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<Paper uid="E99-1012">
  <Title>Ambiguous propositions typed</Title>
  <Section position="3" start_page="86" end_page="88" type="metho">
    <SectionTitle>
2 Two systems
</SectionTitle>
    <Paragraph position="0"> Let us begin with a system of dependent types, confining our attention to three forms of judgments, F context, A type and t:A. (That is, for simplicity, we leave out equations between types and between terms.) Contexts can be formed from the empty sequence ()</Paragraph>
    <Paragraph position="2"> and contexts weaken to the right</Paragraph>
    <Paragraph position="4"> (where O ranges over judgments A type and t : A). Next come formation (F), introduction (I) and elimination (E) rules for dependent functions</Paragraph>
    <Paragraph position="6"> and for dependent sums Y\]. (generalizing Carte-</Paragraph>
    <Paragraph position="8"> Now for the novel part: a second system, with terms t as before, but colons squared, and :types A, B replaced by decorated expressions a, j3 and unadorned expressions .4 generated simultaneously according to</Paragraph>
    <Paragraph position="10"> where a belongs to a fixed countable set X of variables. The intent (made precise in the next section) is that a u-expression .4 describes a set of :-types, while a d-expression a denotes a choice from such a set. D-expressions of the form a~, a p, aq{t} and a/~{t} are said to be non-dependent, and are used, in conjunction with constraints of the form fcn(a,/3), sum(a) and eq(a,/3), to infer sequents relativized to finite sets C of constraints as follows</Paragraph>
    <Paragraph position="12"> where each of the three rules have the side condition that a is non-dependent. 1 In addition, r Fc t::(I'\[z::a)X~ r Fc, u::~r (HE)C/ r FCUC'U{eq(a,'y)} ap(t,u)::~\[x := u\] with the side condition a # % The intuition (formalized in clauses (c2)-(c4) of the next section) is that - the constraint eq(a, 7) is satisfied by a disambiguation equating a with % - fcn(a, i3) is satisfied by a disambiguation of (~ and/3 to :-types of the form (H z : A)B and A respectively and - sum(a) is satisfied by a disambiguation of a to a :-type of the form (~-'~ x: A)B).</Paragraph>
    <Paragraph position="13"> Rules of the previous system translate to</Paragraph>
    <Paragraph position="15"> Proceedings of EACL '99 Further rules provide co-varying choices F l-c t::a z C/ Vat(r) (::c) l-cC, z::a cxt (YIc) l-c r,x::a cxt r,x::a l-o t::t~ l-cuc' r,y::(l'Ix::a)/~ cxt (~c) l-c r,x::a cxt r,x::a t-o t::t3 t-cuc, r,y::(5:~::a)t~ C/xt ' where (Hc) and (~&amp;quot;\].c) each have the side condition y C/ Var(r) u {z}.</Paragraph>
  </Section>
  <Section position="4" start_page="88" end_page="89" type="metho">
    <SectionTitle>
3 Disambiguating ::
</SectionTitle>
    <Paragraph position="0"> Let Ty be the collection of :-type expressions A,  and for every d-expression a, let - X(a) be the set of variables in 2:' occurring in a - D(a) be the set of (sub-)d-expressions/~ occurring in a (including a) and - U(a) be the set of (sub-)u-expressions A occurring in a.</Paragraph>
    <Paragraph position="1">  and for ()-~.=::a)A e U(ao), ((~-~x::a)A) p = {(Zx:p(a))A I A e AP} . Now, call p a disambiguation of ao if the following conditions hold: (i) for every A= E D(a0), p(,4=) E A p (ii) for every (1FIx::a)/3 E D(ao),</Paragraph>
    <Paragraph position="3"> Next, let us pass from a single d-expression ao to a fixed set Do of d-expressions. A disambiguation of the set Do of d-expressions is a function p from U{D(a) \] a E Do} to Ty such that for all a E Do, p restricted to D(a) is a disambiguation of a. 2 A disambiguation p of Do respects a set C of constraints if there is an extension p+ _D p so</Paragraph>
    <Paragraph position="5"> Given a sequence F of the form Xl:el, ...~Xn:an~ let irna(F) = {al,...,an}, and for every disambiguation p of a set Do containing ima(F), let</Paragraph>
    <Paragraph position="7"> Let us say that l-c F cxt can be disambiguated to l- F' context if there is a disambiguation p of ima(F) respecting C such that F' = Fp. Similarly, F l-c a typ (t :: a) can be disambiguated to F' l-A type (t : A) if there is a disambiguation p of irna(F) U {a} respecting C such that F' = Fp and A = p(a).</Paragraph>
    <Paragraph position="8"> 2It is crucial for this formulation that the set Var(F) mentioned in side conditions for various rules in the previous section include all variables in P, whether they occur freely or bound.</Paragraph>
  </Section>
  <Section position="5" start_page="89" end_page="90" type="metho">
    <SectionTitle>
4 Relating the derivations
</SectionTitle>
    <Paragraph position="0"> Observe that to derive a sequent other than }0 context in the first system, or ~C/ 0 cxt in the second, we need to assume a non-empty set 7&amp;quot; of sequents. Let us agree to write F ~_r O to mean that the sequent F }- O is derivable from T, and ~_T F context to mean that }- F context is derivable from 7&amp;quot;. Similarly, for the second system (with ~- replaced by ~-c, context by cxt, etc). As every rule (R) for the first system has a counterpart (R) deg in the second system, it is tempting to seek a natural translation .deg from the first system to the second system validating the following Claim: F ~-?&amp;quot; O implies F deg ~-~'deg 0% For example, if 7&amp;quot; consists of the sequent ~- A type, F is empty, and O is Az.x: (\[i z:A)A, then 7&amp;quot;o is {~-C/ a typ}, F deg is empty, and O deg is Ax.z :: (I\] x :: ax)ax. Replacing F by y:A, and O by ~z.y:(YIx: A)A, we get y :: ay for F deg and ~z.y :: (l'I x :: az)% for 0% To pin down a systematic definition of .deg, it is easy enough to fix a 1-1 mapping X ~4 a x of atomic :-types X to variables a x in ~Y, and set</Paragraph>
    <Paragraph position="2"> While (11) induces a translation F deg of a context F, what about (t : A) deg, where t is not just, as in (11), a variable x? Before revising the definition of d-expressions a to accommodate subscripts t on A deg, let us explore what we can do with (7)(11). Define a simple type base 7&amp;quot; to be a set of sequents of the form F ~- A type. Given a simple type base 7&amp;quot;, let 7&amp;quot;0 be its translation into :: according to equations (11) and (10). By induction on derivations from 7&amp;quot;, we can prove a reformulation of the claim above, where F deg and O deg are replaced by disambiguations.</Paragraph>
    <Paragraph position="3"> Proposition 1. Let 7&amp;quot; be a simple type base.</Paragraph>
    <Paragraph position="4"> (a) r context implies ~0 F' cxt for some F' such that ~-o F' cxt can be disambiguated to F context.</Paragraph>
    <Paragraph position="5">  (b) F ~T A type implies F' ~deg a typ for some r' and a such that F' ~-0 a typ can be disambiguated to F ~ A type.</Paragraph>
    <Paragraph position="6"> (c) F ~_ 7&amp;quot; t : A implies F' ~-o ~ t :: a for some F' and a such that F' ~-o t :: a can be disambiguated to F ~- t:A.</Paragraph>
    <Paragraph position="7">  Moreover, as the rules (1-In), (~\] nv) and (~ nq) can, for disambiguations that meet the appropriate constraints, be replaced by (1&amp;quot;I E), (~\] Ep) and  (~ Eq), it follows that Proposition 2. Let 7&amp;quot; be a simple type base. (a) /f ~-c ~ F cxt and \[-c F cxt can be d/sambiguated to ~- F' context, then ~&amp;quot; F' context. (b) Ifr ~- C/ T~ a typ and r ~-c a typ can be disambiguated to F' ~- A type, then F' ~_T A type. (c) Ifr \[--c rdeg t::a andr ~-c t::a can be disambiguated to r' F- t:A, then F' ~_r t:A.</Paragraph>
    <Paragraph position="8"> Conversely, going from (liE) deg, (~Ep) deg and (E Eq) deg to (\[in), (Y\]~ np) and ()-~ nq), we have Proposition 3. Let 7&amp;quot; be a simple type base. (a) /f ~_r r' context and ~-c r cxt can be disambiguated to ~- F' context, then ~-c ydeg F cxt. (b) IfF' ~_7&amp;quot; A type and P ~-c a typ can be disambiguated to r' S A type, then P ~-~ a typ.</Paragraph>
    <Paragraph position="9"> (c) If F' ~-~&amp;quot; t : A and F ~-c t :: a can be disam- null biguated to F' ~- t:A, then F ~o t::t~.</Paragraph>
    <Paragraph position="10"> Proposition 3(c) is roughly ~ of (3), while Proposition 2(c) approximates =~ of (4). If Proposition 2 says that the system for :: above is sound, Proposition 3 says it is complete. 3 To tie together Propositions 2 and 3 in an equivalence, it is useful to define a set C of constraints to be satisfiable if 0 is a disambiguation (of 0) respecting C. Note that sequents ~-c F and F ~-c e have disambiguations exactly when C is satisfiable. Consequently, Propositions 2 and 3 yield (focussing on ::) Corollary 4. Given a simple type base 7&amp;quot; and a satisfiable set C of constraints, the following are</Paragraph>
    <Paragraph position="12"> which F ~-c t::a can be disambiguated (iii) F' ~_T t : A, for some sequent PS' ~- t : A to which F ~-c t::a can be disambiguated.</Paragraph>
    <Paragraph position="13"> SAs for how this relates to soundness and completeness in say, classical predicate logic, please see the discussion of translation versus entailment in the concluding paragraph below.</Paragraph>
    <Paragraph position="14">  The formulation above of Corollary 4 depends on the possibility of deriving sequents F ~c O where C is not satisfiable. We could have, of course, added side conditions to (1-In), (~-~. nj,) and (~&amp;quot;~ nq) checking that the constraints are satisfiable. By electing not to do so, we have exposed a certain separability of inference from constraint satisfaction, which we will explore in the next section. For now, turning to the general case of a set T of :-sequents, observe that if 7&amp;quot; is to be compatible  with the first system, then (i) whenever F }- Ax.t:C belongs to 7&amp;quot;, C must have the form (rI x:A)B with</Paragraph>
    <Paragraph position="16"> C must have the form (~\[: z: A)B with F \]_r t:A and F }_.7&amp;quot; u:B\[x := t\] whenever F }- ap(t,u):B belongs to 7&amp;quot;, F \]_r t : (1-\[ x : A)B for some x and A such that F \]_'r u: A whenever F ~- p(t) :A belongs to T, F }_7&amp;quot; t:(~\]x:A)B for some x and B whenever P }- q(t):B belongs to T, F ~_r t:(~_,x:A)B for some x and A whenever F ~- e belongs to 7&amp;quot;, ~'r F context whenever ~- F,x : A context or F ~- t : A belongs to T, F ~_7&amp;quot; A type  Thus, a base set T compatible with the first system can be assumed without loss of generality to consist of sequents of two forms: F ~ A type and F }- t: B, where A and t are atomic (i.e. indecomposable by I-i, ~ and A, (,), ap,p, q respectively). By clause (vii) above, it follows that for every sequent F ~- t : B in T, there is some To C_ T such that F ~_7~ B type. So starting with simple type bases To, we can take (for B) the Dexpression/3 which Proposition l(b) returns, given F \[-% B type. We can then define T deg by translating F ~- t:B as F deg }- t ::/3. Alternatively, we might make do with simple type bases by reformulating t as a variable xt, and smuggling zt into enriched contexts F' for which a T-derivation of F' ~- O' is sought (with O' adjusted for zt, rather than t). That is, instead of injecting t on top of \]- (within some superscript 7&amp;quot;), we might add it (along with the context it depends on) to the left of ~-.</Paragraph>
  </Section>
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