# A survey on counting classes

@article{Gundermann1990ASO,
title={A survey on counting classes},
author={Thomas Gundermann and Nasser Ali Nasser and Gerd Wechsung},
journal={Proceedings Fifth Annual Structure in Complexity Theory Conference},
year={1990},
pages={140-153}
}
• Published 8 July 1990
• Mathematics
• Proceedings Fifth Annual Structure in Complexity Theory Conference
Consideration is given to polynomial-time machines. Among these classes are EP and PP. The authors prove P/sup EP(log)/ 25 PP, investigate the Boolean closure BC(EP) of EP, and give a relativization principle which allows them to completely separate BC(EP) in a suitable relativized world and to give simple proofs for known relativization results. Further results concerning the relationships of such classes in unrelativized and relativized worlds are given.<<ETX>>

## Figures from this paper

### Algebraic acceptance mechanisms for polynomial time machines

Several results are presented, whose common theme is the application of (quite basic) algebraic techniques to describe complexity classes, which appear in the many ways nondeterministic polynomial time machines can be used (or abused).

### Extension of Toda's Theorem to Middle Bit Classes

• J. K
• Computer Science
The more recent progress that has been made in extending Toda's Theorem to middle bit classes is reviewed, and how this work has been applied in circuit complexity to establish improved simulations of the class ACC is described.

### PP is closed under intersection

• Computer Science
STOC '91
• 1991
It is shown that PP is closed under a variety of polynomial-time truth-table reductions and in complexity theory include the definite collapse and (assuming P ? PP) separation of certain query hierarchies over PP.

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

• 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.

### Complexity Classes with Finite Acceptance Types

The proof technique uses a key lemma which transforms the inclusionship question for these classes into an existence question for certain hypergraphs, under which one class is contained in another one relative to all oracle seperation.

### A Promise Class at Least as Hard as the Polynomial Hierarchy

It is shown that the polynomial hierarchy is contained in a promise class that naturally corresponds to the class BP \cdot SPP (even though for this class itself, unfortunately, the original problem remains unsolved), and some properties of several related classes defined via operators are studied.

### On the power of deterministic reductions to C=P

It is shown that there exists an oracleA such that there exist techniques that would prove that C=P and PP are polynomial-time Turing equivalent, or that C-C=P is polynomially Turing hard for the polynometric-time hierarchy, would not relativize.

### Polynomial Time Machines Equipped with Word Problems over Algebraic Structures as their Acceptance Criteria

This work investigates the power of polynomial time machines whose acceptance mechanism is defined by a word problem over some finite semigroup, monoid, or group and shows that the according complexity class is PSPACE.

### The Chain Method to Separate Counting Classes

• Mathematics, Computer Science
Theory of Computing Systems
• 1998
All relativizable inclusions between classes NP(k) from the Boolean Hierarchy over NP and other classes defined by what the authors call bounded counting are completely characterized.

### Bounded Queries to Arbitrary Sets

• A. Lozano
• Mathematics
RAIRO Theor. Informatics Appl.
• 1996
We prove that if P A k-T = P A (k +1)-T for some k and an arbitrary set A, then A is reducible to its complement under a relativized nondeterministic conjunctive reduction. By substituting A by

## References

SHOWING 1-10 OF 18 REFERENCES

### On the power of probabilistic polynomial time: P/sup NP(log)/ contained in PP

• Computer Science
[1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference
• 1989
It is shown that probabilistic time is closed under polynomial-time parity reductions. Therefore, every set polynomial-time truth-table reducible to SAT is accepted by a probabilistic polynomial-time

### The power of counting

• U. Schöning
• Psychology
[1988] Proceedings. Structure in Complexity Theory Third Annual Conference
• 1988
An overview of applications and variations of counting in structural complexity theory is provided. It is argued that the capacity for exact counting is closely related with the capacity for

### On the computational power of PP and (+)P

• Seinosuke Toda
• Computer Science
30th Annual Symposium on Foundations of Computer Science
• 1989
It follows that neither PP nor (+)P is a subset of or equivalent to PH unless PH collapses to a finite level, strong evidence that both classes are strictly harder than PH.

### Nondeterministic Turing Machines with Modified Acceptance

• Computer Science
MFCS
• 1986
In this paper an extremely fine hierarchy within BC(NP) is proposed, characterized by nondeter-ministic polynomial time Turing machines with suitably modified acceptance notions.

### The Complexity of Computing the Permanent

• L. Valiant
• Mathematics, Computer Science
Theor. Comput. Sci.
• 1979

### Counting classes with finite acceptance types

• Mathematics
• 1987
Perfectionnement de la hierarchie de Hausdorff generee par NP. Les nouvelles classes permettent une exacte classification de la complexite de certains problemes de denombrement. Les classes sont

### The Boolean Hierarchy II: Applications

• Computer Science
SIAM J. Comput.
• 1989
The Boolean Hierarchy I: Structural Properties explores the structure of the boolean hierarchy, the closure of NP with respect to boolean hierarchies, and the role of symbols in this hierarchy.