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<Paper uid="W91-0108">
  <Title>Structural Non- Correspondence in</Title>
  <Section position="2" start_page="0" end_page="57" type="intro">
    <SectionTitle>
SCOTLAND
ABSTRACT
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    <Paragraph position="0"> This paper explores the options available in the formal definitions of generation and, parasitically, translation, with respect to the assumed necessity for using a single grammar for analysis and synthesis. This leads to the consideratiOn of different adequacy conditions relating the input to the generation process and the products of analysis of its Output.</Paragraph>
    <Paragraph position="1"> I. A SCHEMATIC DEFINITION OF ~GENERATES' We start from the assumption of a constraint-based theory of linguistic description, which supports at least the notions of derivation and underlying form, in that the definition of grammaticality appeals to a relation between surface strings and some formal structure. We will attempt to remain agnostic about the shape of this formal structure, its precise semantics and the mechanisms by which a grammar and lexicon constrain its natUre in any particular case. In particular, we take no stand on whether it is uniform and monolithic, as in the: attribute-value matrices (hereafter AVMs) of PATR-II or HPSG, or varied and partitioned, as in the trees, AVMs and logical formulae of LFG. We will use the phrase products of analysis to refer to the set of underlying structures associated by a grammar and lexicon with a surface string, viz for a grammar G and sentence s e LG, we refer to the set of all products of analysis where we use A for the 'derives' relation. 1 We will also use ~s to refer to an arbitrary member of Xs.</Paragraph>
    <Paragraph position="2"> We will also assume that the formal structures involved support the notions of subsumption and its inverse, extension, as well as unification and generalisation. Whether this is accomplished via appeal to a lattice, or in terms of simlflations, will only become relevant in section IV.</Paragraph>
    <Paragraph position="3"> We can now provide schematic definitions of generation and, with a few further assumptions, translation. We say Definition 1.</Paragraph>
    <Paragraph position="4"> F~(~,s) (a structure ~ generate~ a string s for grammar G) iff 3 ~s ~ ~(~s,~) 2 Most work to date on building generators from underlying forms (e.g. lIn this we follow Wedekind (1988), where we use X/x for an arbitrary underlying form, as he uses C//~ for f-structure and Z/a for sstructure. null 2Again our T is similar to Wedekind (1988)'s adequacy condition C.</Paragraph>
    <Paragraph position="5">  Wedekind 1988, Momma and DSrre 1987, Shieber, van Noord, Pereira and Moore 1990, Estival 1990, Gardent &amp; Plainfoss~ 1990) have taken the adequacy condition T to be strict isomorphism, possibly of some formalismspecific sub-part of the structures Xs and X, e.g. the f-structure part in the case of Wedekind (1988) and Momma and DSrre (1987). In the balance of this paper I want to explore alternative adequacy conditions which may serve better for certain purposes.</Paragraph>
    <Paragraph position="6"> Although some progress has been made towards implementation of generators which embody these alternatives, that is not the focus of this paper. As far as I know, aside from a few parenthetical remarks by various authors, only van Noord (1990) addresses the issue of alternative adequacy conditions--I will place his suggestion in its relevant context below. null II. WEAKER FORMULATIONS Work on translation (Sadler and Thompson 1991) suggests that a less strict definition of T is required.</Paragraph>
    <Paragraph position="7"> Consider the following AVM, from which features irrelevant to our concerns have been eliminated: &amp;quot;cat s pred like</Paragraph>
    <Paragraph position="9"> Under the T is identity approach, this structure will not generate the sentence Robin likes to swim, even though one might expect it to. For although we suppose that somewhere in the grammar and lexicon there will be a constraint of identity between the subject of like and the subject of swim, which should be sufficient to as it were 'fill in' the missing subject, the strict isomorphism definition of T will not allow this.</Paragraph>
    <Paragraph position="10"> II.1 Subsumption and extension If T were loosened to extension, the inverse of subsumption, this would then work 7(ks,Z) iff ~s subsumes thing which translation, straightforwardly (i.e.</Paragraph>
    <Paragraph position="11"> _~ ~, that is, ~s extends ~, ~s). It is just this sort of seems to be required for see for example Sadler and Thompson (1991) and the discussion therein of Kaplan et al. (1989), where X for the desired target arises as a side effect of the analysis of the source, and Xs is additionally constrained by the target language grammar 3.</Paragraph>
    <Paragraph position="12"> Note that for Wedekind (1988) this move amounts to removing the coherence requirement, which prevents the addition of additional information during generation. Not surprisingly, therefore, implementation of a generator for T as subsumption is in some cases straight-forward--for the generator of Momma and DSrre, for example, it amounts to removing the constraints they call COHA and COHB, which are designed to implement Wedekind's coherence requirement. null van Noord (1990) discusses allowing a limited form of extension, essentially to fill in atomic-valued features. This avoids a problem with the unconstrained approach, namely that it has the potential to overgenerate seriously. 3Note that appealing to subsumption assumes that both the inputs to generation (~) and the results of analysis (Xs) are fully instantiated.  For the above example, for instance, the sentence Robin likes to swim on Saturdays could also be generated, on the assumption that temporal phrases are not subcategorised for, as Zs in this case clearly also extends X.</Paragraph>
    <Paragraph position="13"> Rather than van Noord's approach, which is still too strong to handle e.g. the example in Figure 1 above, some requirement of minimality is perhaps a better alternative.</Paragraph>
    <Paragraph position="14"> II.2 Minimal extension I What we want is that not only should ks extend X, but it should do so minimally, that is, there is no other string whose analysis extends X and is in turn properly extended by Xs.</Paragraph>
    <Paragraph position="15"> Formally, we want T defined as 4</Paragraph>
    <Paragraph position="17"> This rules out the over-generation of Robin likes to swim on Saturdays precisely because Xs for this properly extends ~s for the correct answer Robin likes to swim, which in turn extends the input X, as given above in  for extension, subsumption, unification and generalisation, using square-cornered set operators, as follows: ss E ls Ss subsumes ls; ls extends ss ss E ls ss properly subsumes ls; ls properly extends ss SSl U ss2=ls sslandss2unifytols lsl N ls2 = ss llsl and ls2 generalise to ss The intuition appealed to is that of the set operators applying to sets of facts (ssmsmaller set; Is--larger set).</Paragraph>
    <Paragraph position="18"> II.3 Maximal Overlap Unfortunately, the requirement of any kind of extension is arguably too strong. We can easily imagine situations where the input to the generation process is over-specific. This might arise in generation from content systems, and in any case is sure to arise in certain approaches to translation (see section III below). By way of a trivial example, consider the input given below in Figure 2.</Paragraph>
    <Paragraph position="20"> In the case where nouns in the lexicon are not marked for gender, as they might well not be for English, according to Definition 2 no sentence can be generated from this input, as Xs for the obvious candidate, namely Robin swims, will not extend X as it would lack the gender feature. But it seems unreasonable to rule this out, and indeed in our approachto translation to enforce the extension definition as above would be more than an inconvenience, but would rather make translation virtually unachievable. What seems to be required is a notion of maximal overlap, to go along with minimal extension, since obviously the structures in Figures 1 and 2 could be combined. What we want, then, is to define y in terms of minimal extensions to maximal overlaps:  as much as possible of Z with as little left over as possible. Note that we have chosen to give priority to maximal overlap at the potential expense of minimal extension. For example, supposing all proper nouns are marked in the lexicon for person and number, and further that commitative phrases are not sub-categorised for, then given the input  we will prefer Robin swims with Kim, with its extensions for the person and number features, as opposed to the non-extending Robin swims, because the latter overlaps less. Note that in the case of two alternatives with noncompatible overlaps, two alternative results are allowed by the above definition. null  Note that this approach is quite weak, in that it contains nothing like Wedekind's completeness conditionm if the grammar allows it, output may be produced which does not overlap large portions of the input structure, regardless of its status. For example structures which may be felt to be ungrammatical, as in Figure 4 below, may successfully generate surface strings on this account, i.e. Hours elapsed, despite 'leaving out' as 'important' a part of the underlying form as the direct object.</Paragraph>
    <Paragraph position="21">  If it is felt that generating anything at all from such an input is inappropriate, then some sort of completeness-with-respect-to-subcategorised- null for-functions condition could be added, but my feeling is that although this might be wanted for grammar debugging, in principle it is neither necessary nor appropriate.</Paragraph>
    <Paragraph position="22"> Alternatively one could attempt to constrain not only the relationship between Zs and X, but also the nature of itself. In the example at hand, this would mean for instance requiring some form of LFG's coherence restriction for subcategorisation frames. In general I think this approach would be overly restrictive (imposing completeness in addition would, for exam-</Paragraph>
    <Paragraph position="24"> ple, rule out the Z of Figure 1 above as well), and will not pursue it further here.</Paragraph>
    <Paragraph position="25"> It is interesting to note the consequences for generation under this defintion of input at the extremes. For</Paragraph>
    <Paragraph position="27"> grammatical subset), the result will be the empty string, if the language includes that, failing which, interestingly, it will be the set of minimal sentences(-types) of the language, e.g.</Paragraph>
    <Paragraph position="28"> probably just intransitive imperative and indicative in all tenses for English.</Paragraph>
    <Paragraph position="29"> The case of I X = ~ is trickier. If _L is defined such that it extends everything, or alternatively that the general+-sat+-on of anything with +- is the thing itself, then 1) .1_ is infinite so 2) no finite structure can satisfy the maximal overlap requirement; but in any case +- fails to satisfy the first clause of 3, namely the unification of Zs and Z must not be +-, since if Z is +then Xs and Z unify to +- for any Zs.</Paragraph>
    <Paragraph position="30"> Finally note that in cases where substantial material has to be supplied, as it were, by the target grammar (e.g. if a transitive verb is supplied but no object), then Definition 3 would allow arbitrary lexicalisations, giving rise to a very large number of permissible outputs. If this is felt to be problem, then ~estricting (in the sense of (Shieber 1985)) the subsumption test in the second half of Definition 3 to ignore the values of certain features, i.e. pred, would bepstraight-forward. This would have the effect of producing a single, exemplary lexicalisation for each significantly different (i.e. different ignoring differences under pred) structure which satisfies the mini-maximal requirements.</Paragraph>
    <Paragraph position="32"> One potential problem clearly arises with this approach. It stems from its dependence on subsumption and its friends. Since subsumption, in at least some standard formulations (e.g. Definite Clause Grammars) fails to distinguish between contingently and necessarily equivalent sub-structures, we will overgenerate in cases where this is the only difference between two analyses, e.g. for Kim expects to go and Kim expects Kim to go on a straight-forward account of Equi.</Paragraph>
    <Paragraph position="33"> One can respond to this either by saying that this is actually correct, that Equi is optional anyway (wishful thinking, I guess), or by adding side conditions to Definition 3 which amount to strengthening subsumption etc. to differentiate between e.g. the two graphs in Figure 5. As I do not at the moment see any way of expressing these side conditions formally without making more assumptions about the nature of underlying forms than I have so far had to (c.f. for example (Shieber 1986) where subsumption is defined in terms of a simulation plus an explicit requirement on the preservation of token identity), I will leave this point unresolved. ,)deg h h h a a a Figure 5. Two structures not distinguished by subsumption</Paragraph>
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