Some Observations on the Connection Between Counting an Recursion

@article{Wagner1986SomeOO,
  title={Some Observations on the Connection Between Counting an Recursion},
  author={Klaus W. Wagner},
  journal={Theor. Comput. Sci.},
  year={1986},
  volume={47},
  pages={131-147}
}
  • K. Wagner
  • Published 1986
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
Abstract Based on Valiant's class # P of all functions counting the number of accepting computations of nondeterministic polynomial-time Turing machines, the polynomial-time hierarchy of counting functions is introduced. The class PHCF of all functions of this hierarchy and some of its subclasses are characterized by recursion-theoretic means. It turns out that, from the recursion-theoretic point of view, PHCF is an analogue to Kalmar's class E of elementary functions, to the class Pspace of… Expand
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  • [1988] Proceedings. Structure in Complexity Theory Third Annual Conference
  • 1988
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References

SHOWING 1-10 OF 19 REFERENCES
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 algorithmsExpand
On Counting Problems and the Polynomial-Time Hierarchy
  • D. Angluin
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1980
TLDR
The main result is that there exists an oracle set A such that PPA −(Π2P,A ∪ σ2 P,A) ≠ ∅, with the corollary that also D ≠PA − (Π 2P, a ∪σ2 p, A) ≦ ∅. Expand
On Some Natural Complete Operators
  • K. Ko
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1985
TLDR
This work investigates several natural operators which share similar properties and uses the concept of completeness to give a precise classification of the complexity of these operators. Expand
CLASSES OF PREDICTABLY COMPUTABLE FUNCTIONS
0. Introduction. In this paper we study a sequence of classes of computable functions for which a prediction of the complexity of the calculation may be made in a comparatively simple fashion. TheExpand
BPP and the Polynomial Hierarchy
  • C. Lautemann
  • Mathematics, Computer Science
  • Inf. Process. Lett.
  • 1983
TLDR
It is shown by pure counting arguments that BPP is contained in ΣP2, the second level of the hierarchy of the polynomial hierarchy of Meyer and Stockmeyer. Expand
The Polynomial-Time Hierarchy
  • L. Stockmeyer
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1976
TLDR
The problem of deciding validity in the theory of equality is shown to be complete in polynomial-space, and close upper and lower bounds on the space complexity of this problem are established. Expand
The Complexity of Enumeration and Reliability Problems
  • L. Valiant
  • Mathematics, Computer Science
  • SIAM J. Comput.
  • 1979
TLDR
For a large number of natural counting problems for which there was no previous indication of intractability, that they belong to the class of computationally eqivalent counting problems that are at least as difficult as the NP-complete problems. Expand
The Complexity of Computing the Permanent
  • L. Valiant
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1979
Abstract It is shown that the permanent function of (0, 1)-matrices is a complete problem for the class of counting problems associated with nondeterministic polynomial time computations. RelatedExpand
Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version)
  • A. Yao
  • Mathematics, Computer Science
  • FOCS
  • 1985
We present exponential lower bounds on the size of depth-k Boolean circuits for computing certain functions. These results imply that there exists an oracle set A such that, relative to A, all theExpand
The complexity of approximate counting
TLDR
The complexity of computing approximate solutions to problems in #P is classified in terms of the polynomial-time hierarchy (for short, P-hierarchy) in order to study a class of restricted, but very natural, probabilistic sampling methods motivated by the particular counting problems. Expand
...
1
2
...