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<Paper uid="P01-1047">
  <Title>Alain.Lecomte@upmf-grenoble.fr</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 Logico-grammatical rules for merge
</SectionTitle>
    <Paragraph position="0"> and phrasal movement Because of the sub-formula property we need not present all the rules of the system, but only the ones that can be used according to the types that appear in the lexicon. Further more, up to now there is no need to use introduction rules (called hypothetical reasoning in the Lambek calculus): so our system looks more like Combinatory Categorial Grammars or classical ABgrammars. Nevertheless some hypotheses can be cancelled during the derivation by the productelimination rule. This is essential since this rule is the one representing chains or movements.</Paragraph>
    <Paragraph position="1"> We also have to specify how the labels are carried out by the rules. At this point some non logical properties can be taken into account, for instance the strength of the features, if we wish to take them into account. They are denoted by lower-case variables. The rules of this system in a Natural Deduction format are:</Paragraph>
    <Paragraph position="3"> This later rule encodes movement and deserves special attention. The label a107 a85a103 a30a57a3 a78 a7a10a58a11a21a20a87 means the substitution of a103 to the unordered set a3 a78 , a58a11a21 that is the simultaneous substitution of a103 for both</Paragraph>
    <Paragraph position="5"> is. Here some non logical but linguistically motivated distinction can be made. For instance according to the strength of a feature (e.g. weak case a23 versus strong case a24 ), it is possible to decide that only the semantic part that is a41 a103 a43 is substituted with a78 .</Paragraph>
    <Paragraph position="6"> In the figure 1, the reader is provided with an example of a lexicon and of a derivation. The resulting label is a41a63a56a1a0 a35a59a35a3a2a60a43a18a37 a48 a56a40a39 a65 a56a4a0 a35a38a35a3a2 phonological form is a30a34a37 a48 a56a27a39 a65 a30a70a30a38a56a5a0 a35a38a35a3a2a60a30 while the resulting logical form is a41a63a56a6a0 a35a59a35a3a2a60a43 a41a42a37 a48 a56a40a39 a65 a43 . Notice that language variation from SVO to SOV does not change the analysis. To obtain the SOV word order, one should simply use a24 (strong case feature) instead of a23 (weak case feature) in the lexicon, and use the same analysis. The resulting label would be</Paragraph>
    <Paragraph position="8"> ical from a30a38a56a7a0a31a35a59a35a3a2a60a30a73a30a34a37 a48 a56a40a39 a65 a30 and the logical form remains the same a41a63a56a6a0 a35a59a35a3a2a60a43 a41a42a37 a48 a56a40a39 a65 a43 .</Paragraph>
    <Paragraph position="9"> Observe that although entropy which suppresses some order has been used, the labels consist in ordered sequences of phonological and logical forms. It is so because when using [/ E] and [a29 E], we necessarily order the labels, and this order is then recorded inside the label and is never suppressed, even when using the entropy rule: at this moment, it is only the order on hypotheses which is relaxed.</Paragraph>
    <Paragraph position="10"> In order to represent the minimalist grammars of (Stabler, 1997), the above subsystem of partially commutative intuitionistic linear logic (de Groote, 1996) is enough and the types appearing in the lexicon also are a strict subset of all possible types: Definition 1 a8a10a9 -proofs contain only three kinds of steps:  Derivations in this system can be seen as T-markers in the Chomskyan sense. [/E] and [a29 E] steps are merge steps. [a32 E] gives a co-indexation of two nodes that we can see as a move step. For instance in a tree presentation of natural deduction, we shall only keep the coindexation (corresponding to the cancellation of a82 and a83 : this is harmless since the conclusion is not modified, and makes our natural deduction T-markers).</Paragraph>
    <Paragraph position="11"> Such lexical entries, when processed with a8a10a9 -rules encompass Stabler minimalist grammars; this system nevertheless overgenerates, because some minimalist principles are not yet satisfied: they correspond to constraints on derivations. null</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.1 Conditions on derivations
</SectionTitle>
      <Paragraph position="0"> The restriction which is still lacking concerns the way the proofs are built. Observe that this is an algorithmic advantage, since it reduces the search space.</Paragraph>
      <Paragraph position="1"> The simplest of these restriction is the following: the attractor F in the label L of the target a25 locates the closest F' in its domain. This simply corresponds to the following restriction.</Paragraph>
      <Paragraph position="2"> Definition 3 (Shortest Move) : A a8a10a9 -proof is said to respect the shortest move condition if it is such that: a1 the same formula never occurs twice as a hypothesis of any sequent a1 every active hypothesis during the proof process is discharged as soon as possible The consequences of this definition are the following: null</Paragraph>
      <Paragraph position="4"> is a (proper or logical) axiom, a1 then a hypothesis a3 a106 must be introduced, rather than any constant a3 a106 , in order to discharge a3 We may see an application of this condition in the fact that sentences like:  are ruled out. Let us look at the beginning of their derivation (in a tree-like presentation of natural deduction proofs): at the stage where we stop the deduction on figure 2, we cannot introduce a new hypothesis a8a38a94 a80 a23 a32 a14 because there is already an active one (a8a67a93 ), the only possible continuation is to discharge a58 a94 and a78 a94 altogether by means of a &amp;quot;constant&amp;quot;, like a9 a56a57a37a59a58 , so that, in contrast:</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.2 Extension to head-movement
</SectionTitle>
      <Paragraph position="0"> We have seen above that we are able to account for SVO and SOV orders quite easily. Nevertheless we could not handle this way VSO language. Indeed this order requires head-movement.</Paragraph>
      <Paragraph position="1"> In order to handle head-movement, we shall also use the product a32 but between functor types.</Paragraph>
      <Paragraph position="2"> As a first example, let us take the very simple example of: peter loves mary. Starting from the following lexicon in figure 3 we can build the tree given in the same figure; it represents a natural deduction in our system, hence a syntactic analysis. The resulting phonological form is</Paragraph>
      <Paragraph position="4"> a30a6a30a74a54a66a56a27a37a59a58a69a30 while the resulting logical form is a41 a47a49a48a51a50a52a48 a37a67a43a74a41a55a54a4a56a27a37a38a58a60a43a74a41a63a61a63a35a38a64 a48a72a65 a43 -- the possibility to obtain SOV word order with a a24 instead of a a23 also applies here.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 The interface between syntax and
</SectionTitle>
    <Paragraph position="0"> semantics In categorial grammar (Moortgat, 1996), the production of logical forms is essentially based on the association of pairs a10 a65a72a50 a37a5a11 a98a13a12 a7 a50 a58a34a100 a48a15a14 with lambda terms representing the logical form of the items, and on the application of the Curry-Howard homomorphism: each (a30 or a29 ) elimination rule translates into application and each introduction step into abstraction. Compositionality assumes that each step in a derivation is associated with a semantical operation.</Paragraph>
    <Paragraph position="1"> In generative grammar (Chomsky, 1995), the production of logical forms is in last part of the derivation, performed after the so-called Spell Out point, and consists in movements of the semantical features only. Once this is done, two forms can be extracted from the result of the derivation: a phonological form and a logical one.</Paragraph>
    <Paragraph position="2"> These two approaches are therefore very differ-</Paragraph>
    <Paragraph position="4"> ent, but we can try to make them closer by replacing semantic features by lambda-terms and using some canonical transformations on the derivation trees.</Paragraph>
    <Paragraph position="5"> Instead of converting directly the derivation tree obtained by composition of types, something which is not possible in our translation of minimalist grammars, we extract a logical tree from the previous, and use the operations of Curry-Howard on this extracted tree. Actually, this extracted tree is also a deduction tree: it represents the proof we could obtain in the semantic component, by combining the semantic types associated with the syntactic ones (by a homomorphism a0 to specify). Such a proof is in fact a proof in implicational intuitionistic linear logic.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
4.1 Logical form for example 3
</SectionTitle>
      <Paragraph position="0"> Coindexed nodes refer to ancient hypotheses which have been discharged simultaneously, thus resulting in phonological features and semantical ones at their right place3.</Paragraph>
      <Paragraph position="1"> By extracting the subtree the leaves of which are full of semantic content, we obtain a structure that can be easily seen as a composition: (peter)((mary)(to love)) If we replace these &amp;quot;semantic features&amp;quot; by a1 terms, we have:  This shows that necessarily raised constituants in the structure are not only &amp;quot;syntactically&amp;quot; raised but also &amp;quot;semantically&amp;quot; lifted, in the sense that a1a4a3 a19a5a3 a41a100 a48a51a50a52a48 a37a57a43 is the high order representation of the individual peter4.</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
4.2 Subject raising
</SectionTitle>
      <Paragraph position="0"> Let us look at now the example: mary seems to work From the lexicon in figure 4 we obtain the deduction tree given in the same figure.</Paragraph>
      <Paragraph position="1">  a typed lambda term, we must only assume it is of some type freely raised from a24 , something we can represent by a13a25a13a16a24a27a26a29a28a30a22a31a26a29a28a30a22 , where X is a type-variable, here X = a13a16a24a32a26a34a33a35a22 becausea8a10a36a12a11a8a10a21a37a11a38a40a39a42a41a20a43a44a13a16a21a46a45a47a36a37a22 has type a13a16a24a32a26a48a13a16a24a32a26a34a33a49a22a25a22 This time, it is not so easy to obtain the logical representation:</Paragraph>
      <Paragraph position="3"> The best way to handle this situation consists in assuming that: a1 the verbal infinitive head (here to work) applies to a variable a78 which occupies the a14 position, null a1 the semantics of the main verb (here to seem) applies to the result, in order to obtain</Paragraph>
      <Paragraph position="5"> a78 variable is abstracted in order to obtain a1 a78 a19a65a34a48a34a48 a9 a41 a50 a35 a33a36a35a38a37 a2a99a41 a78 a43a10a43 just before the semantic content of the specifier (here the nominative position, occupied by a1a4a3 a19a5a3 a41 a9 a56a27a37a38a58a60a43 ) applies.</Paragraph>
      <Paragraph position="6"> This shows that the semantic tree we want to extract from the derivation tree in types logic is not simply the subtree the leaves of which are semantically full. We need in fact some transformation which is simply the stretching of some nodes. These stretchings correspond to a50 -introduction steps in a Natural deduction tree. They are allowed each time a variable has been used before, which is not yet discharged and they necessarily occur just before a semantically full content of a specifier node (that means in fact a node labelled by a functional feature) applies.</Paragraph>
      <Paragraph position="7"> Actually, if we say that the tree so obtained represents a deduction in a natural deduction format, we have to specify which formulae it uses and what is the conclusion formula. We must therefore define a homomorphism between syntactic and semantic types.</Paragraph>
      <Paragraph position="8"> Let a0 be this homomorphism.</Paragraph>
      <Paragraph position="9"> We shall assume:  determinism but the instantiation of X is always unique. Moreover, when a68 is of type a13a16a69a70a26a29a69a71a22 , it is in fact endowed with the identity function, something which happens everytime a68 is linked by a chain to a higher node.  With this homomorphism of labels, the transformation of trees consisting in stretching &amp;quot;intermediary projection nodes&amp;quot; and erasing leaves without semantic content, we obtain from the derivation tree of the second example, the following &amp;quot;semantic&amp;quot; tree:  where coindexed nodes are linked by the discharging relation.</Paragraph>
      <Paragraph position="10"> Let us notice that the characteristic weak or strong of the features may often be encoded in the lexical entries. For instance, Head-movement from V to I is expressed by the fact that tensed verbs are such that: a1 the full phonology is associated with the inflection component, a1 the empty phonology and the semantics are associated with the second one, a1 the empty semantics occupies the first one6 Unfortunately, such rigid assignment does not always work. For instance, for phrasal movement (say of a a14 to a a23 ) that depends of course on the particular a23 -node in the tree (for instance the situation is not necessary the same for nominative and for accusative case). In such cases, we may assume that multisets are associated with lexical entries instead of vectors.</Paragraph>
    </Section>
    <Section position="3" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
4.3 Reflexives
</SectionTitle>
      <Paragraph position="0"> Let us try now to enrich this lexicon by considering other phenomena, like reflexive pronouns.</Paragraph>
      <Paragraph position="1"> The assignment for himself is given in figure 5 -- where the semantical type of himself is assumed to be a41a10a41 a48 a50 a41 a48 a50 a50 a43a10a43 a50 a41 a48 a50 a50 a43a10a43 . We obtain for paul shaves himself as the syntactical tree something similar to the tree obtained for our first little example (peter loves mary), and the semantic tree is given in figure 5.</Paragraph>
      <Paragraph position="2"> 5 Remarks on parsing and learning In our setting, parsing is reduced to proof search, it is even optimized proof-search: indeed the re6as long we don't take a semantical representation of tense and aspect in consideration.</Paragraph>
      <Paragraph position="3">  striction on types, and on the structure of proof imposed by the shortest move principle and the absence of introduction rules considerably reduce the search space, and yields a polynomial algorithm. Nevertheless this is so when traces are known: otherwise one has to explore the possible places of theses traces.</Paragraph>
      <Paragraph position="4"> Here we did focus on the interface with semantics. Another excellent property of categorial grammars is that they allow -- especially when there are no introduction rules -- for learning algorithms, which are quite efficient when applied to structured data. This kind of algorithm applies here as well when the input of the algorithm are derivations.</Paragraph>
    </Section>
  </Section>
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