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<Paper uid="A00-2010">
  <Title>Generation in the Lambek Calculus Framework: an Approach with Semantic Proof Nets</Title>
  <Section position="2" start_page="0" end_page="72" type="metho">
    <SectionTitle>
2 Multi Usage Proof Nets
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="2" type="sub_section">
      <SectionTitle>
2.1 Proof Nets
</SectionTitle>
      <Paragraph position="0"> (Girard, 1987) introduced proof nets formalism as the natural deduction syntax for linear logic, also studied in (Retor6, 1993). They represent proofs in linear logic with more accuracy than sequential proofs: on one hand they are more compact, on the other hand they identify unessentially different sequential proofs (for instance in the order of the rules introduction).</Paragraph>
      <Paragraph position="1"> From a one-sided sequent and a sequential proof of it, we obtain a proof net by unfolding every formula as a tree (whose nodes are the binary connectives and the leaves are formulas, e.g. atomic ones) and linking together the formulas occurring in the same axiom rule of tile sequent calculus.</Paragraph>
      <Paragraph position="2"> But proof nets have a more intrinsic definition that prevents us to come back every time to sequential proofs. They can be defined as graphs with a certain property (i.e. verifying a correctness criterion) such that every proof net with this property corresponds to a sequential proof and such that every proof net built from a sequential proof has this property. So that we do not present the sequent calculus but only the proof net calculus.</Paragraph>
      <Paragraph position="3"> In this paper, we do not consider all the proof nets, but a part of the multiplicative ones: those of the intuitionistic implicative linear logic. In this case, sequents are made of several antecedent \[brmulas, but only one succedent formula. To deal with tile intuitionistic notion with proof nets (since we consider one-sided sequents), we use the notion of polarities with the input (,: negative) and the output (o: positive) (Danos, 1990; Lamarche, 1995) to decorate formulas. Positive ones correspond to succedent formulas and negative ones to antecedent formulas. null Given the links of table 1, we define proof structures as graphs made of these links such that:  1. any premise of any link is connected to exactly one conclusion of some other link; 2. any conclusion of any link is connected to at most one premise of some other link; 3. input (resp. output) premises are connected to input  (resp. output) conclusions of the same type.</Paragraph>
      <Paragraph position="4"> Note that the two links for tile negative and positive implications correspond to the two connectives of the linear logic: Tensor and Par, so that we name these links after these latter connectives. But in the following, only the graphical forms of the links play a role.</Paragraph>
      <Paragraph position="5"> Proof nets are proof structures that respect the correctness criterion.</Paragraph>
      <Paragraph position="6"> We mentioned the intrinsic definition of proof nets that enables the complete representation of sequential proofs. Tile cut elimination property of sequent calculus also appears intrinsically in the proof net formalism with a sire-</Paragraph>
      <Paragraph position="8"/>
      <Paragraph position="10"> pie rewriting process described in table 2 (in case of complex formulas as in the third rewriting rule, those rules can apply again on the result and propagate until reaching atoms).</Paragraph>
    </Section>
    <Section position="2" start_page="2" end_page="2" type="sub_section">
      <SectionTitle>
2.2 Syntactic Proof Nets
</SectionTitle>
      <Paragraph position="0"> Definitions of proof nets tbr Lambek calculus first appeared in (Roorda, 1991 ). They naturally raised as Lambek calculus is an intuitionistic fragment of non commutative linar logic (with two linear implications: &amp;quot;'\&amp;quot; on the left and &amp;quot;/&amp;quot; on tile right), and the consequences on the proof net calculus we presented in section 2.1 are: . we get two tensor links: one for the formulas (B/A)- (the one in table 1) and one lbr the formula (B\A)- (just inverse the polarities of the premises). And two par links : one for the lbrmula (A\B) + and one for (A/B) + (idem); * formulas in Lambek's sequents are ordered, so that conclusions of the proof nets are cyclically ordered and axiom links may not cross.</Paragraph>
      <Paragraph position="1"> If T v is the set of basic types (e.g. S, NP... ), the set T of syntactic types ~bllows T ::= ~\[T\T\[T/T.</Paragraph>
      <Paragraph position="2"> Note that from a syntactic category, we can untbld the formula to obtain a graph which only lacks axiom links to become a proof structure. So that the parsing process in this framework is, given the syntactic categories of the items and their order, to put non crossing axiom links such that the proof structure is a proof net. It means there is a proof of .b' given types in a certain order. For technical reasons, the order of the conclusions (i.e. the types used) in the proof net to prove S is the reverse order of the words associated to these types.</Paragraph>
      <Paragraph position="3"> As an example, with the lexicon of table 3, proving that John lives in Paris is a correct sentence leads to find axiom links between the atoms in the figure l(a). Figure I(b) shows it actually happens and proves the syntactic correctness of the sentence.</Paragraph>
    </Section>
    <Section position="3" start_page="2" end_page="72" type="sub_section">
      <SectionTitle>
2.3 Semantic Proof Nets
</SectionTitle>
      <Paragraph position="0"> In this section, we present how (de Groote and RetortL 1996) propose to use proof nets as semantic recipes. As a slight difference with this work, we only deal in this paper with semantic recipes that correspond to linear A-terms in the Montague's semantics framework.</Paragraph>
      <Paragraph position="1"> The idea of expressing the semantics with proof nets refers to the fact that both the A-terms (with the Curry-Howard isomorphism) and the proof nets represent prooS; of intuitionistic implicative linear logic. And indeed, the linear A-terms may be encoded as proof nets.</Paragraph>
      <Paragraph position="2"> On the other hand, given an intuitionistic implicative proof net, a simple algorithm (given in (de Groote and  paths), wc can obtain a A-term.</Paragraph>
      <Paragraph position="3"> Then, instead of associating a A-term to a \[exical entry, wc can associate a proof net. For instance, on the semantic side, we can use the Montagovian types e and t and typed constants. Of course, we want to keep the compositionalily principle of Montague's semantics that maps any syntactic association rule with a semantic association rule. We express it in a straightforward way with the ft~llowing homomorphism (for as many basic categories as required):</Paragraph>
      <Paragraph position="5"> And for a lexical item, given its syntactic type, we as-SUlne its semantic proof net to verify: * the type of its unique output conclusion is the homomorplaic image of the syntactic type: * its input conclusions (if any) are decorated with typed constants.</Paragraph>
      <Paragraph position="6"> An example of such a lexicon is given in table 4.  Let us illustrate the process on a short example. We use the lexicon of table 4 to parse the sentence John lives in Paris. The first thing is to define with the syntactic categories of the different lexical items the syntactic proof net of figure 2. It provides the way we should compose the semantic recipes of each lexical item: we take its homomorphic image as in figure 4(a), and we substitute to every input its semantic definition with cut-links.</Paragraph>
      <Paragraph position="7"> Then the cut-elimination on the resulting proof net gives a new proof net (on figure 4(b)) we can use as the semantic analysis of Jolm lives in Paris. If necessary, we can come back to the A-term expression:(in p)(live j).</Paragraph>
      <Paragraph position="8">  i (b) Matching the dual atoms to obtain a proof net of John lives in Paris 3 Generation: Stating the Problem  Let us now consider the problem of generation. We have a given semantic proof net (like the one in figure 4(b)) and we want to gather syntactic entries with axiom links such that: I. this yields a correct (syntactic) proof net; 2. the meaning of the resulting proof net matches the given semantic expression.</Paragraph>
      <Paragraph position="9"> As we already said it, we assume that we have some lexical entries, and we try to make the generation with these entries, each one used once and only once.</Paragraph>
      <Paragraph position="10"> Thus, if we define: * I/0 the semantic proof net of the expression we want to generate; * IIi the semantic proof nets associated to the given lexical entries i we use; * Ti the unfolding in proof structure of the syntactic formula of the lexical item i; * F the forest made of the syntactic trees of all the considered lexical entries plus the output (the type we want to derive).</Paragraph>
      <Paragraph position="11"> The generation problem (see figure 5) is to find a matching M of atomic formulas of F such that:  1. F endowed with M (let us call this proof structure F') is a correct proof net; 2. when cut-linking H(F') with the Hi, and eliminat null ing these cuts, we obtain H0.</Paragraph>
      <Paragraph position="12"> This problem is not an original one: making proof search with proof nets always leads to look for matching between atomic formulas of opposite polarities. So that an answer to this problem would consist in taking f' and try every possible matching. This brute-force technique would of course appear essentially inefficient, and our purpose is to use everything we know to prune tile search domain.</Paragraph>
      <Paragraph position="13"> Nevertheless, note that even with such an algorithm, we already reach the decidability (because the finitness of the number of the matchings) without making any assumption on the form of the semantic entries (neither on the order of the associated A-terms, nor the presence of a free variable). And we want to keep these good properties in our algorithm.</Paragraph>
      <Paragraph position="14">  IIjohn (cf. figure 3(a)) \[IMary (cf. figure 3(b)) IIPari s (cf. figure 3(c)) \[IlivC/ (cf. figure 3(d)) gin (cf. figure 3(e))</Paragraph>
    </Section>
  </Section>
  <Section position="3" start_page="72" end_page="74" type="metho">
    <SectionTitle>
4 Cut-eliminationas Matrix Computation
</SectionTitle>
    <Paragraph position="0"> This section first establishes some equivalent relations between cut-elimination on proof nets and matrix equalions. We then show how to use these equations in the generation process and how we can solve them. It enables us to characterize the properties required by the semantic proof nets to have a polynomial resolution of the generation process.</Paragraph>
    <Section position="1" start_page="72" end_page="73" type="sub_section">
      <SectionTitle>
4.1 Principles
</SectionTitle>
      <Paragraph position="0"> First, as expressed in (Girard, 1989) and refornmlated in (Retord, 1990: Girard, 1993; Girard, 1995), we state the algebraic representation of cuFeliminalion on proof nets. Due to lack of space, we can not develop it, but tile principle is to express cut-elimination between axioms with incidence matrices and paths in graphs.</Paragraph>
      <Paragraph position="1"> Let us consider a proof net U. We denote by (e i ) 1 &lt; i&lt;.~ all the vertices taking place for atoms in ft. We can define U the incidence matrix of axiom links, cr the incidence matrix of cut links (we assume without loss of generality that they happen only between axiom links), and II the incidence matrix of axiom links of-ff where ~is the proof net resulting from all the cut eliminations on  Then we have (Girard, 1989):</Paragraph>
      <Paragraph position="3"> inversible, and its inverse is (1 - crU). The next section make explicit the relation ( I ) with a special choice of the base (~ i ).</Paragraph>
    </Section>
    <Section position="2" start_page="73" end_page="73" type="sub_section">
      <SectionTitle>
4.2 Matrix Relation for Cut Elimination
</SectionTitle>
      <Paragraph position="0"> In tile problem we are dealing with, we know II and some of the axiom links in/~'. Let us assume that Vi E \[1,p\], both C/ i and 13(ei ) i are not cut-linked in L; (this assumption entails no loss of generality).</Paragraph>
    </Section>
    <Section position="3" start_page="73" end_page="73" type="sub_section">
      <SectionTitle>
4.3 Expressing the Reduction of U into II
</SectionTitle>
      <Paragraph position="0"> In this section, we want to give a relation equivalent to ( I ) which focuses on some axiom links we are interested in.</Paragraph>
      <Paragraph position="1"> As mentioned in section 4.2, we can consider the (ei) such that in ~&amp;quot; : * Vi E \[1,p\], ei is not cut-linked (then, because of the hypothesis made in section 4.2, B(ei) is cut-</Paragraph>
      <Paragraph position="3"> Note: Remember we assume that there is no axiom link such that both its conclusions are not cut-linked. So p = III.</Paragraph>
      <Paragraph position="4"> I B(e ) is the atom in \[7 such that there is an axiom link between e and BIe).</Paragraph>
      <Paragraph position="6"> Then in this base, we express tile matrices (every axiom link of t--7 has at least one of its conclusion involved in a cut link):</Paragraph>
      <Paragraph position="8"> Of course, all tire terms are defined.</Paragraph>
      <Paragraph position="9"> We base the proof search algorithm corresponding to the generation process we are dealing with on this third relation, as explained in the next sections.</Paragraph>
    </Section>
    <Section position="4" start_page="73" end_page="74" type="sub_section">
      <SectionTitle>
4.4 Solving the Equations
</SectionTitle>
      <Paragraph position="0"> In this section (proof search oriented), we consider&amp;quot; that the axiom links we are looking for are those whose two conclusions are involved in cut links. That is we want to complete U3. As in the previous section we proceeded  by equivalence, solving the equation (1) correponds to solving the equation</Paragraph>
      <Paragraph position="2"> A consequence of this result is that if o4 = 0, then / = n and we determine X completely with relation (3), and then the same for Ua. This configuration correspond to the fact that in the (given) semantic proof nets, no output contains the two conclusions of a same axiom link.</Paragraph>
      <Paragraph position="3"> In this latter case, the computation is not so simple and should be mixed with word o,'der constraints.</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="74" end_page="76" type="metho">
    <SectionTitle>
5 Example
</SectionTitle>
    <Paragraph position="0"> Let us process on an example the previous results. We still use the lexicon of table 4, and we want to generate (if possible) a sentence whose meaning is given by the proof net of ligure 7: (try(find j))m.</Paragraph>
    <Paragraph position="1"> We first need to associate every atom with all index (in the figures, we only indicate a number i beside the atom to express it is el). Of course, we have to know how to recognize the ei that are the same in U (figure 6) and in 11 (figure 7). This can be done by looking at the typed constants decorating the input conclusions (for the moment. we don't have a general procedure in the complex cases).</Paragraph>
    <Paragraph position="2"> We also assume in this numbering that we know which of the atoms in H(F) are linked to t + (the unique output). In our case where 0.4 = 0, it is not a problem to</Paragraph>
    <Paragraph position="4"> make such a statement. In other cases, tile complexity would increase polynomially.</Paragraph>
    <Paragraph position="5"> Then, the given matrices are:  According to the definition of the (it) and tile (jr) families such that 0.2 = El El+j+, we have:</Paragraph>
    <Paragraph position="7"> and in this case 0&amp;quot;4 = 0, so according to tile preceeding notes .\ is completely determined and</Paragraph>
    <Paragraph position="9"> We can add this matching to the syntactic forest of figure 8(a) (do not forget that//3 represents the edges between ei with i E \[17,22\])and obtain on F the matching of figure 8(b).</Paragraph>
    <Paragraph position="10">  We still have to ensure the correctness of this proof net (because we add all the tensor and par links), but it has a quadratic complexity (less than the matrix computation). In this case, it is correct.</Paragraph>
    <Paragraph position="11"> Note: * Actually, this only gives us the axiom links. It still necessitates to compute the word order to have no crossing axiom link. This can be done from the axiom links easier than quadratic time: it is a wellbracketing check. Here, it is easy to see that putting  the John item on the left would achieve the result of Mary seeks John, * The choice of seeks and its high order type (for intcnsionnality) shows there is no limitation on the order of the A-term.</Paragraph>
  </Section>
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