Oracle Queries Research Articles

Bounding queries in the analytic polynomial-time hierarchy

In a previous paper the present authors (Baier and Wagner, 1996) investigated an ∃-∀-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 analytic 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, Σ k exp or Π k exp ( k ⩾ 1). All these characterizations remain valid if the queries are asked in a nonadaptive form, i.e. in “parallel”.

Relative to a Random Oracle, NP Is Not Small

Resource-boundedmeasure as originated by Lutz is an extension of classical measure theory which provides a probabilistic means of describing the relative sizes of complexity classes. Lutz has proposed the hypothesis that NP does not havep-measure zero, meaning loosely that NP contains a non-negligible subset of exponential time. This hypothesis implies a strong separation of P from NP and is supported by a growing body of plausible consequences which are not known to follow from the weaker assertion P≠NP. It is shown in this paper that relative to a random oracle, NP does not havep-measure zero. The proof exploits the followingindependenceproperty of algorithmically random sequences: ifAis an algorithmically random sequence and a subsequenceA0is chosen by means of abounded Kolmogorov– Loveland place selection, then the sequenceA1of unselected bits is random relative toA0, i.e.,A0andA1are independent. A bounded Kolmogorov–Loveland place selection is a very general type of recursive selection rule which may be interpreted as the sequence of oracle queries of a time-bounded Turing machine, so the methods used may be applicable to other questions involving random oracles.

Functions computable with limited access to NP

It is shown for any constant c < 1 that NP multi-valued functions do not have refinements computable in polynomial time with c log n functional oracle queries to single-valued NP functions unless the polynomial-time hierarchy collapses to its second level.

A new Characterization of Type-2 Feasibility

K. Mehlhorn introduced a class of polynomial-time-computable operators in order to study poly-time reducibilities between functions. This class is defined using a generalization of A. Cobham's definition of feasibility for type-1 functions to type-2 functionals. Cobham's feasible functions are equivalent to the familiar poly-time functions. We generalize this equivalence to type-2 functionals. This requires a definition of the notion poly time in the length of type-1 inputs. The proof of this equivalence is not a simple generalization of the proof for type-1 functions; it depends on the fact that Mehlhorn's class is closed under a strong form of simultaneous limited recursion on notation and requires an analysis of the structure of oracle queries in time-bounded computations.

Adaptive logspace reducibility and parallel time

We discuss two notions of functional oracle for logarithmic space-bounded machines, which differ in whether there is only one oracle tape for both the query and the answer or a separate tape for the answer, which can still be read while the next query is already being constructed. The first notion turns out to be basically nonadaptive, behaving like access to an oracle set. The second notion, on the other hand, is adaptive. By imposing appropriate bounds on the number of functional oracle queries made in this computation model, we obtain new characterizations of the NC and AC hierarchies; thus the number of oracle queries can be considered as a measure of parallel time. Using this characterization of parallel classes, we solve open questions of Wilson.

On learning from queries and counterexamples in the presence of noise

Recently Angluin and Laird have introduced the classification noise process in the Valiant learnability model and proposed an interesting problem to explore the effect of noise in a situation that calls for queries as well as random sampling. In this paper, we present a general method to modify a polynomial-time learning algorithm from a sampling oracle and membership queries to compensate for random errors in the sampling and query responses.

Query complexity, or why is it difficult to separateNP A ∩coNP A fromP A by random oraclesA?

By thequery-time complexity of a relativized algorithm we mean the total length of oracle queries made; thequery-space complexity is the maximum length of the queries made. With respect to these cost measures one can define polynomially time- or space-bounded deterministic, nondeterministic, alternating, etc. Turing machines and the corresponding complexity classes. It turns out that all known relativized separation results operate essentially with this cost measure. Therefore, if certain classes do not separate in the query complexity model, this can be taken as an indication that their relativized separation in the classical cost model will require entirely new principles. A notable unresolved question in relativized complexity theory is the separation of NPA∩ ∩ co NPA fromP A under random oraclesA. We conjecture that the analogues of these classes actually coincide in the query complexity model, thus indicating an answer to the question in the title. As a first step in the direction of establishing the conjecture, we prove the following result, where polynomial bounds refer to query complexity. If two polynomially query-time-bounded nondeterministic oracle Turing machines accept precisely complementary (oracle dependent) languages LA and {0, 1}*∖LA under every oracle A then there exists a deterministic polynomially query-time-bounded oracle Turing machine that accept LA. The proof involves a sort of greedy strategy to selecting deterministically, from the large set of prospective queries of the two nondeterministic machines, a small subset that suffices to perform an accepting computation in one of the nondeterministic machines. We describe additional algorithmic strategies that may resolve the same problem when the condition holds for a (1−e) fraction of the oracles A, a step that would bring us to a non-uniform version of the conjecture. Thereby we reduce the question to a combinatorial problem on certain pairs of sets of partial functions on finite sets.

The complexity of parallel search

This paper studies parallel search algorithms within the framework of independence systems. It is motivated by earlier work on parallel algorithms for concrete problems such as the determination of a maximal independent set of vertices or a maximum matching in a graph and by the general question of determining the parallel complexity of a search problem when an oracle is available to solve the associated decision problem. Our results provide a parallel analog of the self-reducibility process that is so useful in sequential computation. An abstract independence system is specified by a ground set E and a family of subsets of E called the independent sets; it is required that every subset of an independent set be independent. We investigate parallel algorithms for determining a maximal independent set through oracle queries of the form “Is the set A independent?,” as well as parallel algorithms for determining a maximum independent set through queries to a more powerful oracle called a rank oracle. We also study these problems for three special types of independence systems: matroids, graphic matroids, and partition matroids. We derive lower and upper bounds on the deterministic and randomized complexity of these problems. These bounds are sharp enough to give a clear picture of the processor-time trade-offs that are possible, to establish that randomized parallel algorithms are much more powerful than deterministic ones, and to show that even randomized algorithms cannot make effective use of extremely large numbers of processors.

Immunity, Relativizations, and Nondeterminism

It is known [8], [13] that there is a recursive set A such that ${\text{NP }}(A)$ contains a set that is ${\text{P }}(A)$ -immune; that is, there is an infinite set $L \in {\text{NP}}(A)$ such that no infinite subset of L is in ${\text{P}}(A)$. The first result generalizes this fact to situations where the running times of the machines specifying the “larger” class bound the size of the sets of strings queried in the computation trees of the machines specifying the “smaller” class. The second result is of a different type. For relativized complexity classes specified by bounds on the number of oracle queries and the number of nondeterministic steps allowed in computations, it is known [14] that one can describe a set A such that there is an infinite hierarchy of classes relative to A where each new class in the hierarchy is obtained by increasing the amount of nondeterminism. Here it is shown that the conditions allowing infinite hierarchies to exist also allow for each $i \geqq 0$, the existence of a set in the $(i + 1)$st class which has no infinite subset in the ith class.