conclusive at some page will also be conclusive at each page in the subtree rooted at that page (or, rather, at each page reached by a sequence of choices consistent with that choice).</Paragraph>
      <Paragraph position="3"> In keeping with the principle of reporting a fact exactly once along each path, we must avoid reporting the implication at the descendent pages. To this end, after reporting each of the conclusive choices on a page, we report the remainder of the tree below that page under an &amp;quot;Otherwise&amp;quot; choice in which the theory has been refined with the complements of the conclusive choices. This has the effect of dramatically restructuring the tree: each of the non-contingent leaves is promoted to the highest page at which the choice that selects it becomes conclusive.</Paragraph>
      <Paragraph position="4"> Once again this transformation reorganizes the branches of the question tree without changing the set of models it represents.</Paragraph>
      <Paragraph position="5"> To find the conclusive choices we run the theorem prover on the current theory extended, in turn, with each literal upon which it is contingent. If the resulting theory is non-contingent, then that literal is a  e If the current configuration includes the state SUN \[ then... \] * If the current configuration duel not include the state SON ~ teen... \] * If the current configuration includes the state SOP \[ then... \] * If the current configuration does not include the state SOP \[ then... \] * If the event MUTE iS active \[ then.,. \] * If the event MUTE is not active \[ then... \]  The following choices are conclusive:  * If the event OFF is active then: - the next configuration wlll include the state WAITING, but elll not include C/hm states PICTURE or TEXT.</Paragraph>
      <Paragraph position="6"> - the event h'UTE rill not be generated.</Paragraph>
      <Paragraph position="7"> * If the next configuration tncludel the state WAITING then: the event OFF lw active.</Paragraph>
      <Paragraph position="8"> - the next configuration sill not include the states PICTURE or TEXT - the event NUTE will not be generated.</Paragraph>
      <Paragraph position="9">  conclusive choice. To find the remainder of the tree to be reported under the &amp;quot;Otherwise&amp;quot; case we extend the current theory with the negation of each of the conclusive choices. If the resulting theory is inconsistent we will say that the conclusive choices are exhaustive, if the result is a contingent theory we will say that the conclusive choices are non-exhaustive with non-conclusive otherwise, and if the result is a non-contingent theory we will say that the conclusive choices, in this case, are non-exhaustive with conclusive otherwise.</Paragraph>
    </Section>
    <Section position="5" start_page="15" end_page="15" type="sub_section">
      <SectionTitle>
5.4 Aggregating pairs of single conditionals
</SectionTitle>
      <Paragraph position="0"> It frequently happens that, at some page, two conclusive choices lead to the same model. In this case, we would report that each implies (among other things) the other. However, these two implications can be aggregated to form a biconditional. Furthermore, Quantity requires us to select the strongest connective that applies in any such case because if a weaker connective is selected it suggests that no stronger one applies (a scalar implicature). Consequently, we are actually compelled to aggregate these two facts into a single biconditional.</Paragraph>
      <Paragraph position="1"> In practice, we use the theorem prover to either prove or disprove, for every implication, whether its converse is a theorem of the current theory. If proved then the biconditional is reported.</Paragraph>
      <Paragraph position="2"> Biconditional I|plications: - the next configuration rill include the state $OFF if and only if the next configuration will not include the state SON.</Paragraph>
      <Paragraph position="3"> One of ~he following must be the case: Either: - the current configuration includes the ,tats SOFF, but does not include the state TEXT.</Paragraph>
      <Paragraph position="4"> - the event ESOUNU t~ not active.</Paragraph>
      <Paragraph position="5"> - the next configuration rill include the s~ate SOFF, but will not include the state SON.</Paragraph>
      <Paragraph position="6"> - the event ESOUND will not be generated.</Paragraph>
      <Paragraph position="7"> O~:the current configuration Includel the state SUFF, hut does not include the state TEXT.</Paragraph>
      <Paragraph position="8"> - the event ESOUND is active.</Paragraph>
      <Paragraph position="9"> - the next configuration wlll include the state SON, but wlll not include the state SDFF.</Paragraph>
      <Paragraph position="10"> - the event ESOOND .Iii not be generated.</Paragraph>
    </Section>
  </Section>
  <Section position="7" start_page="15" end_page="17" type="metho">
    <SectionTitle>
6 Hyper-text Organization and
Realization Module
</SectionTitle>
    <Paragraph position="0"> The organization of the hyper-text page generated from each node of the question tree visited by the user is shown in Fig. 5. At the top of the page we report (parenthetically) the set of choices that have led to this page. Next we report all of the new facts obtained from the current theory as described in sections 5.1 and 5.2. Then, the propositions that the theory is no longer dependent on (those which no longer occur in the theory ) followed by the list of propositions on which it does depend. Finally we present the choices or, if there are any, the conclusive choices. In the first (Fig. 5), each choice is presented as an implicative sentence with a hyper-text link leading to another page (another node of the question tree). In the second (Fig. 6 top), we present the set of conclusive choices followed by one of the three possible cases (described in Section 5.3) for the &amp;quot;Otherwise&amp;quot; case. If the conclusive choices are exhaustive (the otherwise case is inconsistent), we report the biconditional implications (Section 5.4) followed by the final models (Fig 7). If they are exhaustive with a conclusive otherwise, we report the otherwise as another conclusive choice (Fig 8). Finally, if they are exhaustive with a non-conclusive otherwise, we report only an otherwise hyper-link (Fig 6 bottom).</Paragraph>
    <Paragraph position="1"> The realization module is, in essence, a pattern matching and template filling process. It's basic component simply translates facts into fixed English language sentences. 5 Facts are represented by literals. These are classified into the following categories: current state, current event, next state, and next event and the literais in each category are syn- null The cur~n! configuration includes ti~ state WORKING The next configuration will include the state WAITING, but will not include tbu state PICTURE J The cv(c)nts OFF and TXT an: active  tactically aggregated (Dalianis, 1999). The process is illustrated in Figure 9.</Paragraph>
  </Section>
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