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<Paper uid="P90-1028">
  <Title>ALGORITHMS FOR GENERATION IN LAMBEK THEOREM PROVING</Title>
  <Section position="4" start_page="220" end_page="220" type="metho">
    <SectionTitle>
2 EXTENDING THE CAL-
CULUS
</SectionTitle>
    <Paragraph position="0"> Natural Language Processing as deduction The architectures in this paper resemble the uniform architecture in Shieber (1988) because language processing is viewed as logical deduction, in analysis and generation: &amp;quot;The generation of strings matching some criteria can equally well be thought of as a deductive process, namely a process of constructive proof of the existence of a string that matches the criteria.&amp;quot; (Shieber, 1988, p. 614).</Paragraph>
    <Paragraph position="1"> In the LTP framework a categorial reduction system is viewed as a logical calculus where parsing a syntagm is an attempt to show that it follows from a set of axioms and inference rules. These inference rules describe what the processor does in assembling a semantic representation (representational non-autonomy: Crain and Steedman, 1982; Ades and Steedman, 1982). Derivation trees represent a particular parse process (Bouma, 1989).</Paragraph>
    <Paragraph position="2"> These rules thus seem to be nondeclarative, and this raises the question whether they can be used for generation. The answer to this question will emerge throughout this paper.</Paragraph>
    <Paragraph position="3"> Lexical information As in any categorial grammar, linguistic information in LTP is for the larger part represented with the signs in the lexicon and not with the rules of the calculus (signs are denoted by prosody:syntax:semantlcs). A generator using a categorial grammar needs lexical information about the syntactic form of a functor that is connected to some semantic functot in order to syntactically correctly generate the semantic arguments of this functor. For a parser, the reverse is true. In order to fulfil both needs, lexical information is made available to the theorem prover in the form of in~t6aces of o~ionu. I Axioms then truely represent what should be axiomatic in a lexicalist description of a language: the \]exical items, the connections between form and meaning. 2</Paragraph>
    <Paragraph position="5"> Rules Whenever inference rules are applied, an attempt is made to axiomatize the functor that participates in the inference by the first subsequent of the elimination rules. This way, lexical information is retrieved from the lexicon.</Paragraph>
    <Paragraph position="7"> \[(id*Pros):X:Tarm_X\].</Paragraph>
    <Paragraph position="8"> A prosodic operator connects prosodic elements. A prosodic identity element, id, is necessary because introduction rules are prosodical\]y vacuous. In order to avoid unwanted matching between axioms and id-elements, one special axiota is added for id-elements. Meta-logical checks are included in the rules in order to avoid vsriables occuring in the final derivation, nogenv,2r reeursively checks whether any part of an expression is a variable.</Paragraph>
    <Paragraph position="9"> A sequent in the calculus is denoted with P =&gt; T, where P, called the antecedent, and T, the succedent, are finite sequences of signs. The calculus is presented in (1) . In what follows, X and Y= are categories; T and Z, are signs; R, U and V are possibly empty sequences of signs; @ denotes functional application, a caret denotes ~abstraction, s</Paragraph>
    <Paragraph position="11"> \[loves:(np\s)/np:lows\].</Paragraph>
    <Paragraph position="12"> In order to initiate analysis, the theorem prover is presented with sequents like (2). Inference rules are applied recursively to the antecedent of the sequent until axioms are found. This regime can be called top-down from the point of view ofprob\]em solving and bottom-up from a &amp;quot;parsing&amp;quot; point of view. For generation, a sequent like (3) is presented to the theorem prover. Both analysis and generation result in a derivation like (4). Note that generation not only results in a sequence of lexical signs, but also in a peosodic pl~rasing that could be helpful for speech generation.</Paragraph>
    <Paragraph position="13"> (2) lVem der Linden and Minnen (submitted) contains a more elaborate comparison of the extended cedcu\]tm with the origins\] calculus as proposed in Moortgat (1988). 2A suggestion similar to this proposal was made by K~nig (1989) who stated that lexicsI items are to be seen as axioms, but did not include them as such in her description of the L-calculus.</Paragraph>
    <Paragraph position="14"> SThroughout this paper we will use a Prolog notation because the architectures presented here depend partly on the Prolog un\[i~cstlon mechanism. 221</Paragraph>
    <Paragraph position="16"> Although both (2) and (3) result (4), in the case of generation, (4) does not represent the (4) john:np:john 1or*s: (np\s)/np:loves ma~ry:np:mary =&gt; john*(loves*mary):s:lovesQaary@john &lt;loves: (np\s)/np:loves =&gt; loves: (np\s)/np:1oves &lt;loves: (np\s)/np:loves =I&gt; loves:(np\s)/np:1oves &lt;- true aary:np:aary =&gt; aary:np:aary &lt;ms.ry:np:aa~ry =I&gt; aary:np:aary &lt;- true john: np: J olm loves*mary : np\s : lovea@aary =&gt; j ohn* (loves*mary) : s : loves@aary@j olm &lt;loves*aary : np\s : loves@mary =&gt; loves*aary :np\s : loves@mary &lt;- true john:np:john =&gt; john:np:john &lt;john:np:john -1&gt; john:np:john &lt;- true john* (loves*aary) :s : lovss@aaryQj ohn =&gt; john* (loves*mary) : s : loves@aary@j ohn: &lt;- true  exact proceedings of the theorem prover. It starts applying rules, matching them with the antecedent, without making use of the original semantic information, and thus resulting in an inefficient and nondeterministic generation process: all possible derivations including all hxical items are generated until some derivation is found that results in the succedent. 4 We conclude that the algorithm normally used for parsing in LTP is inefficient with respect to generation.</Paragraph>
  </Section>
  <Section position="5" start_page="220" end_page="224" type="metho">
    <SectionTitle>
3 CALCULI DESIGNED
FOR GENERATION
</SectionTitle>
    <Paragraph position="0"> A solution to the ei~ciency problem raised in the previous section is to start from the origihal semantics. In this section we discuss calculi that make explicit use of the original semantics.</Paragraph>
    <Paragraph position="1"> Firstly, we present Lambek-like rules especially designed for generation. Secondly, we introduce a Cut-rule for generation with sets of categorial reduction rules. Both entail a variant of the crucial starting-point of the semantic-he~d-driven algorithms described in Calder et al. (1989) and Shieber et al. (1989): if the functor of a semantic representation can be identified, and can be refated to a lexical representation containing syntactic information, it is possible to generate the arguments syntactically. The efficiency of this strategy stems from the fact that it is guided by the known semantic and syntactic information, and lexical information is retrieved as soon as possible. null In contrast to the semantic-head-driven al&gt;proach, our semantic representations do not allow for immediate recognition of semantic heads: these can only be identified after all arguments 4ef. Shleber et el. (1989) on top-down generation algorithms. 2 2 2 have been stripped of the functor recursively (loves@mary@john =:&gt; loves@mary =&gt; loves).</Paragraph>
    <Paragraph position="2"> Calder et al. conjecture that their algorithm &amp;quot;(...) extends naturally to the rules of composition, division and permutation of Combinatory Categorial Grammar (Steedman, 1987) and the Lambek Calculus (1958)&amp;quot; (Calder et al., 1989, p. 23 ). This conjecture should be handled with care. As we have stated before, inference rules in LTP describe ho~ a processor operates. An important difference with the categorial reduction rules of Calder et al. is that inference.rules in LTP implicitly initiate the recursion of the parsing and generation process. Technically speaking, Lambek rules cannot be arguments of the rule-predicate of Calder et al. (1989, p. 237). The gist of our strategy is similar to theirs, but the algorithms dilTer.</Paragraph>
    <Paragraph position="3"> Lambek-llke generation Rules are presented in (5) that explicitly start from the known information during generation: the syntax and semantics of the succedent. Literally, the inference rule states that a sequent consisting of an antecedent that unifies with two sequences of signs U and Y, and a succedent that unifies with a sign with semantics Sem_FuQSem_Arg is a theorem of the calculus if Y reduces to a syntactic functor looking for an argument on its left side with the functor-meaning of the original semantics, and U reduces to its argument. This rule is an equivalent of the second elimination rule in (I).</Paragraph>
    <Paragraph position="5"> In this section, we describe an algorithm for the theorem prover that proceeds in a combined bottom-up/top-down fashion from the problem solving point of view. It maintains the same semantics-driven strategy, and enables efficient generation with the bidirectional calculus in (I).</Paragraph>
    <Paragraph position="6"> The algorithm results in derivations like (4), in the same theorem prover architecture, be it along another path.</Paragraph>
    <Paragraph position="7"> A Cut-rule for generation A Cut-rule is a structural rule that can be used within the L-calculus to include partial proofs derived with categorial reduction rules into other proofs. In  The generator regimes presented in this section are semantics-driven: they start from a semantic representation, assume that it is part of the uppermost sequent within a derivation, and work towards the lexical items, axioms, with the recursive application of inference rules. From the point of view of theorem proving, this process should be described as a top-down problem solving strategy. The rules in this section are, however, geared towards generation. Use of these rules for parsing would result in massive non-determinism. Elficient parsing and generation require different rules: the calculus is not bidirectioaal. 223 Bidirectionality There are two reasons to avoid duplication of grammars for generation and interpretation. Firstly, it is theoretically more elegant and simple to make use of one grammar.</Paragraph>
    <Paragraph position="8"> Secondly, for any language processing system, human or machine, it is more economic (Bunt, 1987, p. 333). Scholars in the area of language generation have therefore pleaded in favour of the bidirectionalit~ of linguistic descriptions (Appelt, 1987).</Paragraph>
    <Paragraph position="9"> Bidirectionality might in the first place be implemented by using one grammar and two separate algorithms for analysis and generation (Jacobs, 1985; Calder et el., 1989). However, apart from the desirability to make use of one and the same grammar for generation and analysis, it would be attractive to have one and the same processiag architecture for both analysis and generation. Although attempts to find such architectures (Shieber, 1988) have been termed &amp;quot;looking for the fountain of youth', s it is a stimulating question to what extent it is possible to use the same architecture for both tasks.</Paragraph>
    <Paragraph position="10"> Example An example will illustrate how our algorithm proceeds. In order to generate from a sign, the theorem prover assumes that it is the succedent of one of the subsequeats of one of the inference rules (7-1/2). (In case of an introduction rule the sign is matched with the succedent of the headseq~en~; this implies a top-down step.) If unification with one of these subsequents can be established, the other subsequents and the headsequent can be partly instantiated.</Paragraph>
    <Paragraph position="11"> These sequents can then serve as starting points for further bottom-up processing. Firstly, the headsequent is subjected to bottom-up process-SRon Kaplan during discussion of the $hieber presentation at Coling 1988.</Paragraph>
    <Paragraph position="12"> Generation of nounphrase ~he ~abie. Start with sequent</Paragraph>
    <Paragraph position="14"> l- Assume suecedent is part of an axiom: \[Pros : np: the0t able\] =&gt; \[Pros :np: the@table\] 2- Match axiom with last subsequent of an inference rule:</Paragraph>
    <Paragraph position="16"> 3- Derive instantiated head sequent: \[Pros_Fu: np/Y: the\], \[T \[ R\] =&gt; \[Pros :rip: the0table\] 4- No more applications in head sequent: Prove (bottom-up) first instantiated subsequent: \[Pros_Fu: np/Y: the\] ,,&gt; \[Pros_Fu :np/Y : the\] Unifies with the axiom for &amp;quot;the&amp;quot;: Pros_Fu = the; Y = n.</Paragraph>
    <Paragraph position="17">  ing (7-3), in order to axiomatize the head functor as soon as possible. Bottom-up processing stops when no more application operators can be eliminsted from the head sequent (7-4). Secondly, working top-down, the other subsequents (7-4/5) are made subject to bottom-up processing, and at last the last subsequent (7-6). (7) presents generation of a nounphrsse, the ~able.</Paragraph>
    <Paragraph position="18"> Non-determinism A source for non-determinism in the semantics-driven strategy is the fact that the theorem prover forms hypotheses about the direction a functor seeks its arguments, and then checks these against the lexicon. A possibility here would be to use a calculus where dorainance and precedence are taken apart. We will pursue rids suggestion in future research.</Paragraph>
  </Section>
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