Ralph Loader, The undecidability of #-definability, Church Memorial Volume, A. Anderson, M. Zeleny eds., Kluwer Acad. Press, to appear.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....range problem ) was also investigated by Richard Statman [22] who reduced its decidability to decidability of lambda definability. Moreover, the decidability of higher order matching with so called delta functions is equivalent to the decidability of lambda definability. Unfortunately Ralf Loader [13] gave a negative answer to the later question. Thus matching with delta functions is undecidable. Some other extensions of matching are also known to be undecidable, e.g. matching in G odel s system T and Girard s system F [7] Statman is also the author of a theorem stating that deciding ....

Ralph Loader, The undecidability of #-definability, Church Memorial Volume, A. Anderson, M. Zeleny eds., Kluwer Acad. Press, to appear.

....full type hierarchies: given an element a of a finite structure, it turns out to be co r.e. whether the pair ha; ai belongs to all binary relations which are obtainable by closing logical relations under intersection and projection (and hence by closure under composition) while by Prop. 5. 9 and [Loa9 ] it is not co r.e. whether ha; ai belongs to all binary pre logical relations. In the case of arbitrary full type hierarchies, the question is open: the proof of Theorem 7.1 fails if we take a full type hierarchy in place of A[X ] and we conjecture that Corollary 7.2 does not hold. For ....

R. Loader. The undecidability of -definability. Church Memorial volume, to appear (199?).

....are not closed under composition by Example 5.4, and since composition is definable in terms of product, intersection and projection it follows that they are not closed under projection. Failure of closure under restriction to a substructure follows from decidability and undecidability results in [Loa9 ]. A consequence of closure under intersection is that given a property P of relations that is preserved under intersection, there is always a least pre logical relation satisfying P . We then have (see Example 3.4 above) Proposition 5.7. The least pre logical predicate over a given lambda Sigma ....

....finite full type hierarchies: given an element a of a finite structure, it turns out to be co r.e. if the pair ha; ai belongs to all binary relations which are obtainable by closing logical relations under intersection and projection (and hence by closure under composition) while by Prop. 5. 7 and [Loa9 ] it is not co r.e. if ha; ai belongs to all binary pre logical relations. In the case of arbitrary full type hierarchies, the question is open: the proof of Prop. 6.1 fails if we take a full type hierarchy in place of A[X ] and we conjecture that Corollary 6.2 does not hold. For non extensional ....

R. Loader. The undecidability of -definability. Church Memorial volume, to appear (199?).

....under product, permutation and 8 but not under intersection, projection or restriction to a substructure. 2 Other classes of relations satisfy different closure properties. For instance, Irelations are closed under intersection as well as product, permutation and 8. According to Prop. 14 of [Loa9 ], given a full type hierarchy A over a finite base type and an element a in A, it is decidable whether a belongs to all I predicates over A. It follows that I relations are not closed under composition or projection, or restriction to a substructure. Otherwise the pre logical predicate ....

....type hierarchies. In fact, given an element a of a finite structure, it turns out to be decidable if the pair ha; ai belongs to all binary relations which are obtainable by closing logical relations under intersection and projection (and hence by closure under composition) while by Prop. 5. 7 and [Loa9 ] it is not decidable if ha; ai belongs to all binary pre logical relations. This is not an easy result, but it is peripheral to the main topic of this paper so we omit its proof and other interesting results about full type hierarchies for lack of space. For non extensional structures the ....

R. Loader. The undecidability of -definability. Church Memorial volume, to appear (199?).

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC