Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S 2 = S 2 . Building on this, we strengthen the Kamper-- AFK Theorem, namely, we prove that if NP coNP)/poly then the polynomial hierarchy collapses to S 2 . We also strengthen Yap's Theorem, namely, we prove that if NP coNP/poly then the polynomial hierarchy collapses to S 2 . Under the same assumptions, the best previously known collapses were to ZPP respectively ([KW98, BCK 94], building on [KL80, AFK89, Kam91, Yap83]). It is known that S 2 [Cai01]. That result and its relativized version show that our new collapses indeed improve the previously known results. Since the Kamper-- AFK Theorem and Yap's Theorem are used in the literature as bridges in a variety of results---ranging from the study of unique solutions to issues of approximation---our results implicitly strengthen all those results.

We show the following new lowness results for the probabilistic class ZPP NP . { The class AM \ coAM is low for ZPP NP . As a consequence it follows that Graph Isomorphism and several group-theoretic problems known to be in AM \ coAM are low for ZPP NP . { The class IP[P=poly], consisting of sets that have interactive proof systems with honest provers in P=poly, is also low for ZPP NP . We consider lowness properties of nonuniform function classes, namely, NPMV=poly, NPSV=poly, NPMV t =poly, and NPSV t =poly. Specifically, we show that { Sets whose characteristic functions are in NPSV=poly and that have program checkers (in the sense of Blum and Kannan [8]) are low for AM and ZPP NP . { Sets whose characteristic functions are in NPMV t =poly are low for p 2 .

We show that the class AM\coAM is low for ZPP . As a consequence, it follows that Graph Isomorphism and several group-theoretic problems are low for ZPP . We also

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy [BBS86] than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL2, yet their join is in EL2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL2 and prove that EL2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL2 lower bounds for certain notions generalizing Selman’s P-selectivity [Sel79], which may be regarded as an interesting result in its own right. 1

Assuming that k 2 and \Delta P k does not have p-measure 0, it is shown that BP \Delta \Delta P k = \Delta P k . This implies that the following conditions hold if \Delta P 2 does not have p-measure 0. (i) AM " co-AM is low for \Delta P 2 . (Thus BPP and the graph isomorphism problem are low for \Delta P 2 .) (ii) If \Delta P 2 6= PH, then NP does not have polynomial-size circuits. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International, Microware Systems Corporation, and Amoco Foundation. 1 Introduction Many widely believed conjectures in computational complexity are "strong" in the sense that they are known to imply that P 6= NP, but are not known to follow from the P 6= NP hypothesis. Recent investigations have shown that a number of these conjectures do follow from the (apparently) stronger hypothesis that NP does not have p-measure 0. (This hypothesis, written ¯ p (NP) 6= 0, is defined in...

For a given set of strings, the problem of obtaining a succinct description becomes an important subject of research, related to several areas of theoretical computer science. In structural complexity theory, researchers have developed a reasonable framework for studying the complexity of these problems. In this paper, we survey how such investigation has proceeded, and explain the current status of our knowledge.

Abstract. A partial information algorithm for a language A computes, for some fixed m, for input words x1,..., xm a set of bitstrings containing χA(x1,..., xm). E.g., p-selective, approximable, and easily countable languages are defined by the existence of polynomial-time partial information algorithms of specific type. Self-reducible languages, for different types of self-reductions, form subclasses of PSPACE. For a self-reducible language A, the existence of a partial information algorithm sometimes helps to place A into some subclass of PSPACE. The most prominent known result in this respect is: P-selective languages which are self-reducible are in P [9]. Closely related is the fact that the existence of a partial information algorithm for A simplifies the type of reductions or self-reductions to A. The most prominent known result in this respect is: Turing reductions to easily countable languages simplify to truth-table reductions [8]. We prove new results of this type. We show: 1. Self-reducible languages which are easily 2-countable are in P. This partially confirms a conjecture of [8]. 2. Self-reducible languages which are (2m − 1, m)-verbose are truthtable self-reducible. This generalizes the result of [9] for p-selective languages, which are (m + 1, m)-verbose. 3. Self-reducible languages, where the language and its complement are strongly 2-membership comparable, are in P. This generalizes the corresponding result for p-selective languages of [9]. 4. Disjunctively truth-table self-reducible languages which are 2-membership comparable are in UP. Topic: Structural complexity 1