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<Paper uid="C92-1061">
  <Title>PROOF-NETS AND DEPENDENCIES</Title>
  <Section position="2" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2. The Method of Proof-Nets in the Lambek
Calculus
</SectionTitle>
    <Paragraph position="0"> The problem of spurious ambiguities in Categorial Grammar is very often discussed (see for instance Hendriks and Roorda (1991)). A proof-net is a device which contains all the equivalent proofs of the same result. As Roorda (1990) says: &amp;quot;A proof-net can be viewed as a parallellized sequent proof \[...\] It is a concrete structure, not merely an abstract equivalence class of derivations, and surely not a special derivation with certain constraints on the order in which the rules must be applied.&amp;quot; The principles of construction of proof-nets are related to the inference rules of the Lambek Calculus, when it is viewed as a sequent calculus. If we here omit the product, we have the following rules, which belong to two different types: 1 1 am indebted to Dirk Roorda for fruitful discussions during a brief visit I made in Amsterdam in Spring 1991 Ac'rv_s DE COL1NG-92, NANTES, 2.3-28 hofrr 1992 3 9 4 Pgoc. OF COLING.92, NANTES, AUG. 23-28, 1992 Binary rules (or type-2 rules): (where O is a non-empty sequence of categories, and F and A are arbitrary sequences of categories) 2  \[L/I: O--4B F, A, A -4 C .....................</Paragraph>
    <Paragraph position="1"> F, A/B, O, A --~ C \[L\\]: O -~ B F, A, A --7 C F, O, B\A, A --~ C Unary rules (or type-I rules): (F is non empty) \[R/l: 0, B --~ A \[R\\]: B, @ ~ A</Paragraph>
  </Section>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
@~A/B O~BXA
</SectionTitle>
    <Paragraph position="0"> In these rules, contexts (O, F, A) are &amp;quot;static&amp;quot;. That means that they can neither be contracted, nor expanded, nor permuted. They play no role in the application of rules. So, it is convenient to &amp;quot;forget&amp;quot; them and to represent the rules according to schemes similar to: A*B Such a scheme is called a link. Other types of links are provided by the identity axiom: \[axl A---~A, which  according to which the argument category is always under the slash, in such a way that Aft\] means a category which becomes an A if a B is met on file right, and BXA a category which becomes an A if a B is met on tlie left.</Paragraph>
    <Paragraph position="1"> If the rules \[L\\], \[L/J, JR\\], \[R/I are represented by the following links:</Paragraph>
    <Paragraph position="3"> then a sequent F --~ A is a theorem of the (Product free) Lambek Calculus if and only if we can build, starting with this sequent and applying links re, cursively, a connected planar graph having the following property: for each application of a typed link, every suppression of one of the two dashed lines leaves the graph connected.</Paragraph>
    <Paragraph position="4">  A dependency structure associated to a sentence is a tree on the words of this sentence. Edges represent dependency links such that the source of an edge is considered as the head and the target as a dependant. Hudson (1984) givcs criteria to distinguish heads and dependants. It is an open question whether a head can be vicwed as a functor, the dependant being viewed as an argumenL The facts that criteria involve agreement and ACTES DE COLING-92, NANTES, 23-28 AO'tYr I992 3 9 5 Paoc. OF COL1NG-92, NANTES, AUG. 23-28, 1992 that according to the Keenan's thesis: &amp;quot;functors agree with their argument&amp;quot; seem in favour of identification. But other scholars disagree, like Moortgat and Morrill, who introduce in their recent works, four notions: head, dependant,functor and argument. Nevertheless, we will accept the first thesis in the following, adopting the conception of Barry and Pickering (1990) on this subjecL Another problem appears in the necessity of accounting for slructums with multiplicity of heads (in Ihe case of the control of infinitives for instance) because this necessity leads to graphs which are no longer trees, but dags.</Paragraph>
    <Paragraph position="5"> We assume that a dependency structure is a graph on words, In a first step, we will consider only trees. The approach will be that of a semantic interpretation in terms of wee.s, similar to what we do when we give a semantic interpretation of logic formulae in terms of sets. The usual operators like / and \ will be interpreted as connection operations in an algebra of trees. In a second step, we will have to modify this interpretation in order to obtain not only application and composition but division too.</Paragraph>
    <Paragraph position="6"> 4.Operations on Trees We start with a set of directed trees associated to lexical entries. (see figure (3) below).</Paragraph>
    <Paragraph position="8"> trees are called initial trees. The initial state of a representation of the structure of a sentence consists in an ordered sequence of these initial trees, Then, at each ~aep. we build a new tree obtained by connection of previous trees, These operations are: (cf Lecomte 1990) - left-linkage - right-linkage A tree GI is right (resp. left) linkable to a tree G2 iff: I) G 1 and G2 are adjacent, G2 being adjacent to the right (w.ap, left) ofG 1 2) GI has a rightmost (resp leftmost) branch the first edge of which is right (resp left) directed and the maximal sub-tree attached to this first edge entirely covers a continuous subtree of G2.</Paragraph>
    <Paragraph position="9"> The by-product of the right-linkage (resp left-linkage) of GI with G2, when GI is right (resp left) linkable to G2 is the tree G3 obtained as the union of G 1 and G2, modified in the following way: The rightmost dght-direeted (resp. leftmost left-directed) first-levd edge of G 1 is connected to the root of G2, and the subtree of G2 covered by the maximal subtree attached to this edge is said to be marked in G2. Left (res'p. right)-directed edges of G 2 which are not marked romainfree and take precedence in the left-to-right (resp right-to-left) order of first-level edges over those remaining free in G 1.</Paragraph>
    <Paragraph position="10"> We can introduce restrictions on these operations: we will call restriction-AB the following constraint: the subtree of G 2 covered by the subtre, e of G1 must be identical to the whole tree G2, restriction-C: at most the rightmost (rcsp leftmost) branch of G2 may be uncovered.</Paragraph>
    <Paragraph position="11"> restriction-Crec: a right (resp left) subtree of G2 may be uncove~d restriction-Cmix: at most the rightmost (resp leftmos0 or the leftmost (resp rightmost) branch of G2 may be uncovered Definition: we call connection tree every initial tree and every tree obtained by the application of linkage operations on earlier connection trees (according to eventual restrictions).</Paragraph>
    <Paragraph position="12"> We claim that such a system gives an interpretation of very simple categorial grammars, depending on the restrictions we select. Like similar constructions (Stoedman 1991) where general principles such as Adjacency, Directional Consistency and Directional Inheritance arc explained in terms of a more detailed analysis of categories, this system is suited to express such generalities. Because of the structure of linkage operations, these principles are obvious. Adjacency and Directional Consistency are contained in the definition. Directional Inheritance comes from the fact that we never allow to change anything in the labels of edges (the fact that they are left or right directed). We only allow m change tile status of an edge (free to bound or marked). In so doing, we reach, like Steedman does. the conclusion that so-called Dysharmonic Composition Rules are consistent with these principles (even if they are not with the Lambek Calculus\[).</Paragraph>
    <Paragraph position="13"> A connection system eliminates spurious ambiguities because when they are bound, links are undefeasable : there is no way of re-doing something that was primilarly done with success. In this respect, the calculus on trees concurs with the well known method of chart-parsing. (see figure (4): there is only one tree for two reductions by means of Cancellation Schemes).</Paragraph>
    <Paragraph position="14">  It is obvious that the previous system does not include any kind of Division Rules or any kind of Type-Raising Rule. So, it cannot provide any analysis for sentences with extraction, as for instance: le livre dont je connais le titre est sur la table (the book the title of which I know is on the table) because in such an analysis, we have to transform a regular n (titre) into a functorial category which requires a nonn-tandifier on its right (n/(nkn)).</Paragraph>
    <Paragraph position="15"> We shall define a new connection system which is a conservative extension of the previous one (except for the admissibility of Dysharmonic Rules). We will call it: the Connection Net System.</Paragraph>
    <Paragraph position="16"> As for the proof-nets, we want to demonstrate theorems that have a sequent form like: F---~ X, where F is a non empty sequence of categories and X is a category. We distinguish two kinds of connection Irees: those which are on the right-hand side of the sequent we want to demonstrate, and those which are on the left-hand side. When we are viewing the problems in a naturaldeduction way, we can say that the first are the trees to build and the second are those which are used in this task. We will call the firstright-trees and the second lefttrees. The set of left-trees and right-trees at any stage will be called a Construction Net.</Paragraph>
    <Paragraph position="17"> Schematically, operations are not merely connections because connections can only expand elementary trees towards more complex ones. And we need operations to reduce the complexity of a tree. For instance, to show the usual rule of Type-Raising: a ~ b/(a~b) we have to show that the fight-tree associated to b/(a~b) reduces to something isomorphic to a. The fact that, generally, the converse (b/(a~)-oa) is not true results from the fact that the same reduction is not possible when the same tree is put on the left-hand side. This exemplifies the fundamental dissymetry of the calculus.</Paragraph>
    <Paragraph position="18"> 5.2.Type-I Edges and Type-2 Edges We will then distinguish two sorts of edges and two sorts of nodes in a connection tree: typed edges and nodes and type-2 edges and nodes.</Paragraph>
    <Paragraph position="19"> Definition: A type-2 edge in a connection tree is: - an odd level edge in a left-tree, or - an even level edge in a right-tree A type-1 edge in a connection tree is: - an even level edge in a left-tree, or - an odd level edge in a right-tree A type-i (i =1.2) node is the target of a type-i edge.</Paragraph>
    <Paragraph position="20"> Roots are type-1 nodes if in a left-tree, and tyl~2 nodes if in a right-tree.</Paragraph>
    <Paragraph position="21"> Two nodes are mid to be complementary if they have not the same type.</Paragraph>
    <Paragraph position="22">  Definition: we call identification link either a nondirected edge which links two identical nodes which are complementary, one in a left-tree, the other in a righttree, or a type-I directed edge linking two complementary nodes having same label.</Paragraph>
    <Paragraph position="23"> We call connection link every link we shall be able to establish, according to the following conventions, between a typo-I node, which is the ending point of a ACRES DE COLING-92, NANTFm, 23-28 Aotrr 1992 3 9 7 PRec. OF COLING-92, NANTES, AUG. 23-28, 1992 type-2 edge, and a type-2 node which does not belong to the same tree.</Paragraph>
    <Paragraph position="24"> 5.3.Nodes-numbering Rule: each node of the initial construction net receives a number, called its degree, according to the following roles: -for a type-2 edge: if it is right directed, the degree of its source is less than the degree of all the nodes below it, if it is left directed, the degree of its source is greater than the degree of all the nodes below it, for two type-2 edges, children's degrees of the leftmost branch are less than those of the rightmost branch.</Paragraph>
    <Paragraph position="25"> -for a type-1 edge: if it is right directed, the degree of its source is greater than the degree of all the nodes below it, if it is left directed, the degree of its source is less than the degree of all the nodes below it, for two type-I edges, children's degrees of the rightmost branch are less than those of the leftmost branch.</Paragraph>
    <Paragraph position="26"> The lowest degree of the right successor of an initial tree is the successor of the greatest degree of this latter tree.</Paragraph>
    <Paragraph position="27"> Example of such a numbering: figure (6)</Paragraph>
    <Paragraph position="29"> s-s: \[5_61 Each link is now associated to a pair of degrees, called its interval.</Paragraph>
    <Paragraph position="30"> From now on, L and R will denote respectively: the left hand side and the right hand side of a Construction Net. The Construction Net will be denoted by: &lt;L I R&gt;. 5.4.Linking the Nodes Nodes will be linked according to the following principles: COMPLEMENTARITY: two nodes are linked only if they have the same label and they are of complementary types.</Paragraph>
    <Paragraph position="31"> NON-OVERLAP: the linking of all the nodes in the Construction Net must meet the non-overlap convention, which stipulates that given two arbitrary intervals, either one contains the other or they are disjoint.</Paragraph>
    <Paragraph position="32"> Theorem: (Conservativity of Connection Operations) The Non-Overlap condition is a conservative extension of the conditions on connection (restriction C rec) stipulated in ~4. That means: every connection system based on C rec, when translated in the Connection Net System, follows this convention.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
5.5.Building a Correct Net
</SectionTitle>
      <Paragraph position="0"> Definition: Given an ordered sequence of left-trees L and a right-tree R, we will say that L and R yield a correct net iff there is a linkage of all the nodes in the Construction Net &lt;L I R&gt;, which gives a connected graph, respects the complementarity principle and the non-overlap principle, and is such that: when all the rype-I edges are removed, the graph remains connected.</Paragraph>
      <Paragraph position="1"> The fundamental result is the following: Theorem: (Soundness and Completeness w.r.t. A 3) Let F --~ A be a sequent expressed in the Product-Free Lambek Calculus, where F is a non empty sequence of categories and A is a category, let L be the sequence of left connection trees associated to the elements of F and R be the right tree associated to A, the sequent is a theorem if and only if L and R yield a correct neL In other terms: the Connection Net System is sound and complete w.r.t, the Product-Free Lambek Calculus, Examples: figure (6) shows that: s/(s/np) s/((s/np)ks) I-- s is a theorem of the Lambek Calculus.</Paragraph>
      <Paragraph position="2"> Figure (7) below gives a correct net for the analysis of the sentence: le livre dont je connais le litre est sur la table  Theorem: (Categorization of Links) In a correct net, links are either identification links or connection links. Corollary: A net is correct if and only if all nodes are either identified or connected.</Paragraph>
      <Paragraph position="3"> Definition: we call Tree on words the tree obtained from a correct net by merging connected nodes and removing identification links. In the case of an identification link consisting in a type-1 edge, the link and the nodes linked by it are removed, and the adjacent type-2 edges are merged.</Paragraph>
      <Paragraph position="4"> Example: figure (g): Pierre promet d Marie de venir  amounts to doing little transformations on the correct net obtained by &amp;quot;equating&amp;quot; an ordered sequence of initial trees to a node representing a primitive category. These transformations involve indexing nodes in such a way that: - indices of two different initial trees constitute two disjoint sets.</Paragraph>
      <Paragraph position="5"> - indices inside an initial tree may be identical (if we want to express a coreference) - linking two nodes results in identifying their indices.</Paragraph>
      <Paragraph position="6">  After getting a tree on words, we identify two distant nodes having the same index: we call the new node obtained: a shared node.</Paragraph>
      <Paragraph position="7"> Finally, we can say that dependencies are obtained in the following stages: 1 ~ indexing the nodes having the same label and belonging to different initial trees by different variables, taken in a set {i, j, ...} (the distribution of indices inside an initial tree being set by the lexicon) \[INDEXINGstep\]. null 2-building the net corresponding to the assertion that the sequence is of type s \[NET-step\]. 3- suppressing the nodes identified by type-1 edges of the left-hand side and all the identification links \[COLLAPSING-step\].</Paragraph>
      <Paragraph position="8"> 4- if the same index appears on distinct nodes having the same label: merging them \[MERGING-step\]  This method of Connection Nets has many advantages over other methods.</Paragraph>
      <Paragraph position="9"> Firstly, compared to classical strategies in the sequent calculus, it avoids spurious ambiguities and in so doing, it improves efficiency of searching the solution. Secondly, compared to the method of Proof-Nets, it gives more clarity to the resulting structures. It is more efficient too, because the stage of checking the coanexity when suppressing a branch of a type-1 link is replaced by a stage where the connexity is checked only once: when we have removed all the type-1 edges. The corresponding stage in Proof-Nets is usually named switching. In the early method by Girard which used the &amp;quot;long lrip condition&amp;quot;, there was a switch for each link and that gave an exponential-time algorithm (in the number of links). In the method defined by Roorda, only type-1 links lead to switches. The reason lies in the necessity of checking that a type-1 link is not used to connect two subsets of the net, which would not be connected without it. (Let us recall that a type-I link refers to a unary rule). In our method, switches are completely avoided.</Paragraph>
      <Paragraph position="10"> Thirdly. it can be done incrementally. The reason is that the numbering of nodes is consistent with the order of initial trees. Thus, at each stage of the processing from left to right, we may have a beginning net which represents the present state of the processing. Here, the properties of left-associative grammars (Hausser 1990) are reeL Finally, a very few transformations are needed in order to obtain graphs on words which can be really interpreted as Dependency Structures.</Paragraph>
    </Section>
  </Section>
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