A taxonomy of problems with fast parallel algorithms
Information and Control
Computational limitations of small-depth circuits
Computational limitations of small-depth circuits
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
The Boolean formula value problem is in ALOGTIME
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Short propositional formulas represent nondeterministic computations
Information Processing Letters
Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
On the decomposability of NC and AC
SIAM Journal on Computing
Non-uniform automata over groups
Information and Computation
Journal of Computer and System Sciences - 3rd Annual Conference on Structure in Complexity Theory, June 14–17, 1988
An optimal parallel algorithm for formula evaluation
SIAM Journal on Computing
Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
A Uniform Circuit Lower Bound For the Permanent
SIAM Journal on Computing
Size--Depth Tradeoffs for Threshold Circuits
SIAM Journal on Computing
Counting quantifiers, successor relations, and logarithmic space
Journal of Computer and System Sciences - special issue on complexity theory
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
The Shrinkage Exponent of de Morgan Formulas is 2
SIAM Journal on Computing
Reductions in circuit complexity: an isomorphism theorem and a gap theorem
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
Zero knowledge and the chromatic number
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
Nondeterministic NC1 computation
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
Time—space tradeoffs for satisfiability
Journal of Computer and System Sciences - Eleventh annual conference on computational learning theory&slash;Twelfth Annual IEEE conference on computational complexity
Introduction to Circuit Complexity: A Uniform Approach
Introduction to Circuit Complexity: A Uniform Approach
Time-space trade-off lower bounds for randomized computation of decision problems
Journal of the ACM (JACM)
Uniform constant-depth threshold circuits for division and iterated multiplication
Journal of Computer and System Sciences - Complexity 2001
Lower Bounds for Deterministic and Nondeterministic Branching Programs
FCT '91 Proceedings of the 8th International Symposium on Fundamentals of Computation Theory
A Non-Linear Time Lower Bound for Boolean Branching Programs
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Time-Space Tradeoffs in the Counting Hierarchy
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
On the approximability of clique and related maximization problems
Journal of Computer and System Sciences
Number-theoretic constructions of efficient pseudo-random functions
Journal of the ACM (JACM)
Verifying and decoding in constant depth
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
A survey of lower bounds for satisfiability and related problems
Foundations and Trends® in Theoretical Computer Science
SIAM Journal on Computing
On the constant-depth complexity of k-clique
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Separating the polynomial-time hierarchy by oracles
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Minimizing Disjunctive Normal Form Formulas and $AC^0$ Circuits Given a Truth Table
SIAM Journal on Computing
Circuit Complexity of Regular Languages
Theory of Computing Systems - Special Issue: Computation and Logic in the Real World; Guest Editors: S. Barry Cooper, Elvira Mayordomo and Andrea Sorbi
Cracks in the defenses: scouting out approaches on circuit lower bounds
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
A New Characterization of ACC0 and Probabilistic CC0
Computational Complexity - Selected papers from the 24th Annual IEEE Conference on Computational Complexity (CCC 2009)
Better inapproximability results for maxclique, chromatic number and min-3lin-deletion
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Hi-index | 0.00 |
We observe that many important computational problems in NC1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial-size TC0 circuits if and only if it has TC0 circuits of size n1+&epsis; for every &epsis; 0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC1 and has the self-reducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC0 circuits of size n1+&epsis;d. If one were able to improve this lower bound to show that there is some constant &epsis; 0 (independent of the depth d) such that every TC0 circuit family recognizing BFE has size at least n1+&epsis;, then it would follow that TC0 ≠ NC1. We show that proving lower bounds of the form n1+&epsis; is not ruled out by the Natural Proof framework of Razborov and Rudich and hence there is currently no known barrier for separating classes such as ACC0, TC0 and NC1 via existing “natural” approaches to proving circuit lower bounds. We also show that problems with small uniform constant-depth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known time-space tradeoff lower bounds to show that SAT requires uniform depth d TC0 and AC0[6] circuits of size n1+c for some constant c depending on d.