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<Paper uid="E99-1012">
  <Title>Ambiguous propositions typed</Title>
  <Section position="6" start_page="90" end_page="91" type="evalu">
    <SectionTitle>
5 Variations and refinements
</SectionTitle>
    <Paragraph position="0"> The sequent rules for :: chosen above lie between two extremes. The first is obtained by dropping the side conditions of (I-In), (~-~. np) and (~-'~. nq), rendering the four rules (\[i E) deg, (~-\] Ep) deg, (~ nq) deg and (H E)C/ redundant. The idea is to put off constraint satisfaction to the very end. Alternatively, the side conditions of (I'\[n), (~-~. np), (~-~ n~) and (l-I E)# might be strengthened to check that the constraints are satisfiable (adding to (1-In), for example, the requirement that sum(a) ~ C U C' and eq(a,~') C/ C U C' for all 8' 6 D(/3)). Assuming that we did, we might as well rewrite the relevant d-expressions, and dispense with the subscript C. (For example, with the appropriate side conditions, (\[In) might be revised to r t::a I&amp;quot; F- u::# r\[a := (1J=::#)a\] F- ap(t,=)::a\[x := =1 where F\[a := (I-I x::B)a\] is F with a replaced by (\[i z ::/3)a.) An increase in complexity of the side conditions is a price that we may well be willing to pay to get rid of subscripts C. Or perhaps not.</Paragraph>
    <Paragraph position="1"> Among the considerations relevant to the interplay between inference and constraint satisfaction are: (z) the diffficulty/ease of applying/abusing inference rules (D) the difficulty of disambiguating (i.e. of verifying the assumption in Corollary 4 of a &amp;quot;satisfiable set C&amp;quot; ) (W) wasted effort on spurious readings (i.e. sequents F ~-c O with non-satisfiable C).</Paragraph>
    <Paragraph position="2"> Designing sequent rules balancing (I), (D) and (W) is a delicate language engineering problem, about which it is probably best to keep an open mind.</Paragraph>
    <Paragraph position="3"> Consider again the binary connective * mentioned in the introduction (which we set aside to concentrate instead on certain underspecified representations). It is easy enough to refine the notion of a disambiguation to an e-disambiguation, where e is a function encoding the readings specified by o. In particular, example (1) can be re-conceptualized in terms of (i) the instance F ~-o z::a r I-o y::~ r F{fcn(c~,~)} ap(z,y)::a~{y} of the rule (1&amp;quot;I n) where F is the context x :: a,y::/3, and say, a is % and/3 is a'~ (against the base set of sequents }-e a typ and ~-$ a' typ)  Proceedings of EACL '99 and (ii) an c-disambiguation of a~{y}, where ~(a) = {A --+ B, C} and e(/3) = {A, D}.</Paragraph>
    <Paragraph position="4"> Given a (partial) function e from some set Do of d-expressions to Pow(Ty) - {0}, an e-disambiguation of Do is a disambiguation p of Do such that for every a in the domain of C/, p(a) E e(a). 4 Now, there are at least two ways to incorporate e-disambiguations into Corollary 4. The first is to leave the sequent rules for :: as before, but to relativize the notion of a satisfiable set C of constraints to e (adding to the definition of &amp;quot;p respects C&amp;quot; the requirement that the extension p+ be an e-disambiguation). A more interesting approach is to bring e into the sequent rules by forming constraints to guarantee that disambiguations are e-disambiguations (the general point being that all kinds of information might be encoded within the subscripts C on ~-). For starters, we might change the rule (0c) deg to (Oc)deg I-o, 0 cxt where the subscript 0, e denotes a constraint set requiring that for every a in the domain of e, a can only be disambiguated into an element of e(a). The rules (l-in), (~nv) , (~'~ nq) and (FI E)C/ might then be modified to trim the sets e(a) so that in example (1), for instance, the application of (Fin) reduces e(a) = {A -~ B, C) to</Paragraph>
    <Paragraph position="6"> with the side condition that ~x is non-dependent, and e is consistent with 4 (i.e. for every a in the domain of both e and d, ~(a) n e'(a) # 0) and where C&amp;quot; is C t3 C'U {fcn(a,B)} and e&amp;quot; combines e and e' in the obvious way (e.g. mapping every a in the domain of both C/ and e' to e(a)nd(a)). (Subscripts C, e may, as in the case of 0, C/, be construed as single constraint sets, which are convenient for certain purposes to decompose into pairs C, e.) We could put a bit more work into (Fin) as follows. Given an integer k &gt; 0, let Du(/3) be 4We can also introduce, as a binary connective on u-expressions and/or on d-expressions, although this would require a bit more work and would run against the spirit of underspecified representations, insofar as * spells out possible disambiguations.</Paragraph>
    <Paragraph position="7"> the subset of the set D(~) of sub-d-expressions of B, from which ~ can be constructed with &lt; k applications of d-expression formation rules. (For example, D1 ((~ x :: a)(It Y ::/3)7) is with ~ and 7 buried too deeply to be included.) Now, for a fixed k, add to the side condition of (l'\]n) the requirement that sum(a) 9~ C U C' and eq(a, ff) 9~ C U C' for all/3' e Dk(/~); and choose e&amp;quot; to also rule out the possibility that a is ff for some ff E Dk(~). Clearly, the larger k is, the stronger the rule becomes. But so long as a satisfiability check is made after inference (as suggested by Corollary 4), it is not necessary that the constraint set C in a sequent F I-c O that has been derived be reduced (to make all its consequences explicit) any more than it is necessary to require that C be satisfiable. (Concerning the latter, notice also that spurious sequents may drop out as further inferences are made, eliminating the need there to ever disambiguate.) To establish (the analog of) Corollary 4, a crucial property for a sequent rule rl t-cl O1 --- r, t-c. O, (,) r -cO to have is monotonicity: for every disambiguation p respecting C, p respects Ci for 1 &lt; i &lt; n. s (This is a generalization of Ci _C C, suggested by the encoding above of e-disambiguations/, in terms of constraints.) To weed out spurious readings (consideration (W) above), side conditions might be imposed on (*), which ought (according to (I)) to be as simple as possible. The trick in designing C (and (*)) is to make inference }- just complicated enough so as, (D), not to put an undue burden on disambiguating at the end. The whole idea is to distribute the work between inferring sequents and (subsequently) checking satisfiability. The claim is that the middle ground between the two extremes mentioned at the beginning of this section (i.e. between lax side conditions that leave the bulk of the work to disambiguation at the end, and strict side conditions that essentially reduce:: to :) is fertile.</Paragraph>
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