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<Paper uid="J99-2003">
  <Title>Tree Adjoining Grammars in a Fragment of the Lambek Calculus</Title>
  <Section position="5" start_page="218" end_page="225" type="metho">
    <SectionTitle>
5. TAG Analysis Using Noncommutative Proofnets
</SectionTitle>
    <Paragraph position="0"> A proof in sequent calculus contains many useless properties in its contexts. Girard (1987) has defined, in a purely geometric way, a class of graphs of formulas, called  Computational Linguistics Volume 25, Number 2 proofnets: for each proof of a sequent t- F in the one-sided sequent calculus for multiplicative linear logic, there is a corresponding proofnet whose conclusions are exactly the formulas in F, and for each proofnet, there is at least one corresponding proof of the sequent t- F in the one-sided sequent calculus for multiplicative linear logic (where P is a sequence of all the conclusions of the proofnet). Similarly, Abrusci (1991) defined in a purely geometric way a class of graphs, called noncommutative proofnets, relative to multiplicative noncommutative linear logic. Roorda (1992) also described proofnets for Lambek calculus. Other criteria exist by now for characterizing proofnets for commutative or noncommutative, intuitionistic or nonintuitionistic linear logic. We present here Abrusci's criteria.</Paragraph>
    <Section position="1" start_page="219" end_page="219" type="sub_section">
      <SectionTitle>
5.1 Noncommutative Proofnets
</SectionTitle>
      <Paragraph position="0"> Proofnets are defined on one-sided sequent calculi. Presentations of the one-sided sequent calculus, and of proofnets are given in the appendix. Let us recall that ~ is the &amp;quot;or&amp;quot; connective associated to (r) (the &amp;quot;and&amp;quot; connective), such that A --o B = A+-~gB. To every proof 7r of a sequent F- F in the one-sided sequent calculus for multiplicative noncommutative linear logic, we can associate (by induction on the construction of the proof zr) a noncommutative proofnet with conclusions P, i.e., an oriented planar graph 7r' of occurrencies of formulas such that:  and every occurrence of formula is a premise of at most one link and is a conclusion of exactly one link.</Paragraph>
      <Paragraph position="1"> The translation ~r I of 7r is a proofnet, i.e., it admits no shorttrip. A shorttrip is a trip that does not contain each node twice. A trip is a sequence of nodes, going from one node to another according to the graph and to a switch for each Q-link and each =?-link, in a bideterministic way: the traversal of nodes is done according to Figure 8.</Paragraph>
      <Paragraph position="2"> Every assignment for 7r' is total: two integer variables are associated to each label (one for each &amp;quot;side&amp;quot; of the variable). Constraints are imposed on variables with respect to how trips are done throughout the net. The assignment is total if the set of constraints has a solution.</Paragraph>
      <Paragraph position="3">  sequent k P in the sequent calculus for multiplicative noncommutative linear logic such that 7r I is associated to 7r.</Paragraph>
      <Paragraph position="4"> Note that every noncommutative proofnet is a planar graph.</Paragraph>
    </Section>
    <Section position="2" start_page="219" end_page="223" type="sub_section">
      <SectionTitle>
5.2 Parse Examples
</SectionTitle>
      <Paragraph position="0"> In this section, we give two simple examples of parses. The aim of this section is to show the strong connection between the structure of proofs of sequents and a standard TAG derived structure. Moreover, it emphasizes the interest of a proofnet approach as the syntax (and parsing process) is concretely designed as a logical manipulation of logical structures. In the next section, we develop this approach and show how lexical rules can be integrated into it. Finally, we briefly mention that this can also give a logical formalization of D-trees (Vijay-Shanker 1992).</Paragraph>
      <Paragraph position="1"> The first example requires only substitution, i.e., the cut rule in the logical point of view. We first give the sequents (provable in .4) associated to the lexical items. Their meanings are straightforward, e.g., &amp;quot;John and Mary are noun phrases (NP)&amp;quot; or &amp;quot;saw requires a complement NP to obtain a verb phrase (VP) and a subject NP to obtain a sentence (S).&amp;quot; Note that VP is an adjunction node so the sequent associated to the item saw includes the formula VP o-- VP. The next example uses this specification.</Paragraph>
      <Paragraph position="2">  A(G) proofs of John saw Mary.</Paragraph>
      <Paragraph position="3"> The proof associated to the analysis of John saw Mary requires two cuts. The two sequent proofs given in Figure 9 are the only two possibilities for this sentence in the fragment A(G). This pinpoints the fact that the order in which the cuts are done is not significant with respect to the derived structure. Proofnets allow the expression of this equivalence. Hence the two proofs have the same associated proofnet, given in Figure 10. For the sake of clarity, the cut rules are bold lines, and subnets associated to lexical items are circled. Obviously, if we delete the two cut lines, we are left with three proofnets referring to (provable) sequents. The proofnet in Figure 10 still contains some superfluous information, namely, nodes that cannot be targeted by the only available operations in A(G)--the cut rule and the adjunction rule on a propositional variable.</Paragraph>
      <Paragraph position="4"> In fact, we only need to keep nodes (i) that refer to conclusions of the proofnet that are propositional variables or negation of propositional variables (a cut can be done on such a literal), and (ii) that belong to subgraphs of the following form (corresponding to the existence of a formula A o- A in the left part of a sequent, i.e., its negation  We can then simplify the graph and replace the internal logical machinery by black boxes (shown in the figures as solid black circles). The conclusions of each basic proofnet are labeled: outputs (i.e., conclusions that are propositional variables) are drawn as closed half circles, inputs (i.e., conclusions that are the negation of propositional variables) are drawn as open half circles. Plain lines link black boxes to black boxes or conclusions, and subgraphs corresponding to adjunction points are drawn as dashed lines. The previous proofnet is then redrawn as in Figure 11. We obviously find the derived tree (neglecting some minor differences). The logical proofnet can then be seen as an &amp;quot;explanation&amp;quot; of the structure of the tree, that is to say the operations available on the tree are the result of some focus of what can be done on the proofnet. On the one hand, the use of black boxes is necessary to clarify the structure of the analysis; on the other hang this hides proof details that can be useful for some linguistic operations (as is the case for adjunction with respect to the classical structure of a derived tree). We show in the next subsection another application of such a (logical) refinement.</Paragraph>
      <Paragraph position="5"> The last example discussed in this section is the analysis of the sentence John saw Mary today. The sequent associated to the adverb today is the following one: today VP o-- VP (r) today, VP, today F- VP  The logical analysis includes the two operations substitution and adjunction, i.e., two cut rules and an adjunction rule. In Figure 12 the adjunction rule is shown as a double-thick dashed line: this (logically) mimics the adjunction as it is shown in the derived tree given in Figure 13. Note that the adverb has to be placed after the complement (rightmost in the proofnet) in order to keep the graph planar. The proofnet in Figure 14 is the proofnet corresponding to a cut-free proof.</Paragraph>
    </Section>
    <Section position="3" start_page="223" end_page="225" type="sub_section">
      <SectionTitle>
5.3 On Some Extensions
</SectionTitle>
      <Paragraph position="0"> As usual in lexicalized formalisms, TAG states rules to generate the lexicon from a basic set of descriptions. Among these, we find rules for passivization, interrogative forms or wh-sentences. We focus here on one example (namely who) to show to what extent the previous paradigm can be used also to logically interpret these lexical rules.</Paragraph>
      <Paragraph position="1"> We expect this will help in understanding the underlying mechanisms. The formulation we propose is the simplest one. This is also closely related to the approach used in categorial grammars (the raising rule is simply the introduction of an implication; see also Joshi and Kulick \[1995\] for such a relation and the way who can be defined).</Paragraph>
      <Paragraph position="2"> Figures 15, 16, and 17 present proofnets and simplified proofnets for the two noun adjuncts who John meets and who meets John. The analysis of complete sentences including these adjuncts is then similar to the process developed in the previous section.</Paragraph>
      <Paragraph position="3"> The corresponding (provable) sequents are given below. The basic lexical descriptions are the following (we have deleted the adjunction declarations for sake of clarity; the (logical) adjunction rule has to be slightly extended in order to take care of these new structures): John NP o- John, John t- NP meets S o-- NP (r) VP, NP, VP o-- V (r) NP, V o-- meets, meets, NP k S who N o-- N (r) who @ (S o-- NP), N, who, S o-- NP t- N Let M(~) denote the set of the three previous sequents. From these basic descriptions, the following entries are computed, i.e., the part of the lexicon relevant to these words  Abrusci, Fouquer6, and Vauzeilles Tree Adjoining Grammars</Paragraph>
      <Paragraph position="5"/>
      <Paragraph position="7"> But they are not provable with the cut and adjunction rules from M(~). In other words, we should consider the construction of the language in two steps. The first step is the construction of the lexicon (a TAG grammar) from a basic set of descriptions using complex rules. The second step is the closure of the TAG grammar with the cut and adjunction rules. This point of view needs to be further developed but could be a first approach to a complete integration of lexicon and grammar.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="225" end_page="225" type="metho">
    <SectionTitle>
6. Conclusion
</SectionTitle>
    <Paragraph position="0"> The use of logic as a framework to describe natural language is not a new idea. Works on Lambek calculus and logic programming are famous examples. However, linguistic formalisms have fundamentally evolved in the past two decades. Though theoretical research has been done on unification and attribute-value structures, operations on syntactic trees have been investigated mainly by comparing different solutions (Vijay-Shanker and Weir 1994a, 1994b). In this paper, we consider another way to look at these operations. We focus on the adjunction operation available in Tree Adjoining Grammars, as it seems to be the simplest way to augment the expressive power of a formalism. We prove that noncommutative intuitionistic linear logic is a good framework and we define a fragment equivalent to TAG. We show, furthermore, to what extent geometric representations of proofs (proofnets) may be useful in understanding how black boxes (i.e., relations between nodes in a syntactic tree) help simplify a parse but also hide interesting mechanisms. There is still a lot to do in this direction. For one thing, generalized categorial grammars also have to be logically investigated, the objective being to relate GCG operations to logical operations (completed if necessary).</Paragraph>
    <Paragraph position="1"> The preceding discussions also show the relationship between our point of view and the idea of quasi trees developed by Vijay-Shanker (1992). He proposes to consider partial descriptions of trees, i.e., adjunction nodes represented by means of loose relations whose meaning is a domination relation. In this case, the adjunction operation is identified by a pair of substitution operations. The strong relation with what precedes is clean However, in order to take into account exactly this presentation, the axiom of identity A t- A, where A is a propositional variable, must be added to the calculus ,4(G) given in Section 4. In this way, adjunction nodes can be deleted from sequents.</Paragraph>
    <Paragraph position="2"> In this new calculus, the following rule is satisfied:</Paragraph>
  </Section>
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