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<Paper uid="C00-2091">
  <Title>Generation, Lambek Calculus, Montague's Semantics and Semantic Proof Nets</Title>
  <Section position="3" start_page="0" end_page="629" type="metho">
    <SectionTitle>
2 Proof Nets for Linear Logic
</SectionTitle>
    <Paragraph position="0"> Linear logic (Girard, 1987) proposes for proofs a morn compact and accurate syntax than sequent calculus: proof nets (they group distinct sequential proofs that only have inessential differences). They have both a related to sequential proof definition and a geometrical definition: they can be defined as a class of graphs (proof structures) satisfying a geometrical property so that every proof net corresponds to a sequential proof and every proof structure built from a sequential proof has this prop-erty (Retor6, 1998).</Paragraph>
    <Paragraph position="1"> In this paper, we only consider proof nets of the intuitionistic implicative linear logic: sequents are made of several antecedent formulas, but only one succedent formula. To deal with the intuitionistic notion for proof nets (since we consider one-sided sequents), we use the notion of polarities with the input (o: negative) and the output (o: positive) (Danes, 1990; Lamarche, 1995) to decorate formulas. Positive ones correspond to succedent formulas and negative ones to antecedent formulas.</Paragraph>
    <Paragraph position="2"> Given the links of table 1, we define proof structures (we consider implicative fragment) 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="3"> Proof nets are proof structures that respect the correctness criterion.</Paragraph>
    <Paragraph position="5"> The last link of table 1, the Cut link, allows the combination of proofs of 17 I- A and of A, A t- /3 into a single proof of I', A I- /3. In sequential calculs, tile cut-elimination property states that there exists a normal (not using the Cut rule) proof for the same sequent only IYom premises of 17 and A (and builds it).</Paragraph>
    <Paragraph position="6"> Of course, this property hokls for proof nets too.</Paragraph>
    <Paragraph position="7"> And to enforce the intrinsic definition of these latter, a simple rewriting process (described in table 2) actually performs the cut-elimination (in case of complex fornmlas as in the third rewriting rule, those rules can apply again on the result and propagate until reaching atoms).</Paragraph>
    <Section position="1" start_page="628" end_page="628" type="sub_section">
      <SectionTitle>
2.1 Proof Nets for Lambek Calculus
</SectionTitle>
      <Paragraph position="0"> As Lambek calculus is an intuitionistic l'ragment of non commutative linar logic (with two linear implications: &amp;quot;\&amp;quot; on the left and &amp;quot;/&amp;quot; on the right), proof nets for it naturally appeared in (Roorda, 1991).</Paragraph>
      <Paragraph position="1"> They slightly differ from those of table 1 : * we get two tensor links: one for the fornmla (/3/A)- (the one in table 1) and one for the formula (/3\A)- (just inverse the polarities of the premises). And two par links: one for the fommla (A\B) + and one for (A/B) + (idem); * formulas in Lambek's sequents arc ordered, so that conclusions of the proof nets are cyclically ordered and axiom links may not cross.</Paragraph>
      <Paragraph position="2"> From a syntactic category, we can unfold 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 ,5' given types in a certain order. Proving that John lives in Palls is a correct sentence w.r.t, the lexicon of table 3 (the two first columns) is finding axiom links between the atoms in the figme 1 (a) so that the proof structure is correct. Figure l(b) shows it actually happens (for technical reasons, ill the proof net, the order ot' the syntactic categories is the inverse of the order of the words in the sentence to be analysed.</Paragraph>
      <Paragraph position="3"> Figure 1 (c) shows John lives Palls in cannot be successfully parsed).</Paragraph>
    </Section>
    <Section position="2" start_page="628" end_page="629" type="sub_section">
      <SectionTitle>
2.2 Proof Nets for Montague's Semantics
</SectionTitle>
      <Paragraph position="0"> Capitalizing on tile fact that both A-torms (with the Curry-Howmzl isomorphism) and proof nets represent proofs of intuitionistic implicative linear logic, (de Groote and Retor6, 1996) propose to use proof nets as semantic recipes: since proof nets encode linear A-terms, instead of associating a )~-term in tile Montagovian style to a lexicai entry, they associate a proof net (decorated with typed constants). An example of such a lexicon is given in table 31 (par links encode abstraction, and tensor links encode application). null Of course, to respect semantic types based on Montagovian basic types e and t, they use the following homomorphism:</Paragraph>
      <Paragraph position="2"> Let us illustrate the process in parsing tile sentence John lives in Paris. First we have to find the syntactic proof net of figure l(b) as explained in 2.1. It provides the way syntactic componants combine, hence how semantic recipes of each lexical item combine: we take its homomorphic image I Unlike in (de Groote and Retor6, 1996), we restrict ourselves for llle moment to linear ~-terms.</Paragraph>
      <Paragraph position="4"> (a) Unfolding of the syntactic typos ,d :itrl ........ j l 'S'+ (b) Matching the dual atoms to obtain a prool' net i! N/')ri! ;~';i'\r~ht S+ (c) Incorrect prool' structure  for parsing Jolm lives I'aris in Figure 1 : Parsing of John lives in Paris as in figure 3(a). The substitution of every input with its semantic definition we would like to perform on the ~-calculus side appears on the logical side as plugging selnantic proof nets with cut-links. Then, the fl-reduction we would like to perforln has its logical counterpart in the cut-elimination on the resulting proof net. it gives a new proof net (on figure 3(b)) we can use as the semantic analysis of John lives in Paris. If necessary, we can come back to the k-term expression: (in p)(live j). In other words, the syntactic proof net yields a term t expressing how the elementary parts combine (in this case t = (ab)(cd)). Then the resulting proof net of figure 3(b) corresponds to the/3-normal form of t\[)~x.*y.(in x )y/a, p/b, Az.live z / c, j/d\].</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="629" end_page="630" type="metho">
    <SectionTitle>
3 What is Generation?
</SectionTitle>
    <Paragraph position="0"> We can now state the problem we arc dealing with: given a semantic proof net (like the one in figure 3(b)), we want to put together syntactic entries with axiom links such that:  1. this yields a correct (syntactic) proof net; 2. the meaning of the resulting proof net matches  the given semantic expression.</Paragraph>
    <Paragraph position="1"> Thus, if we define: * l\]o the semantic proof net of the expression we want to generate; * Hi the semantic proof nets associated to the given lexical entries i we use; . Ti the nnfolding in proof slructure of the syntactic formula of the lexical item i (as in figure 1 (a)); * F the forest made of the syntactic trees (7~) of all the considered lexical entries plus the output (the type we want to derive), the generation problem (see figure 4) 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 7\[(1 v/) with tile lIi, and eliminating these cuts, we obtain 110.</Paragraph>
    <Paragraph position="2"> We note that the problem is intrinsically decidable (because the finitness of the number of the matchings) without making any assumption on the form of tile semantic entries. Of course, we want to keep these good properties in our algorithm.</Paragraph>
  </Section>
  <Section position="5" start_page="630" end_page="631" type="metho">
    <SectionTitle>
4 Cut-eliminationas Matrix Computation
</SectionTitle>
    <Paragraph position="0"> U:dng proof nets resulling from a cut-elimination to guide a proof search on proof nets b@)re cut-elimination relies on the algebraic representation of cut-elimination on proof nets expressed in (Girard, 1989) and rel'ormulated in (Retor6, 1990; Girard, 1995). Due Io lack of space, we can not developp il, but tile principle is to express cut-elimination between axioms with incidence matrices and paths in graphs.</Paragraph>
    <Paragraph position="1"> Let us consider a proof net U. We can define U the incidence matrix of axiom links', c, tile incidence matrix of cut links (we assume wilhout loss of generality that llley happen only between axiom links), and \]\[ the incidence matrix of axiom links of 1I where 111 is lhe proof net resulting from all lhe cuteliminations on U. Then we have (Girard, 1989): \]l-- (:l - ~2)U(:l - ~l:;)-'(:l - ~) (l) We want to give an equiwflent relation to (1) focusing on some axiom links we are interested in.</Paragraph>
    <Paragraph position="2"> Without loss of generality, we assume tile lack of any axiom link in U such that none of its conclusions are involved in cut links.</Paragraph>
    <Paragraph position="3"> Then we can choose an el'tier for tile atoms (from lhe proof net before the cut-elimination, there is three subsets of atoms: those not involved in a cut link, those involved in a cut link and whose dual is not involved in a cut link, and those involved in a cut link and their dual as well) such that:</Paragraph>
    <Paragraph position="5"> Note that all the atoms belonging to the matching we are looking for in the generation process (see ligure 4) are in U:I.</Paragraph>
    <Paragraph position="6"> If we detine A = ( llJ 1111 - o~ eU 1 l/J1 and X = U:~(1 - ~r.l U3)- J, we can state tilt theorem: Theorem 1 Lel g/ be a correcl proof net reducing in Res(o, U) after cul-eliminalion. These relations are equivalenl:</Paragraph>
    <Paragraph position="8"> Of course, all the terms are defined.</Paragraph>
    <Paragraph position="9"> We bast the proof search algorithm corresponding to the generation process we are dealing with on this third relation.</Paragraph>
    <Paragraph position="10"> Indeed, the axiom links we are looking for are those whose two conclusions are involved in cut links. That is we want Io complete U3 (knowing all the other matrices). The previous theorem states that solving tile equation (1) correponds to solving  the equation A = cr2X tcr2 in X with X inversible. Then, we have to solve U3 = X -1 + or4 such that</Paragraph>
    <Paragraph position="12"> then the latter is unique and completely defined (as matrices product)from A and ~2.</Paragraph>
    <Paragraph position="13"> If cq 7~ 0 we generally have many solutions, and we have to investigate this case to obtain good computational properties for example in adding word order constraints.</Paragraph>
    <Paragraph position="14"> Nevertheless, we can decide the case we are handling as soon as we are given the lexical entries.</Paragraph>
  </Section>
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