Parity, circuits, and the polynomial-time hierarchy

  title={Parity, circuits, and the polynomial-time hierarchy},
  author={Merrick L. Furst and James B. Saxe and Michael Sipser},
  journal={Mathematical systems theory},
A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy. 

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.

A Depth 3 Circuit Lower Bound for the Parity Function

It is proved that for any depth 3 circuit with top fan-in t, computing the n-variable parity function must have at least t n t 1 2 − wires.

Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version)

  • A. Yao
  • Computer Science
  • 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 the

On the Correlation of Parity and Small-Depth Circuits

We prove that the correlation of a depth-$d$ unbounded fanin circuit of size $S$ with parity of $n$ variables is at most $2^{-\Omega(n/(\log S)^{d-1})}$.

Lower bounds for modular counting by circuits with modular gates

It is proved that constant depth circuits, with one layer of M O Dm gates at the inputs, followed by a fixed number of layers of MO Dp gates, require exponential size to compute the M O O Dq function, if q is a prime that divides neither p nor m.

Circuit and Decision Tree Complexity of Some Number Theoretic Problems

It is proved that deciding if a given integer is square-free and testing co-primality of two integers by unbounded fan-in circuits of bounded depth requires superpolynomial size.

Uniform characterizations of complexity classes

Results from generalized operators in the context of polynomial-time machines, and gates computing arbitrary groupoidal functions in the contexts of Boolean circuits are surveyed, and relationships to a generalized quantifier concept from finite model theory are presented.

With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy

  • Jin-Yi Cai
  • Computer Science, Mathematics
    STOC '86
  • 1986
It is shown that a random oracle set A separates PSPACE from the entire polynomial-time hierarchy with probability one as a consequence of how much error a fixed depth Boolean circuit must make in computing the parity function.

Relations Among Parallel and Sequential Computation Models

This paper relates uniform small-depth circuit families as a parallel computation model with the complexity of polynomial time Turing machines. Among the consequences we obtain are: (a) a collapse of

Algebraic methods in the theory of lower bounds for Boolean circuit complexity

It is proved that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm.



On Counting Problems and the Polynomial-Time Hierarchy

The Polynomial-Time Hierarchy

  • L. Stockmeyer
  • Computer Science, Mathematics
    Theor. Comput. Sci.
  • 1976

Complexity of the realization of a linear function in the class of II-circuits

AbstractIt is proved that the linear function gn(x1,..., xn) = x1 + ... + xnmod 2 is realized in the class of II-circuits with complexity Lπ(gn) ≥n2. Combination of this result with S. V.

A 2.5 n-lower bound on the combinational complexity of Boolean functions

  • W. Paul
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1977
A new Method for proving linear lower bounds of size 2n is presented and a trade-off result between circuit complexity and formula size is derived.

Method of determining lower bounds for the complexity of P-schemes

A method of calculating lower bounds, quadratic in the arguments of the function, for the complexity of P-schemes.

A second step toward the polynomial hierarchy

  • T. BakerA. Selman
  • Mathematics
    17th Annual Symposium on Foundations of Computer Science (sfcs 1976)
  • 1976

A 3n-Lower Bound on the Network Complexity of Boolean Functions

  • C. Schnorr
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1980

Word problems requiring exponential time(Preliminary Report)

A number of similar decidable word problems from automata theory and logic whose inherent computational complexity can be precisely characterized in terms of time or space requirements on deterministic or nondeterministic Turing machines are considered.


This work defines alternating Turing Machines which are like nondeterministic Turing Machines, except that existential and universal quantifiers alternate, and shows that while n-state alternating finite automata accept only regular sets that can be accepted by 22n-O(logn) state deterministic automata, alternating pushdown automata acceptance all languages accepted by Turing machines in deterministic exponential time.

Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question

Relativized versions of the open question of whether every language accepted nondeterministically in polynomial time can be recognized deterministic in poynomial time are investigated.