<?xml version="1.0" standalone="yes"?>
<Paper uid="E93-1016">
  <Title>Parsing with polymorphism *</Title>
  <Section position="3" start_page="120" end_page="123" type="intro">
    <SectionTitle>
2 Polymorphism
</SectionTitle>
    <Paragraph position="0"> Despite the great simplicity of Lambek grammars, a surprising amount of coverage is possible \[Moortgat, 1988\]. Two aspects of this are embryonic ac -. counts of extraction, and scope-ambiguity, the latter arising from the fact that there may be more than one proof of a given sequent. However, the accounts possible have remained only partial. Non-peripheral extraction remainsd unaccounted for (eg.</Paragraph>
    <Paragraph position="1"> the (man)/ who Dave told ei to leave) and only the scope-ambiguities of peripheral quantifiers are covered (as in the structure QNP TV QNP). A simple account of cross-categorial coordination has also often been cited as an attractive feature of Lambek grammars (\[Moortgat, 1988\]). However, the analyses are never in a purely Lambek grammar. Belonging to Lambek grammar proper is a part assigning some category to the strings to be coordinated, and then lying without Lambek grammar, a coordination schema, such as x, and, x ::~ x.</Paragraph>
    <Paragraph position="2">  To overcome these deficits in coverage, I have proposed a polymorphic extension of the calculus.</Paragraph>
    <Paragraph position="3"> Added to the categorial vocabulary are category variables and their universal quantification, allowing such categories as: X, X/X, VX.X/(X\np). To L(/,~ \~ are added left and right rules for V, to give what I will call L(/,\,v)(I given straightaway the term-associated calculus):</Paragraph>
    <Paragraph position="5"> Notation: the terms are drawn from the language of 2nd Order Polymorphic A-calculus \[Girard, 1972\], \[Reynolds, 1974\]. Here, terms carry their type as a superscript, and one can have variables in these types (eg. Axr.x~), one can abstract over such variable types, deriving terms of quantified type (eg.</Paragraph>
    <Paragraph position="6"> A~r.Ax ~.z ~, of type Vr(Tr--*~r)), and terms of quantified type can be applied to types (eg. Ar.Axr.=x(t), of type (Z-+Z)). In the (VL) rule above, the type, a, that a is applied to, is the type that corresponds to the category, y, that is being substituted for the cate~ory variable, Z. 2 An equivalent slight variant on L (/,\,v) takes as axioms only those z ::~ x sequents where z is basic or a variable, something I will call L~/'\'v). It is easy to show L~/'\'v)~-r iff L(/,\,v)~--r (see \[Emms and Leiss, forthcoming\]).</Paragraph>
    <Paragraph position="7"> By assigning conjunctions to YX.((X\X)/X), negation to VX.X/X, and quantifiers to VX.X/(X\np) and VX.X\(X/np), one obtains coverage of cross-categorial coordination and negation, as well as a comprehensive account of quantifier scope ambiguity \[Emms, 1989\],\[Emms, 1991\]. Assigning relativisers to VX.((cn\cn)/(s\X)/(X/np)), non-peripheral extraction can also be handled \[Emms, 1992\]. The meanings that go along with these categories are as follows. Where PS is Q, ff or A f, let/:G vary over the conventional meanings of quantifiers, junctions and negation, with PS:p the polymorphic version.</Paragraph>
    <Paragraph position="9"> who(a)(P~ a)(p~ t)(Qe t)(xe ) = P2(P~x) AQx I will give two illustrations. The proof below would allow the embedded quantifier, every man, to be assigned a de-re interpretation in John believes every man walks. Note (s\np)\((s\np)/s) : X.</Paragraph>
    <Paragraph position="10"> 2The (VR) given is a cut-down version of the 'official' version, which allows a change of bound variable np, s\np ~ X nP, (s\np)/s, X =~s s\np =~.''~npl: np, (s\np)/s, X/(X'\np) s\np ::~ s ,Y=L np, (s\np)/s, VX.X/(X\np), s\np ::~ s Now assuming j, bel, em and walk were the terms associated with the antecedents of the root sequent, the term for the proof is: emp (tel, et ) ( AxA f A y\[f ( walk( z ) )( y) \] ) ( bel)(j ) We obtain as a possible denotation for John believes every man walks:</Paragraph>
    <Paragraph position="12"> As an illustration of non-peripheral extraction, the proof below allows the string who John told to go to be recognised as a postmodifier of a common noun: s/vpc, vpc =~ s __ \R r vpc =~ s\X D (c.\c.)/(s\X), vpc ca\ca np, V, np, vpc ::~ s /L _ ./L np, V :~ X/np /L rip, v, vpc cn\cn VL VX.((cn\cn)/(s\X)/(X/np)), rip, V, vpc cn\cn Here r = cn\cn ~ cn\cn, V -- ((s\np)/vpc)/np, = s/vpc. Assuming who, j, told, and go were associated with the antecedents of the root, the term for the proof is: who( (et, t ) )( AzAy\[told(z)(y)(j)l)( A f\[f (go)\]) We obtain for the denotation of the string who John told to go:</Paragraph>
    <Paragraph position="14"> For the further discussion of the analyses within an L (/,\,v) grammar that cover a significant range of data, see the earlier references. I turn now to the main problem which this paper addresses: is there an automatic procedure able to find these analyses ?</Paragraph>
    <Section position="1" start_page="121" end_page="122" type="sub_section">
      <SectionTitle>
2.1 Cut Elimination for L (/,\,v)
</SectionTitle>
      <Paragraph position="0"> We want a procedure to decide whether G ~-s E z, where G is an L (/,\,v) grammar. As with L(/'\) grammars, this problem reduces to deciding L(/'\'v)~ - r if it can be shown both that Cut can be eliminated, and without the loss of any significant semantic diversity. This has recently been shown (\[Emms and Leiss, forthcoming\]). I make some remarks on the proof. The strategy of the proof of Cut elimination for L (/A) starts from the observation that a proof, 7,  using Cut must contain at least one use of Cut which dominates no further uses of Cut - a 'topmost' use of Cut. Suppose this use of Cut derives r. Then one defines two things: a degree of the Cut leading to r, and a transformation taking the proof of r to an alternative proof of r, such that either the transformed proof of r is Cut-free, or it is a proof with 2 or less cuts of lesser degree. After a finite number of iterations of the transformation, one must have a cut free proof.</Paragraph>
      <Paragraph position="1"> In the proof for L (/'\), the degree of a Cut inference is simply the sum of the numbers of connectives in the two premises. This cannot be the degree for L (/'\'v). For example, a cases to be considered is where one has a cut of the kind shown in (2). The natural rewrite is (3) (that T ~ y\[a/Z\] is provable relies on the fact that Z is not free in T and substitution for free variables preserves derivability \[Emms  and Leiss, forthcoming\]) (2) T ~ v VR U, v\[~/Z\], V ~ WVL T ~ VZ.y U, VZ.v, V =~ w Cut V,T,V =~ w (3) T ::~ y\[a/Z\] U, y\[a/Z\], V =~. w .Cut  U, T, V =C, w With degree defined by number of connectives, we need that the number of connectives in y\[a/Z\] is strictly less than the number in VZ.y, and that is often false. The proof goes through instead by taking the degree of a cut to be the sum of sizes of the proofs of its two premises, where the size is the number of nodes in the proof. 3</Paragraph>
    </Section>
    <Section position="2" start_page="122" end_page="123" type="sub_section">
      <SectionTitle>
2.2 Difficulties in deciding L(/,\,v)}-T ::C/, x
</SectionTitle>
      <Paragraph position="0"> So the problem reduces to one of L (/'\'v) derivability. Whether L (/'\'v) derivability is decidable I do not know. The nearest to an answer to this that the logical literature comes is a result that quantified intuitionistic propositional logic is undecidable \[Gabbay, 1974\]. The difference between L(/,\,v) and logic of this result is the presence of the further connectives (V, A), and the availability of all structural rules. I will describe below some of the problems that arise when some natural lines of thought towards a decision procedure are pursued.</Paragraph>
      <Paragraph position="1"> One might start by considering the logic that is L(/'\)+ (VR). This can be argued to be decidable in the same fashion as L(/'\): read (VR) backwards as a rewrite, adding another way to build deduction trees. As for L((/'\) a sequent has only finitely many deduction trees, and provability is equivalent to the existence of a deduction tree with axiom leaves.</Paragraph>
      <Paragraph position="2"> ~In fact nodes above axiom form sequents are not counted in the size, and the proof relies on changes of bound variable and substitutions not changing the size of L(/'\'Y ) proofs However, when (VL) is added this simple argument will not work: if (VL) is read backwards as a further claus- ill tile definition of deduction trees, then a leaf containing an antecedent V could be rewritten infinitely many different ways. A natural move at.</Paragraph>
      <Paragraph position="3"> this point is to redefine deduction trees, reading the (VL) rule as an instruction to substitute all unknown.</Paragraph>
      <Paragraph position="4"> One hopes then that: (i) the set of so-defined deduction trees for a given sequent, r, is finite (ii) there is some easy to check property, P, of these trees such that the existence of a P-tree in the set would be equivalent to L(/,\,v)~-r. Now, if we were considering the combination of first-order quantification with the Lambek calculus, this strategy works, but whether it works for n (/'\,v) remains unknown.</Paragraph>
      <Paragraph position="5"> I will go through the application of the strategy in the first-order case to highlight why g(/,\,v) does not yield so easily. The first-order quantification plus the Lambek calculus, I will call L (/'\,v'). It is the end-point of a certain line of thought concerning agreement phenomena. One first reanalyses basic categories, such as s and np, as being built up by the application of a predicate to some arguments, giving categories such as np(3rd,sing), s(fin). It is natural then to consider quantification over the first order positions, such as Vp. s(fin)\np(p,pl), which could be used when, as in English, the plural forms of a verb are not distinguished according to person. Now L(/,\,v~) is decidable, which can be shown by adapting an argument that shows that when the contraction rule is dropped from classical predicate logic, it becomes decidable \[Mey, 1992\]. Deduction trees for a sequent, r, of L (/'\'v~) are defined so that the rewrite associated with the (VL) rule substitutes an unknown. There are then only finitely many deduction trees (the absence of the structural rule of contraction is essential here). Now, if L(/'\'v')~--r, and r has a complex first order term, one can be sure that this term is present in an axiom, because no rules build complexity in the places in categories where a bound variable can occur. For this reason, the so-defined deduction trees for r cover all the possible patterns for a proof of r. Provability is therefore equivalent to the existence of a substitution making one of the deduction trees have axiom leaves, and this can be checked using resolution.</Paragraph>
      <Paragraph position="6"> This situation does not wholly carry over to g(/,\,v). The 'substitute an unknown' rewrite reading of (VL) defines only finitely many deduction trees for a sequent, r. However, these so-defined deduction trees for r do not cover all the possible palterns for a proof of r: unlike g (/,\'v~), there are rules that build complexity in the places in categories where a bound variable can occur. So, for example, L(/'\,v)~ no, VX.X/(X\np), (s\np)\np, but none of the deduction trees represents the pattern of the proof. So to check for the existence of a deduction tree (as above defined) that by a substitution would have ax- null iom leaves is not sufficient to decide derivability. It seems we must defined the looked for property, P, of deduction trees recursively, so that a tree has P if (1) the leaves by a substitution become axioms, or (2) by hypothesising a connective in one of the unknowns, and extending the tree by rewrites licensed by this connective, one obtains a P-tree.</Paragraph>
      <Paragraph position="7"> It would amount to the same thing if the definition of deduction tree was extended (by hypothesising a connective in an unknown), and the looked for property, P, kept simple: a tree whose leaves by a substitution become axioms. However, the extended definition of deduction tree now allows infinitely many trees for a sequent. This may seem surprising, but is seen one considers a leaf such as T ==~ X. One can hypothesis X = Y/Z, extend the deduction tree by the rewrite associated with a slash Right rule, obtaining once again a leaf with a succedent occurrence of an unknown. By imposing a control strategy which would systematically consider all deduction trees of height h, before deduction trees of height h + 1, one can be sure that any provable sequent would sooner or later be accepted by the decision procedure (because its provability would entail the existence of a deduction tree of a certain finite height). However, there is no reason to expect the procedure to terminate when working on an underivable sequent. 4</Paragraph>
    </Section>
  </Section>
class="xml-element"></Paper>