On the Power of Probabilistic Polynomial Time:PNP[log] ⊆ PP

  title={On the Power of Probabilistic Polynomial Time:PNP[log] ⊆ PP},
  author={L. Hemachandra and Gerd Wechsung},
We show that every set in the 0~ level of the polynomial hierarchy-every set polynomial­ time truth-table reducible to SAT~is accepted by a probabilistic polynomial-time Turing machine: pNP[log] ~ PP. Relatedly, we show that probabilistic polynomial time is closed under polynomial-time parity reductions. 

On the power of parity polynomial time

It is proved that the complexity class ⊕P, parity polynomial time [PZ], contains the class of languages accepted byNP machines with few accepting paths, and that theclass of nondeterministic path-restricted languages is closed under bounded truth-table reductions.

Restricted Relativizations of Probablistic Polynomial Time

The polynomial method in circuit complexity

  • R. Beigel
  • Computer Science
    [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference
  • 1993
The basic techniques for using polynomials in complexity theory are examined, emphasizing intuition at the expense of formality and closure properties, upper bounds, and lower bounds obtained.

Exponential-Time and Subexponential-Time Sets

Counting classes are at least as hard as the polynomial-time hierarchy

  • Seinosuke TodaM. Ogihara
  • Mathematics, Computer Science
    [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference
  • 1991
It is shown that many natural counting classes are at least as computationally hard as PH in the following sense: for each K of the counting classes, every set in K(PH) is polynomial-time randomized many-one reducible to a set inK with two-sided exponentially small error probability.

Polynomial Time 1-Turing Reductions from #PH to #P

Perceptrons, PP, and the polynomial hierarchy

  • R. Beigel
  • Computer Science
    [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference
  • 1992
AbstractWe construct a predicate that is computable by a perceptron with linear size, order 1, and exponential weights, but which cannot be computed by any perceptron having subexponential

A Taxonomy of Complexity Classes of Functions

  • A. Selman
  • Mathematics
    J. Comput. Syst. Sci.
  • 1994

On polynomially many queries to NP or QMA oracles

This work shows that for any verification class C, any P C machine with a query graph of “separator number” s can be simulated using deterministic time exp, and shows how to combine Gottlob’s “admissible-weighting function’ framework with the “flag-qubit” framework for embedding P C computations directly into APX-SIM instances in a black-box fashion.

PP is as Hard as the Polynomial-Time Hierarchy

It is shown that every set in PH is polynomial-time Turing reducible to a set in PP, and PH is included in BP, which implies a collapse of PH.



Two remarks on the power of counting

The relationship between the polynomial hierarchy and Valiant's class #P is at present unknown. We show that some low portions of the polynomial hierarchy, namely deterministic polynomial algorithms

A Comparison of Polynomial Time Reducibilities

On truth-table reducibility to SAT and the difference hierarchy over NP

  • S. BussL. Hay
  • Computer Science
    [1988] Proceedings. Structure in Complexity Theory Third Annual Conference
  • 1988
It is shown that polynomial-time truth-table reducibility by Boolean circuits to SAT is the same as log-space truth-table reducibility via Boolean formulas to SAT and the same as log-space Turing

Computational complexity of probabilistic Turing machines

It is shown how probabilisticlinear-bounded automata can simulate nondeterministic linear-bounding automata and an example is given of a function computable more quickly by Probabilistic Turing machines than by deterministic Turing machines.

Bounded query computations

  • K. Wagner
  • Computer Science
    [1988] Proceedings. Structure in Complexity Theory Third Annual Conference
  • 1988
The main topics are: the relationship between the number of adaptive and parallel queries, connections to the closure of NP under polynomial-time truth-table reducibility, the Boolean hierarchy, the power of one more query, sparse oracles versus few queries, and natural complete problems for the most important bounded query classes.

P^(NP[O(log n)]) and Sparse Turing-Complete Sets for NP

  • Jim Kadin
  • Computer Science, Mathematics
    J. Comput. Syst. Sci.
  • 1987

On some central problems in computational complexity

In this thesis we examine some of the central problems in the theory of computational complexity, like the trade-offs between time and memory, the power of nondeterminism and parallelism, and the

On the Difference Between One and Many (Preliminary Version)

It is shown, that in specific cases, the question, ‘Given a problem, is it more difficult to tell how many solutions the problem has than just deciding whether it has a solution?’ can be put into a mathematically meaningful form.

The complexity of optimization problems

The central result is that any FPSAT function decomposes into an OptP function followed by polynomial-time computation, and it quantifies "how much" NP-completeness is in a problem, i.e., the number of NP queries it takes to compute the function.

The Boolean Hierarchy I: Structural Properties

The complexity of sets formed by boolean operations (union, intersection, and complement) on NP sets are studied, showing that in some relativized worlds the boolean hierarchy is infinite, and that for every k there is a relativization world in which the Boolean hierarchy extends exactly k levels.