<?xml version="1.0" standalone="yes"?>
<Paper uid="E99-1009">
  <Title>Geometry of Lexico-Syntactic Interaction</Title>
  <Section position="4" start_page="61" end_page="64" type="metho">
    <SectionTitle>
2 Lambek sequent calculus
</SectionTitle>
    <Paragraph position="0"> In the sequent calculus of Lambek (1958) a sequent F ~ A consists of a sequence F of 'input' category formulas (the antecedent) and an 'output' category formula A (the succedent). A sequent states that the ordered concatenation of expressions in the categories F yields an expression of the category A. The valid sequents are the theorems derivable from the following axiom and rule schemata)  (5) a.</Paragraph>
    <Paragraph position="2"> ZThe completeness of the calculus with respect to the intended interpretation was proved in Pentus (1994).</Paragraph>
    <Paragraph position="3">  F(n) and A(n) range over context sequences of category formulas; A, B, and A*B are referred to as the active formulas. The calculus L lacks the usual structural rules of permutation, contraction and weakening. Adding permutation collapses the two divisions into a single non-directional implication and yields the multiplicative fragment of intuitionistic linear logic, known as the Lambek-van Benthem calculus LP. 3 The validity of the id axiom and the Cut rule follows from the reflexivity and the transitivity respectively of set containment. The calculus enjoys the property of Cut elimination whereby every proof has a Cut-free equivalent (indeed, one in which only atomic id axioms are used: what we shall call \[3rl-long sequent proofs). 4 Thus, processing can be performed using just the left (L) and right (R) rules. These rules all decompose active formulas A*B in the left or the right of the conclusions into subformulas A and B in the premises, and have exactly one connective occurrence less in the premises than in the conclusion; therefore one can compute all the (Cut-free) proofs of any sequent by traversing the finite space of proof search without Cut.</Paragraph>
    <Paragraph position="4"> By way of illustration of the sequent calculus, the following is a proof of a theorem of lifting, or (subject) type raising:</Paragraph>
    <Paragraph position="6"> Where a labels the antecedent, the coding of this proof as a lambda term ---what we 3Adding also contraction and weakening we obtain the implicational and conjunctive fragment of intuitionistic logic. Thus every Lambek proof can be read as an intuitionistic proof and has a constructive content which can be identified with its intuitionistic normal form natural deduction proof (Prawitz 1965) or, what is the same thing under the Curry-Howard correspondence, its normal form as a typed lambda term.</Paragraph>
    <Paragraph position="7">  shall call the derivational semantics--- is Xx(x a). The converse of lifting, lowering, in (7) is not derivable. A proof of a theorem of composition (it has as its semantics functional composition) is given in (8).</Paragraph>
    <Paragraph position="9"> A grammar contains a set of lexical assignments C/x: A. An expression wl+...+Wm is of category A just in case wl +...+win is the concatenation oq+...+CCn of lexical expressions such that ai: Ai, l&lt;i&lt;n, and A1 ..... An ~ A is valid. For instance, assuming the expected lexical type assignments to proper names and intransitive and transitive verbs, there are the following derivations:</Paragraph>
    <Paragraph position="11"> Ungrammaticality occurs when there is no validity of the sequents arising by lexical insertion, as in the following:  There is a reading &amp;quot;it is surprising that sometimes it rains&amp;quot; and another &amp;quot;sometimes the manner in which it rains is surprising&amp;quot;. As would be expected there are in such a case distinct derivations corresponding to alternative scopings of the adverbials:  (13) a.</Paragraph>
    <Paragraph position="13"> However, sometimes a non-ambiguous expression also has more than one sequent proof (even excluding Cut); thus the sequent in (14a) has the proofs (14b)  and (14c).</Paragraph>
    <Paragraph position="14"> (14) a.</Paragraph>
    <Paragraph position="15"> N/CN, CN, NiS ~ S the+man+runs: S b.</Paragraph>
    <Paragraph position="17"> As the reader may check, N/CN, cN S/(N~S) has three Cut-free proofs; in general the combinatorial possibilities multiply exponentially. This feature is sometimes referred to as the problem of spurious ambiguity or derivational equivalence. It is regarded as problematic computationally because itmeans that in an exhaustive traversal of the proof search space one must either repeat  subcomputations, or else perform book-keeping to avoid so doing.</Paragraph>
    <Paragraph position="18"> The problem is that different \[3rl-long sequent derivations do not necessarily represent different readings, and this is the case because the sequent calculus forces us to choose between a sequentialisation of inferences ---in the case of (14)/L and kL--- when in fact they are not ordered by dependency and can be performed in parallel.</Paragraph>
    <Paragraph position="19"> The problem can be resolved by defining stricter normalised proofs which impose a unique ordering when alternatives would otherwise be available (K6nig 1990, Hepple 1990, Hendriks 1993). However, while this removes spurious ambiguity as a problem arising from independence of inferences, it signally fails to exploit the fact that such inferences can be parallelised. Thus we prefer the term 'derivational equivalence' to 'spurious ambiguity' and interpret the phenomenon not as a problem for sequentialisation, but as an opportunity for parallelism. This opportumty is grasped in pro@nets.</Paragraph>
    <Paragraph position="20">  Roorda (1991), adapting their original introduction for linear logic in Girard (1987). In proof-nets, the opposition of formulas arising from their location in either the antecedent or the succedent of sequents is replaced by assignment of polarity: input (negative) for antecedent and output (positive) for succedent. A In the id and Cut links X and -X proof-net is a kind of graph of polar schematise over occurrences of the same formulas, category with opposite polarity. Note that the nodes of links are also marked First we define a more general concept (implicitly) as being either conclusions of proof structure. These are graphs (looking down) or premises (looking up). assembled out of the following links: In the i- and ii-links the middle nodes are the conclusions and the outer nodes the (15) a. premises. The i-links correspond to unary I I sequent rules and the ii-links to binary I I sequent rules. Observe that in the output, but not in the input, unfoldings the order X -X of subformulas is switched between id link: premises and conclusion; this is essential zero premises, to the characterization of ordering by two conclusions graph planarity.</Paragraph>
    <Paragraph position="22"> Proof structures are assembled by identifying nodes of the same polar category which are the premises and conclusions of differentcomponents; premises and conclusions not fused in this way are the premises and conclusions of  the proof structure as a whole. For example, in (16a) four links are assembled into a proof structure (16b) with no premises and two conclusions, N- null and S/(N~S)+: (16) a.</Paragraph>
    <Paragraph position="24"> Proof-nets are proof structures which arise, essentially, by forgetting the contexts of the sequent rules and keeping only the active formulas, but not all proof structures are well-formed as proofs.</Paragraph>
    <Paragraph position="25"> There must exist a global synchronization of the partitioning of contexts by rules (the long trip condition of Girard 1987).</Paragraph>
    <Paragraph position="26"> Eschewing the (somewhat involved) details (Danos and Regnier 1990; Bellin and Scott 1994) it suffices here to state that a proof structure is well-formed, a module (partial proof-net), iff every cycle crosses both edges of some i-link. A module is a proof-net iff it contains no premises. The structure (16b) is a proofnet, in fact it is the proof-net for our instance (6) of lifting since its conclusions are the polar categories for this sequent:</Paragraph>
    <Paragraph position="28"> The structure in (18) is not a module because it contains the circularity indicated: it corresponds to the lowering (7), which is invalid.</Paragraph>
    <Paragraph position="30"> The structure of figure 1 is a module with two premises and three conclusions; the latter are the polar categories of our composition theorem (8). Adding the remaining id axiom link makes it a proof-net for composition.</Paragraph>
    <Paragraph position="31"> For L, proof-nets must be planar, i.e.</Paragraph>
    <Paragraph position="32"> with no crossing edges. This corresponds to the non-commutativity of L. In LP, linear logic, which is commutative, there is no such requirement.</Paragraph>
    <Paragraph position="33"> Like the sequent calculus, proof-nets enjoy the Cut elimination property whereby every proof has a Cut-free equivalent. The evaluation of a net to its Cut-free normal form is a process of graph reduction. The reductions are as shown in figure 2.</Paragraph>
  </Section>
  <Section position="5" start_page="64" end_page="65" type="metho">
    <SectionTitle>
5 Language processing
</SectionTitle>
    <Paragraph position="0"> As is the case for the sequent calculus, with proof-nets every proof has a Cut-free equivalent in which only atomic id axiom links are used: what we shall call \[3q-long proof-nets. However, whereas some ~r Ilong sequent proofs are equivalent, leading to spurious ambiguity/derivational equivalence, distinct \[3q-long proof-nets always have distinct readings.</Paragraph>
    <Paragraph position="1"> The analysis of an expression as search for \[3rl-long proof-nets can be construed in three phases, 1) selection of lexical categories for elements in the expression, 2) unfolding of these categories into a .fi'ame of trees of i- and ii-links with atomic leaves (literals), and 3) addition of (planar) id axiom links to form proofnets. For example, 'John walks' has the following analysis:</Paragraph>
    <Paragraph position="3"> The ungrammaticality of 'walks John' is attested by the non-planarity of the proof structure (20).</Paragraph>
    <Paragraph position="5"> As expected, where there is structural ambiguity there are multiple derivations; see figure 3. But now also, when there is no structural ambiguity there is only one derivation, as in figure 4. This property is entirely general: the problem of spurious ambiguity is resolved.</Paragraph>
  </Section>
  <Section position="6" start_page="65" end_page="67" type="metho">
    <SectionTitle>
6 Proof-net semantic extraction
</SectionTitle>
    <Paragraph position="0"> Until now we have not been explicit about how a proof determines a semantic reading. We shall show here how to extract from a proof-net a functional term representing the semantics (see de Groote and Retor6 1996, who reference Lamarche 1995). This is done by travelling through a proof-net and constructing a lambcla term following deterministic instructions. (The proof-nets are the proof structures m which following these instructions visits each node exactly once.) First one assigns a distinct variable index to each i-link; then one starts travelling upwards through the unique positive conclusion. Thereafter the function L mapping proof-nets to lambda terms is as follows (for brevity we exclude product): (21) a.</Paragraph>
    <Paragraph position="1"> Going up through the conclusion of a i-link, make a functional abstraction for the corresponding variable and continue upwards through the positive premise:</Paragraph>
    <Paragraph position="3"> Going up through one id conclusion, go down through the other:</Paragraph>
    <Paragraph position="5"> Going down through one premise of Cut, go up through the other: d.</Paragraph>
    <Paragraph position="6"> Going down through one premise of a \i-link, make a functional application and continue going down through the conclusion (function) and going up through  Let us observe that the following lexical type assignments capture the paraphrasing of (la) and (lb); a-C/ := A signifies the assignment to category A of expression a with lexical semantics C/.  marks the point at construction and Roman numerals indicate the argument traversals, performed after the function traversals, triggered by entry into ii-links.  This is not the same semantic term as that in (23) but it reduces to the same by 13conversion, showing that the semantic content in the two cases is identical, that is, that there is paraphrase: (25) (()vx)vy((in x) (live y)) b) f) = )vy((in b) (live y)) f) = ((in b) (live\])) Although such lambda conversion only calculates what the grammar defines and is not part of the grammar itself, computationally it is an on-line process. The following section shows how this can be rendered, in virtue of proof-nets, an off-line process of lexical compilation.</Paragraph>
  </Section>
  <Section position="7" start_page="67" end_page="67" type="metho">
    <SectionTitle>
7 Off-line semantic evaluation
</SectionTitle>
    <Paragraph position="0"> In the processing as presented so far semantic evaluation is, as is usual, normalisation of the result of substituting lexical semantics into derivational semantics. Logically speaking, this substitution at the lexico-syntactic interface is a Cut, and the normalisation is a process of Cut elimination. Currently the substitution and Cut elimination is executed after the proof search. However, if lexical semantics is represented as a proof-net, one can calculate off-line the module resulting from connecting the lexical semantics with a Cut to the module resulting from the unfolding of the lexical &amp;quot; categories.</Paragraph>
    <Paragraph position="1"> Lexical semantics expressed as a linear (=single bind) tambda term is unfolded into an (unordered) proof-net by the algorithm (26): (26) adeg Start with the )v-term go at a + node: q~+. b.</Paragraph>
    <Paragraph position="2"> To unfold Kxnq)+, make it the conclusion of a i-link with index n and unfold C/p+ at the positive premise:</Paragraph>
    <Paragraph position="4"/>
  </Section>
class="xml-element"></Paper>