Bounded-depth, polynomial-size circuits for symmetric functions
Theoretical Computer Science
Definability by constant-depth polynomial-size circuits
Information and Control
Log depth circuits for division and related problems
SIAM Journal on Computing
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
Parallel computation with threshold functions
Journal of Computer and System Sciences - Structure in Complexity Theory Conference, June 2-5, 1986
Finite monoids and the fine structure of NC1
Journal of the ACM (JACM)
Journal of the ACM (JACM)
Journal of Computer and System Sciences - 3rd Annual Conference on Structure in Complexity Theory, June 14–17, 1988
Fast parallel arithmetic via modular representation
SIAM Journal on Computing
Information Processing Letters
An arithmetic model of computation equivalent to threshold circuits
Theoretical Computer Science
On threshold circuits and polynomial computation
SIAM Journal on Computing
Circuit definitions of nondeterministic complexity classes
SIAM Journal on Computing
A very hard log-space counting class
Theoretical Computer Science - Special issue on structure in complexity theory
Depth reduction for noncommutative arithmetic circuits
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Why is Boolean complexity theory difficult?
Poceedings of the London Mathematical Society symposium on Boolean function complexity
Bits and relative order from residues, space efficiently
Information Processing Letters
Gap-definable counting classes
Journal of Computer and System Sciences
Depth reduction for circuits of unbounded fan-in
Information and Computation
Some results on uniform arithmetic circuit complexity
Mathematical Systems Theory
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
The complexity of iterated multiplication
Information and Computation
The complexity of matrix rank and feasible systems of linear equations (extended abstract)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Space-Efficient Deterministic Simulation of Probabilistic Automata
SIAM Journal on Computing
Non-commutative arithmetic circuits: depth reduction and size lower bounds
Theoretical Computer Science
A combinatorial algorithm for the determinant
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Verifying the Determinant in Parallel
ISAAC '94 Proceedings of the 5th International Symposium on Algorithms and Computation
Space-Efficient Deterministic Simulation of Probabilistic Automata (Extended Abstract)
STACS '94 Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science
A Note on Uniform Circuit Lower Bounds for the Counting Hierarchy (Extended Abstract)
COCOON '96 Proceedings of the Second Annual International Conference on Computing and Combinatorics
Non-commutative Computation, Depth Reduction, and Skew Circuits (Extended Abstract)
Proceedings of the 14th Conference on Foundations of Software Technology and Theoretical Computer Science
Nondeterministic NC1 Computation
CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Counting quantifiers, successor relations, and logarithmic space
SCT '95 Proceedings of the 10th Annual Structure in Complexity Theory Conference (SCT'95)
Bounded Depth Arithmetic Circuits: Counting and Closure
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
Counting and Locality over Finite Structures: A Survey
ESSLLI '97 Revised Lectures from the 9th European Summer School on Logic, Language, and Information: Generalized Quantifiers and Computation
Arithmetic Circuits and Polynomial Replacement Systems
FST TCS 2000 Proceedings of the 20th Conference on Foundations of Software Technology and Theoretical Computer Science
CIAC '00 Proceedings of the 4th Italian Conference on Algorithms and Complexity
On the complexity of inducing categorical and quantitative association rules
Theoretical Computer Science
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Continuing a line of investigation that has studied the function classes P, we study the class of functions AC/sup 0/. One way to define AC/sup 0/ is as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fanin addition and multiplication gates. In contrast to the preceding function classes, for which we know no nontrivial lower bounds, lower bounds for AC/sup 0/ follow easily from established circuit lower bounds. One of our main results is a characterization of TC/sup 0/ in terms of AC/sup 0/: A language A is in TC/sup 0/ if and only if there is a AC/sup 0/ function f and a number k such that x/spl isin/A/spl hArr/f(x)=2/sup |x|k/. Using the naming conventions, this yields: TC/sup 0/=PAC/sup 0/=C=AC/sup 0/. Another restatement of this characterization is that TC/sup 0/ can be simulated by constant-depth arithmetic circuits, with a single threshold gate. We hope that perhaps this characterization of TC/sup 0/ in terms of AC/sup 0/ circuits might provide a new avenue of attack for proving lower bounds. Our characterization differs markedly from earlier characterizations of TC/sup 0/ in terms of arithmetic circuits over finite fields. Using our model of arithmetic circuits, computation over finite fields yields ACC/sup 0/. We also prove a number of closure properties and normal forms for AC/sup 0/.