A second step toward the strong polynomial-time hierarchy

Theoretical Computer Science, 1992

Bruschi, D., Strong separations of the polynomial hierarchy with oracles: constructive separations by immune and simple sets, Theoretical Computer Science 102 (1992) 215-252. In this paper we show that the techniques introduced by Furst et al. (1984), which connected oracle separation results for the relativized polynomial-time hierarchy to the problem of proving lower bounds for constant-depth circuits, and the subsequent probabilistic arguments introduced by Yao (1985), Hastad (1986), and Ko (1989) in order to prove the existence of relativized polynomial-time hierarchies with different structures, can be adapted for resolving the main problems related to the existence of immune and simple sets in the relativized polynomial-time hierarchy. In particular, we construct oracles which witness:

Information and Computation, 1989

Mathematical Logic Quarterly, 1998

Motivated by results on interactive proof systems we investigate an 3-V-hierarchy over P using word quantifiers as well as two types of set quantifiers. This hierarchy, which extends the (arithmetic) polynomial-time hierarchy, is called the analytic polynomial-time hierarchy. It is shown that every class of this hierarchy coincides with one of the following Classes: El;, II; (k 2 O) , PSPACE, C y p or IIyp (k 2 1). This improves previous results by Orponen [6] and allows interesting comparisons with the above mentioned results on interactive proof systems.

Problem statement: In this study we discuss the relationship between the known classes P and NP. We show that the difficulties in solving problem "P versus NP" have methodological in nature. An algorithm for solving any problem is sensitive to even small changes in its formulation. Conclusion/Recommendations: As we will show in the study, these difficulties are exactly in the formulation of some problems of the class NP. We construct a class UF that contains P and that simultaneously is strictly contained in NP. Therefore, a new problem arises "P versus UF".

Lecture Notes in Computer Science, 2007

Existing definitions of the relativizations of NC 1 , L and NL do not preserve the inclusions NC 1 ⊆ L, NL ⊆ AC 1 . We start by giving the first definitions that preserve them. Here for L and NL we define their relativizations using Wilson's stack oracle model, but limit the height of the stack to a constant (instead of log(n)). We show that the collapse of any two classes in {AC 0 (m), TC 0 , NC 1 , L, NL} implies the collapse of their relativizations. Next we develop theories that characterize the relativizations of subclasses of P by modifying theories previously defined by the second two authors. A function is provably total in a theory iff it is in the corresponding relativized class. Finally we exhibit an oracle α that makes AC k (α) a proper hierarchy. This strengthens and clarifies the separations of the relativized theories in . The idea is that a circuit whose nested depth of oracle gates is bounded by k cannot compute correctly the (k + 1) compositions of every oracle function.

Theoretical Computer Science, 1983

We consider under the assumption PINP questions concerning the structure of the lattice of NP sets together with the sublattice P. We show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize. The two questions which we consider are whether every infinite NP set contains an infinite P subset and whether there exists an NP-simple set. We construct several oracles, all of which make P I NP, and which in addition make the above-mentioned statements either true or false. In particular we give a positive answer to the question, raised by Bennett and Gill (1981), whether an oracle B exists making PBINPt and such that every infinite set in NPB has an infinite subset in PB. The constructions of the oracles are finite injury priority arguments.

A new class UF of problems is introduced, strictly included in the class NP, which arises in the analysis of the time verifying the intermediate results of computations. The implications of the introduction of this class are considered. First of all, we prove that P  NP and establish that it needs to consider the problem "P vs UF" instead the problem "P vs NP". Also, we determine the set-theoretical of properties of one-way functions that used in cryptology.

Proceedings of Computational Complexity (Formerly Structure in Complexity Theory)

We study two properties of a complexity class-whether there exists a truthtable hard p-selective language for , and whether polynomially-many nonadaptive queries to can be answered by making O log n-many adaptive queries to (in symbols, is PF tt PF O log n). We show that if there exists a NP-hard p-selective set under truth-table reductions, then PF NP tt PF NP O log n. As a consequence, it follows that if there exists a tt-hard p-selective set for NP, then for all k 0 SAT DTIME 2 n log k n. We show that if ZPP NP , then these two properties are equivalent. Also, we show that if there exists a truth-table complete standard-left cut in NP, then the polynomial hierarchy collapses to P NP. We prove that P = NP follows if for some k 0, the class PF NP tt is effectively included in PF NP k log n 1 .

Lecture Notes in Computer Science, 1994

We introduce and study two classifications refining the polynomial hierarchy. Both extend the difference hierarchy over NP and are analogs of some hierarchies from recursion theory. We answer some natural questions on the introduced classifications, e.g. we extend the result of J.Kadin that the difference hierarchy over NP does not collapse (if the polynomial hierarchy does not collapse). * The work was done at the University of Heidelberg and was supported by the Alexander von Humboldt Foundation. I would like to thank K.Ambos-Spies and B.Borchert for discussions and pointing out to me some literature in complexity theory.

Theoretical Computer Science, 1998

In a previous paper the present authors (Baier and Wagner, 1996) investigated an S-V-hierarchy over P using word quantifiers as well as two types of set quantifiers, the so-called analytic polynomial-time hierarchy. The fact that some constructions there result in a bounded number of oracle queries and the recent PCP results which can be expressed by set quantifiers with a bounded number of queries motivated us to examine a hierarchy which extends the analytic polynomial-time hierarchy by considering restrictions on the number of oracle queries. This hierarchy is called bounded anulytic polynomial-time hierarchy. We show that every class from this hierarchy having a certain normal form coincides with one of the classes NP, coNP, PSPACE, Cy or II? (k> 1). All these characterizations remain valid if the queries are asked in a nonadaptive form, i.e. in "parallel".