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Optimal proof systems imply complete sets for promise classes
Information and Computation
SIAM Journal on Computing
Reductions between disjoint NP-Pairs
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Fine hierarchy of regular aperiodic ω-languages
DLT'07 Proceedings of the 11th international conference on Developments in language theory
Inverting onto functions and polynomial hierarchy
CSR'07 Proceedings of the Second international conference on Computer Science: theory and applications
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We study the shrinking and separation properties (two notions well-known in descriptive set theory) for and and show that under reasonable complexity-theoretic assumptions, both properties do not hold for and the shrinking property does not hold for . In particular we obtain the following results. 1and do not have the shrinking property, unless is finite. In general, and do not have the shrinking property, unless is finite. This solves an open question from [25].1The separation property does not hold for , unless .1The shrinking property does not hold for , unless there exist -hard disjoint -pairs (existence of such pairs would contradict a conjecture by Even, Selman, and Yacobi [6]).1The shrinking property does not hold for , unless there exist complete disjoint -pairs.Moreover, we prove that the assumption is too weak to refute the shrinking property for in a relativizable way. For this we construct an oracle relative to which , , and has the shrinking property. This solves an open question by Blass and Gurevich [2] who explicitly ask for such an oracle.