<?xml version="1.0" standalone="yes"?>
<Paper uid="E95-1018">
  <Title>Mixing Modes of Linguistic Description in Categorial Grammar</Title>
  <Section position="4" start_page="127" end_page="128" type="metho">
    <SectionTitle>
2 The substructural hierarchy
</SectionTitle>
    <Paragraph position="0"> I will address only logics (or levels) having three connectives: a 'product' connective (a form of conjunction, corresponding to 'matter-like addition' of substructures), plus two implicational connectives (the left and right 'residuals' of the product), notated as deg-L, and o for a product o.</Paragraph>
    <Paragraph position="1"> The minimal set of sequent rules for any group o 0 of connectives {o,~,.---} is as in .(1,2): 3</Paragraph>
    <Paragraph position="3"> The Identity (id) and Cut rules express the reflexivity and transitivity of the derivability relation. Each connective has a Right \[R\] and Left \[L\] rule, showing, respectively, how to prove and how to use a type containing that connective. Note that this formulation includes a system of term</Paragraph>
    <Paragraph position="5"> mula A can be derived from the structured configuration of antecedent formulas F. F\[(I )1\] represents represents the result of replacing (I) with ~l in F\[(I)\].</Paragraph>
    <Paragraph position="6"> labelling, whereby each type is associated with a lambda term (giving objects TYPE:term) in accordance with the well known Curry-Howard interpretation of proofs, with the consequence that complete proofs return a term that records the proof's functional or natural deduction structure.</Paragraph>
    <Paragraph position="7"> Such terms play an important role in the approach to be developed. The system of term labelling has the following features. All antecedent formulas are associated with variables. Cut inferences are interpreted via substitution (with a\[b/v\] representing the substitution of b for v in a). For implicational connectives, Left and Right inferences are interpreted via functional application and abstraction, respectively. A different abstraction and application operator is used for each implicational connective, so that terms fully record the proof structure. The implication o (resp. o_%) has application operator ~ (resp. ~ ), giving aT b (resp.</Paragraph>
    <Paragraph position="8"> b-~ a) for 'a applied to b', and abstraction operator \[g-\] (resp. \[-~\]), e.g. \[~\]v.a (resp. \[-~\]v.a) for abstraction over v in a. Product Right inferences are interpreted via system specific pairing. For product Left inferences, a term such as \[z/vow\].a represents the substitution of z for v+w in a. 4 We must next consider the issue of resource structure and its consequences for linguistic derivation. If we assume for the above sequent system that antecedents are (non-empty) binarily bracketted sequences of types then we have a version of the non-associative Lambek calculus (NL: Lambek 1961), where deduction is sensitive to the order and bracketting of assumptions, each of which must be used precisely once in a deduction. NL is a system whose implicit notion of linguistic structure is binary branching tree-like objects, and this rigidity of structure is reflected in the type combinations that the system allows. 5 However, it is possible to undermine sensitivity to aspects of resource structure by inclusion of structural rules, which act to modify the structure of the antecedent configuration. For example, the following rules of Permutation (\[P\]) and Association (\[A\]) undermine sensitivity to the linear order and bracketting of assumptions, respectively:  r\[(S:b, C:c)*\] ~ A:a \[P\] r\[(c:c. B:b)*\] ~ A:a F\[(B:b, (C:c, D:d)deg) deg\] =,, A:a \[A\] F\[((B:b,C:c) deg, D:d) deg\] =~ A:a Adding \[P\] to NL gives NLP, a system whose implicit notion of linguistic structure is binary  the systems of algebraic semantics that are provided for such logics. Discussion of such issues, however, is beyond the scope of the present paper.</Paragraph>
    <Paragraph position="9">  branching mobiles (since order is undermined only within the confines of the given bracketting).</Paragraph>
    <Paragraph position="10"> Adding \[A\] to NL gives a version of associative Lambek calculus (L: Lambek 1961), which views language purely in terms of strings of tokens. If both \[A\] and \[P\] are added, we have the system LP (van Benthem 1983), corresponding to a fragment of linear logic (Girard 1987), which views language in terms of unordered multisets of tokens.</Paragraph>
    <Paragraph position="11"> I will adopt special notations for the operators of these systems: NL:{(r),~,~}, NLP:{O,O--,--~}, L:{*,\,/}, LP:{(r),-o,o--}.</Paragraph>
    <Paragraph position="12"> The proof below illustrates this formulation, showing the composition of two implicationals (a combination which requires associativity). If we simplify the resulting proof term, using \[~'\] for A and left-right juxtaposition for application, we get the familiar composition term Az.x(yz).</Paragraph>
    <Paragraph position="13">  X:v=vX:v Y:w=v Y:w i\[ ~L\] (X--Y:z, Y:w)* ~ X:(~.w Z:z =~ Z:z \[:L\] (X--Y:z, (Y--Z:y, Z:z)*)* ~ X:(x.(yTz)) ~ . .-:-:7:--7-, :---W, ..... \[A\] ((X~-Y:,,Y--Z:y) ,Z:z) =~ X:(xT(y*~z))_~ _ --~. V - -- ..-ZT.~--Z - .-~o - - -T-g.- ....... \[ R\] (X Y:x, (Y~Z:y) ~ X*--Z:\[7\]z.(x;(y;z))</Paragraph>
  </Section>
  <Section position="5" start_page="128" end_page="128" type="metho">
    <SectionTitle>
3 Structural modalities
</SectionTitle>
    <Paragraph position="0"> Structural modalities are unary operators that allow controlled involvement of structural rules which are otherwise unavailable in a system, 6 e.g. a modified structural rule might be included that may only apply where one of the types affected by its use are marked with a given modality. For example, a unary operator /k, allowing controlled permutation, might have the following rules (where/kF indicates a configuration in which all types are of the form/kX):</Paragraph>
    <Paragraph position="2"> The Left and Right rules are as for necessity in $4. The restricted permutation rule \[/kP\] allows any formula of the form AX to permute freely, i.e.</Paragraph>
    <Paragraph position="3"> undermining linear order for just this assumption.</Paragraph>
    <Paragraph position="4"> The left rule \[/kL\] allows a/k-marking to be freely discarded. Such a modality has been used in treatments of extraction. The calculus L 'respects' linear order, so that s/up or np\s corresponds to a sentence missing a NP at its right or left periphery. However, a type s/(/knp) corresponds to a sentence missing NP at some position, and so 6The original structural modalities are linear logic's 'exponentials'. See Barry et at. (1991) for some structural modalities having suggested linguistic uses.</Paragraph>
    <Paragraph position="5"> is suitable for use in the general case of extraction, where a NP extraction site may occur non peripherally within a clause. Proof A in Figure 1 illustrates (proof terms are omitted to simplify).</Paragraph>
    <Paragraph position="6"> Structural modalities allow that stronger logics may be embedded within weaker ones, via embedding translations, i.e. so that a sequent is derivable in the stronger logic iff its translation into the weaker logic plus relevant modalities is also derivable. For example, using /k, a fragment of LP may be embedded within L.</Paragraph>
  </Section>
  <Section position="6" start_page="128" end_page="128" type="metho">
    <SectionTitle>
4 Relating substructural levels
</SectionTitle>
    <Paragraph position="0"> hnagine how an LP formula X(r)Y might be 'translated' into the system 'L plus /~' ('LA').</Paragraph>
    <Paragraph position="1"> This formula shows the interderivability X(r)Y C/~ YQX. A corresponding 'reordering' interderivahility would be allowed if X(r)Y translated to any of (/kX).(/XY) or X.(AY) or (AX).Y, i.e. with either or both of the product subcomponents modalised, (indicating that subcomponents X and Y may legitimately appear in either order). Such /ks may be 'dropped', e.g. (/kX)o(/kY) =C/, XoY, a step corresponding to selection of one of the permitted orders. This latter transformation suggests X(r)Y ~ X*Y as a theorem of a mixed logic, revealing a natural relation between XQY and X.Y, as if the former were in some sense 'implicitly modalised' relative to the latter.</Paragraph>
    <Paragraph position="2"> Consider next the implicational Xo-Y, which exhibits the interderivability Xo--Y C/:~ Y--oX.</Paragraph>
    <Paragraph position="3"> This suggests the translation X/(AY), for which we observe X/(/kY) C/::, (/kY)\X. L/k allows X/Y :C/. X/(AY), suggesting X/Y ~ Xo-Y as a 'linking' theorem of a mixed logic revealing the natural relation between Xo--Y and X/Y.</Paragraph>
    <Paragraph position="4"> The above discussion suggests how the systems L and LP might be interrelated in a logic where they coexist. Such relation might be justified in terms of allowing transitions involving forgetting of information, i.e. X(r)Y indicates that both orders are possible for its subeomponents, and the move to XeY (or YoX) involves forgetting one of these possibilities. Generalising from this ease, we expect that for any two sublogics in a mixed system, with products oi and oj, where the former is the stronger logic (including more structural rules), we will observe transformations: oi XoiY ::C/, XoiY and Xdeg~(Y :=~ X*--Y.</Paragraph>
  </Section>
  <Section position="7" start_page="128" end_page="130" type="metho">
    <SectionTitle>
5 A hybrid system
</SectionTitle>
    <Paragraph position="0"> Consider how we might formulate a mixed logic of the kind just suggested, what I term a hybrid system -- one which includes the logics that arise by choices from just \[A\] and \[P\]. The sequent rules shown in (2) may still be used for each of the levels (with o serving as a placeholder for the various product operators), as may the axiom and Cut rule in (1). In addition, we require the following</Paragraph>
    <Paragraph position="2"> With only the rules (1,2,3), we would have a system where different substructural levels coexist, but without interrelation. Such interrelation is effected by the rule (4), which allows a bracket pair of one system (oj) to be replaced by that of another system (oi), just in case the latter's system exhibits greater freedom of resource usage (as indicated by the relation &lt;, which orders the sub- null systems thus: (r) &lt; {O, e} &lt; (r)).</Paragraph>
    <Paragraph position="3"> (4) r\[(B: b, C: c) degi\] =~ A:a \[&lt;\] F\[(B:b, C:c) deg,\] ~ A:.</Paragraph>
    <Paragraph position="4">  The following proofs are for the two transformations discussed in the previous section, illustrating 'natural relations' between levels.</Paragraph>
    <Paragraph position="6"> The converse transitions are not derivable, since the converse substitution of brackets under \[&lt;\] is not allowed. Corresponding transformations may be derived for the connectives of any two appropriately related subsystems, e.g. A(r)B =&gt; AOB, A~B ~ A/B.</Paragraph>
    <Paragraph position="7"> Proof terms have been used in categorial work for handling the natural language semantic consequences of type combinations. The above terms, however, encode distinctions unwanted for this purpose, but can easily be simplified to terms using only a single abstractor (A) and with application notated by left-right juxtaposition, e.g.:</Paragraph>
    <Paragraph position="9"> A standard method for handling the word order consequences of categorial proofs uses the linear order of formulas in the proven sequent in the obvious way. This method cannot be used for the hybrid approach, because for any theorem, there exist other theorems for combining the same antecedent types under any possible ordering thereof. 7 The word order consequences of proofs are instead determined from the normal forms of proofs terms, s which encode all the relevant information from the proof, and in particular, the directional, etc, information encoded by the connectives of the types combined. Consider the labelled theorem: (A/B: x, C~B: y)e =~ Ao-C: \[~\]z.x; (z~ y) rAny proof of r :0 A may be extended by multiple \[&lt;\] inferences to give a proof of F' =v A, where F' is just like F except all bracket pairs are 0 (r). Extending this proof with repeated uses of \[P\] and \[A\], we can attain any desired reordering of the component types.</Paragraph>
    <Paragraph position="10"> SNormalisation of proof terms is defined by the following conversion rules:</Paragraph>
    <Paragraph position="12"> For the result label's subterm x~(z~y), the directionality of applications suggests the ordering x -&lt; z -&lt; y. Abstraction discounts z as an 'orderable element', leaving just x -4 y, i.e. with A/B preceding C~B, as we would expect. For a term x~y, the permutativity of (r) suggests that both orderings of x and y are possible. Note however that word order determination must be sensitive to the specific modes of structuring and their properties, e.g. the non-associativity of (r) implies an 'integrity' for y, z in x~ (y~ z) excluding y -&lt; x -&lt; z as a possible order, despite the permutativity of (r). To determine word order, a normalised proof term is first transformed to give a yield term, in which its orderable elements are structured in accordance with their original manner of combination, e.g.</Paragraph>
    <Paragraph position="13"> xZ(z-;y) ~ x.(zOy) \[(v'~ w)/x.v\].(=, v) deg ~ (v(r)w) Yield terms may be restructured in ways appropriate to the different operators (e.g. subterms p(r)q may be rewritten to q(r)p, etc.). Possible linear orders can simply be 'read off' the variants a yield term under restructuring, e.g. x'~(y~z) gives orders xyz and yzx, since its yield term is x(r)(y(r)z), whose only variant is (y(r)z)(r)x</Paragraph>
  </Section>
  <Section position="8" start_page="130" end_page="131" type="metho">
    <SectionTitle>
7 The linguistic model
</SectionTitle>
    <Paragraph position="0"> I noted earlier that extensive use of structural modalities tends to result in very complex analyses. This fact tends to favour the selection of stronger systems for the base level logic, a move which is associated with loss of possibly useful resource sensitivity. This problem does not arise for the hybrid approach, which freely allows us to use weaker logics for constructing lexical types that richly encode linguistic information.</Paragraph>
    <Paragraph position="1"> Consider firstly a hybrid system that includes only the two levels L and LP, of which clearly L will in general be more appropriate for linguistic description. Under the view of how levels are related that I have argued for, the linkage between these two levels is such that X(r)Y ::~ XeY is a theorem, alongside which we will find also (e.g.) X/Y ::~ Xo-Y. Note that it is the latter theorem, and its variants, that most crucially bear upon what is gained by the move to a mixed system, given that the lexical encoding of linguistic information predominantly involves the assignment of functional types. Hence, a lexical functor constructed with L connectives may be transformed to one involving LP connectives, allowing us to exploit the structural freedom of that level. 9 For 9Note that with the converse direction of linkage, as advanced by Moortgat &amp; Oehrle (1993), but with lexical functors still constructed using L connectives, no practical use could be made of the permutative LP level in this minimal mixed system.</Paragraph>
    <Paragraph position="2"> example, in handling extraction, a 'sentence missing NP somewhere' may be derived as so-np, as in proof B of Figure 1.</Paragraph>
    <Paragraph position="3"> Consider next a system that includes also the non-associative level NL. This additional level might be adopted as the principal one for lexieal specification, giving various advantages for linguistic analysis. For example, by having a lexieal element subcategorise for a complement that is some 'non-associative functor' (i.e. of the form At~B or B~A), we could be sure that the complement taken was a 'natural projection' of some lexical head, and not one built by composition (or other associativity based combination). On the other hand, where the freedom of associative combination is required, it is still available, given that we have (e.g.) XtgY ==v X/Y. Some categorial treatments of non-constituent coordination have depended crucially (either implicitly or explicitly) on associativity allowing, for example, subject and verb to be combined without other verb complements, making possible a 'like-with-like' coordination treatment of non-constituent coordination as in e.g. (i) Mary spoke and Susan whispered, to Bill (where the conjuncts are each analysed as s/pp).</Paragraph>
    <Paragraph position="4"> In a purely non-associative system, such as NL, such an analysis is excluded. In the hybrid approach, however, this treatment is still possible even with non-associative lexical types, provided coordination is done at the associative level, e.g.</Paragraph>
    <Paragraph position="5"> the conjuncts of (i) can be derived and coordinated as s/np since: np, (np~s)t~pp =~ s/pp is a theorem as in (5). Furthermore, since we have also Xt~Y =~ Xo--Y, such non-associative lexical specification is still compatible with the treatment of extraction described above.</Paragraph>
    <Paragraph position="6"> (5) np ~ np s =~ s \[~L\] (up, np~s) q) ::~ s pp :=~ pp (np, ((np~s)~pp, pp)O)o =~ s \[&lt;\] (np, ((np~s)~pp, pp)*)e =~ s .\[&lt;1 (np, ((np~s)~pp, pp)')&amp;quot; =l~ s \[A\] ((np, (np~s)~pp)*, pp)* =~ s \[/a\] (np, (np~s)~pp)' =~ s/pp It is hoped that the above simple examples of linguistic uses will serve to give a feeling for the general character of the linguistic model that the hybrid approach would favour, i.e. one with very rich lexical encoding of syntactic information, achieved using predominantly the implicational connectives of the weakest available logic, with the stronger logics of the mixed system allowing less informative (but hence also more 'flexible') descriptions of (functional) linguistic objects. The above example systems clearly do not exhaust the possibilities for 'rich lexical encoding'. For example, it seems likely that lexical assign- null ments should specify headedness or dependency information, as in the calculi of Moortgat &amp; Morrill (1991).</Paragraph>
  </Section>
  <Section position="9" start_page="131" end_page="131" type="metho">
    <SectionTitle>
8 Parsing with hybrid grammars
</SectionTitle>
    <Paragraph position="0"> It is well known that parsing (theorem proving) with sequent formalisms suffers efficiency problems as a consequence of derivational equivalence (or 'spurious ambiguity'), i.e. from the existence of multiple proofs that assign the same meaning for a given type combination. Alternative but equivalent formalisations of the above system are possible. Hepple (1993), for example, provides a natural deduction formalisation. Such a formalisation should readily provide the basis for a chart based approach to parsing hybrid logic grammars, after the manner of existing chart methods for use with L (KSnig 1990; Hepple 1992). A further promising possibility for efficient parsing of hybrid system grammars involves proof net reformulation, following a general scheme for such reformulation described in Moortgat (1992). However, the precise character of either chart or proof net based methods for parsing hybrid system grammars is a topic requiring further research.</Paragraph>
  </Section>
class="xml-element"></Paper>