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<Paper uid="P83-1003">
  <Title>Crossed Serial Dependencies: i low-power parseable extension to GPSG</Title>
  <Section position="3" start_page="0" end_page="16" type="metho">
    <SectionTitle>
II. AN EXTENSION TO GPSG
II.I Extendin G the s~ntax
</SectionTitle>
    <Paragraph position="0"> GPSG includes the idea of compound non-terminals, composed of pairs of standard category labels. We can extend this trivially to finite sequences of category labels. This in itself does not change the weak generative capacity of the grammar, as the set of non-terminals remains finite. CPSG also includes the idea of rule schemata - rules with variables over categories. If we further allow variables over sequences, then we get a real change.</Paragraph>
    <Paragraph position="1"> At this point I must introduce some notation. I will write \[a,b ,c\] for a non-terminal label composed of the categories a, b, and c. I will write Za b* to indicate that the schematic variable Z ranges over sequences of the category b. We can then give the following grammar for anb n with crossed  where we allow variables over sequences to appear not only alone, but in simple, that is with constant terms only, concatenation, notated with a vertical bar (I). This grammar gives us the following analysis for a3b 5, where I have used subscripts to record the dependencies, and the marginal numbers give the rule which admits the</Paragraph>
    <Paragraph position="3"> With the aid of this example, we see that rule I generates a's while accumulating b's, rule 2 brings this process to an end, and rule 5 successively generates the accumulated b's, in the correct, 'crossed', order. This is essentially the structure we will produce for the Dutch examples as well, so it is important to point out exactly how the crossed dependencies are captured. This must come out in two ways in GPSG - subcategorisation restrictions, and interpretation. That the subcategorisation is handled properly should be clear from the above example. Suppose that the categories a and b are pre-terminals rather than terminals, and that there are actually three sorts of a's and three sorts of b's, subcategorised for each other. If one used the standard GPSG mechanism for recording this dependency, namely by providing three rules, whose rule number would then appear as a feature on those pre-terminals appearing in them directly, we would get the above structure, where we can reinterpret the subscripts as the rule numbers so introduced, and see that the dependencies are correctly reflected.</Paragraph>
    <Paragraph position="4">  As for the semantics no actual extension is required - the untyped lambda calculus is still sufficient to the task, albeit with a fair amount of work. We can use what amounts to apa ...... 6 and unpacking approach. The compound b nodes have compound interpretations, which are distributed appropriately higher up the tree. For this, we need pairs and sequences of interpretations.</Paragraph>
    <Paragraph position="5"> Following Church, we can represent a pair &lt;l,r&gt; as ~f(1)(r)\]. If P is such a pair, then PO P(~x~x\[x\]) and PI = P(kxXx\[y\]). Using pairs we can of course produce arbitrary sequences, as in Lisp. In what follows I will use a Lisp-based shorthand, using CAR, CDR, CONS, and so on. These usages are discharged in Appendix I.</Paragraph>
    <Paragraph position="6"> Using this shorthand, we can give the following example of a set of semantic rules for association with the syntactic rules given above, which preserves the appropriate dependency, assuming that the b'(a',S') is the desired result at each level:</Paragraph>
  </Section>
  <Section position="4" start_page="16" end_page="19" type="metho">
    <SectionTitle>
CONS(CADR (Q')(a' )(CA~(Q' )),CDDR (Q ' )) (~
</SectionTitle>
    <Paragraph position="0"> where Q' is short for SI, Z~,b ' , CO~S(CAR (Q ' )(a') (S') ,CDR(Q ' )) (2 where Q' is short for Ziqh ' , ADJOIN(Z' ,b' ). (3 These rules are most easily understood in reverse order. Rule 3 simply appends the interpretation of the immediately dominated b to the sequence of interpretations of the dominated sequence of b's. Rule 2 takes the first interpretation of such a sequence, applies it to the interpretations of the immediately dominated a and S, and prepends the result to the unused balance of the sequence of b interpretations. We now have a sequence consisting of first a sentential interpretation, and then a number of h interpretations. Rule I thus applies the second (b type) element of such a sequence to the interpretation of the immediately dominated a, and the first (S type) element of the sequence.</Paragraph>
    <Paragraph position="1"> The result is again prepended to the unused balance, if any. The patient reader can satisfy himself that this will produce the following  (crossed) interpretation:</Paragraph>
    <Paragraph position="3"> As for parsing context-free grammars with the non-terminals and schemata this proposal allows, very little needs to be added to the mechanisms I have provided to deal with non-sequence schemata in GPSG, as described in (Thompson 1981 b). We simply treat all non-terminals as sequences, many of only one element. The same basic technique of a bottom-up chart parsing strategy, which substitutes for matched variables in the active version of the rule, will do the job. By restricting only one sequence variable to occur once in each nonterminal, the task of matching is kept simple and deterministic. Thus we allow e.g. SIZIb but not ZlblZ. The substitutions take place by concatenation, so that if we have an instance of rule (~) matching first \[a\] and then \[3,b,b,b\] in the course of bottom-up processing, the Z on the right hand side will match \[b,b\], and the resulting substitution into the left hand side will cause the constituent to be labeled \[S,b,b\].</Paragraph>
    <Paragraph position="4"> In making this extension to my existing system, the changes required were all localised to that part of the code which matches rule parts against nodes, and here the price is paid only if a sequence variable is encountered. This suggests that the impact of this mechanism on the parsing complexity of the system is quite small.</Paragraph>
    <Paragraph position="5"> III. APPLICATION TO DUTCH Given the limited space available, I can present only a very high-level account of how this extension to GPSG can provide an account of crossed serial dependencies in Dutch. In particular I will have nothing to say about the difficult issue of the precise distribution of tensed and untensed verb forms.</Paragraph>
    <Paragraph position="6"> III. 1 The Dutch data Discussion of the phenomenon of crossed serial dependencies in Dutch subordinate clauses is bedeviled by considerable disagreement about just what the facts are. The following five examples form the core of the basis for my analysis: I) omdat ik probeer Nikki te leren Nederlands  spreken.</Paragraph>
    <Paragraph position="7"> With the proviso that (I) is often judged questionable, at least on stylistic grounds, this pattern of judgements seems fairly stable among native speakers of Dutch from the Netherlands.</Paragraph>
    <Paragraph position="8"> There is some suggestion that this is not the pattern of judgements typical of native speakers of Dutch from Belgium.</Paragraph>
    <Paragraph position="9"> III.2 Grammar rules for the Dutch data This pattern leads us to propose the following basic rules for subordinate clauses:  Taken straight, these give us (I) only. For (2) - (4), we propose what amounts to a verb lowering approach, where verbs are lowered onto VPs, whence they lower again to form compound verbs. (5) is ruled out by requiring that a lowered verb must have a target verb to compound with. The resulting compound may itself be lowered, but only as a unit. This approach is partially inspired by Seuren's transformational account in terms of predicate raising (Seuren 1972).</Paragraph>
    <Paragraph position="10"> So the interpretation of the compound labels is that e.g. \[V,V\] is a compound verb, and \[VP,V,V! is a VP with a compound verb lowered onto it. It follows that for each VP rule, we need an associated compound version which allows the lowering of (possibly compound) verbs from the VP onto the verb, so we would have e.g.</Paragraph>
    <Paragraph position="11"> Di) VPIZ -&gt; NP ZIV, where we now use Z as a variable over sequences of VS. The other half of the process must be  reflected in rules associated with each VP rule which introduces a VP complement, allowing the verb to be lowered onto the complement. As this rule must also expand VPs with verbs lowered onto them, we want e.g.</Paragraph>
    <Paragraph position="12"> cii) vPlz -&gt; ~P wlzlv.</Paragraph>
    <Paragraph position="13"> Rather than enumerate such rules, we can use metarules to conveniently express what is wanted:</Paragraph>
    <Paragraph position="15"> (I) will apply to all three of (B) - (D), allowing compound verbs to be discharged at any point. (II) will apply to (B) and (C), allowing the lowering (with compounding if needed) of verbs onto complements. We need one more rule, to unpack the compound verbs, and the syntactic part of our effort is complete: E) wlz -&gt; W Z, where W is an ordinary variable whose range consists of V. This slight indirection is necessary to insure that subcategorisation information propagates correctly.</Paragraph>
    <Paragraph position="16"> By suitably combining the rules (A) - (E), together with the meta-generated rules (Bi) - (Di), (Bii) and (Cii), we can now generate examples (2) (4). (4), which is fully crossed, is very similar to the example in section II.1, and uses meta-generated expansions for all its VP nodes:  Once again I include the relevant rule name in the margin, and indicate with subscripts the rule name feature introduced to enforce subcategorisation. Sentences (2) and (3) each involve two meta-generated rules and one ordinary one. For reasons of space, only (3) is illustrated below. (2) is similar, but using rules (B), (Cii), and (Di).</Paragraph>
    <Paragraph position="18"> pro~eer ~c . !preken te leren Nederlands te III.3 Semantic rules for the Dutch data The semantics follows that in section II.2 quite closely. For our purposes simple interpretations  of (B) - (D) will suffice: B') v'(vP') c') v' (NP' ,~') D') v'(NP').</Paragraph>
    <Paragraph position="19">  The semantics for the metarules is also reasonably straightforward, given that we know where we are going: I') F(V') ==&gt; CONS(F(CAR(Z:V')),CDR(Z',V')) II') F(V',VP') ==&gt; CONS(F(CADR(Q'),CAR(Q')), cm~(Q')), where Q' is short for VPlZl, V '. (I') will give semantics very much like those of rule (2) in section II.2, while (II') will give semantics like those of rule (I). (E deg) is just like (3): E') ADJ01N(Z' ,W ' ) It is left to the enthusiastic reader to work through the examples and see that all of sentences (I) - (4) above in fact receive the same interpretation.</Paragraph>
    <Paragraph position="20"> III.4 Which structure is right - evidence from conjunction The careful reader will have noted that the structures proposed are not the same as those of BKPZ. Their structures have the compound verb depending from the highest VP, while ours depend from the lowest possible. With the exception of BKPZ's example (~3), which none of my sources judge grammatical with the 'root Marie' as given, I  believe my proposal accounts for all the judgements cited in their paper. On the other hand, I do not believe they can account for all of the following conjunction judgement, the first three based on (4), the next two on (3), whereas under the standard GPSG treatment of conjunction they all fall out of our analysis: 6) omdat ik Nikki Nederlanda wil leren spreken en Frans wil laten schrijven because I want to teach Nikki to speak Dutch and let \[Nikki\] write French 7) * omdat ik Nikki Nedrelands wil leren spreken en Frans laten schrijven 8) omdat ik Nikki Nederlands wil leren spreken en Carla Frans wil laten schrijven because I want to teach Nikki to speak Dutch and let Carla write French.</Paragraph>
    <Paragraph position="21"> 9) omdat ik Nikki wil leren Nederlands te spreken en Frans te schrijven because I want to teach Nikki to speak Dutch and to write French IO) * omdat ik Nikki wil leren Nederlands te spreken en Carla Frans te schrijven or ... en Frans (ts) laten schrijven (6) contains a conjoined \[VP,V,V\], (8) a conjoined \[VP,V\], and (7) fails because it attempts to conjoin a \[VP,V,V\] with a \[VP,V\]. (9) conjoins an ordinary VP iaside a \[VP,V\], and (10) fails by trying to conjoin a VP with either a non-constituent or a \[VP,V\].</Paragraph>
    <Paragraph position="22"> It is certainly not the case that adding this small amount of 'evidence' to the small amount already published establishes the case for the deep embedding, but I think it is suggestive. Taken together with the obvious way in which the deep embedding allows some vestige of compositionality to persist in the semantics, I think that at the very least a serious reconsideration of the BKPZ proposal is in order.</Paragraph>
  </Section>
  <Section position="5" start_page="19" end_page="19" type="metho">
    <SectionTitle>
IV. CONCLUSIONS
</SectionTitle>
    <Paragraph position="0"> It is of course too early to tell whether this augmentation will be of general use or significance. It does seem to me to offer a reasonably concise and satisfying account of at least the Dutch phenomena without radically altering the grammatical framework of GPSG.</Paragraph>
    <Paragraph position="1"> Further work is clearly needed to exactly establish the status of this augmented GPSG with respect to generative capacity and parsability. It is intriguing to speculate as to its weak equivalence with the tree adjunction grammars of Joahi et al. Even in the weakest augmentation, allowing only one occurence of one variable over sequences in any constituent of any rule, the apparent similarity of their power remains to be formally established, but it at least appears that like tree adjunction grammars, these grammars cannot generate anbncn with both dependencies crossed, and like them, it can generate it with any one set crossed and the other nested. Neither can it generate WW, although it can with a sequence variable ranging over the entire alphabet, if it can be shown that it is indeed weakly equivalent to TAG, then strong support will be lent to the claim that an interesting new point on the Chomsky hierarchy between CFGs and the indexed grammars has been found.</Paragraph>
  </Section>
  <Section position="6" start_page="19" end_page="19" type="metho">
    <SectionTitle>
ACKNOWLEDGEMENTS
</SectionTitle>
    <Paragraph position="0"> The work described herein was partially supported by SERC Grant GR/B/93086. My thanks to Han Reichgelt, for renewing my interest in this problem by presenting a version of Seuren's analysis in a seminar, and providing the initial sentential data; to Ewan Klein, for telling me about Church's 'implementation' of pairs and conditionals in the lambda calculus; to Brian Smith, for introducing me to the wonderfully obscure power of the Y operator; and to Gerald Gazdar, Aravind Joshi, Martin Kay and Mark Steedman, for helpful discussion on various aspects of this work.</Paragraph>
  </Section>
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