Upward Separation for FewP and Related Classes

This paper studies the range of application of the upward separation technique that has been introduced by Hartmanis to relate certain structural properties of polynomial-time complexity classes to their exponential-time analogs and was rst applied to NP [Har83]. Later work revealed the limitations of the technique and identi ed classes defying upward separation. In particular, it is known that coNP as well as certain promise classes such as BPP, R, and ZPP do not possess upward separation in all relativized worlds [HIS85; HJ93], and it had been suspected that this was also the case for other promise classes such as UP and FewP [All91]. In this paper, we refute this conjecture by proving that, in particular, FewP does display upward separation, thus providing the rst upward separation result for a promise class. In fact, this follows from a more general result the proof of which heavily draws on Buhrman, Longpr e, and Spaan's recently discovered tally encoding of sparse sets. As consequences of our main result, we obtain upward separations for various known counting classes such as P, coC=P, SPP, and LWPP. Some applications and open problems are discussed.