Bounded-depth, polynomial-size circuits for symmetric functions
Theoretical Computer Science
Definability by constant-depth polynomial-size circuits
Information and Control
Logarithmic depth circuits for algebraic functions
SIAM Journal on Computing
Log depth circuits for division and related problems
SIAM Journal on Computing
Efficient parallel circuits and algorithms for division
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
Very fast parallel polynomial arithmetic
SIAM Journal on Computing
On the decomposability of NC and AC
SIAM Journal on Computing
Journal of Computer and System Sciences - 3rd Annual Conference on Structure in Complexity Theory, June 14–17, 1988
IDTC Second international conference on Database theory
Optimal size integer division circuits
SIAM Journal on Computing
Fast parallel arithmetic via modular representation
SIAM Journal on Computing
On threshold circuits and polynomial computation
SIAM Journal on Computing
On Optimal Depth Threshold Circuits for Multiplication andRelated Problems
SIAM Journal on Discrete Mathematics
Bits and relative order from residues, space efficiently
Information Processing Letters
Logarithmic depth circuits for Hermite interpolation
Journal of Algorithms
Optimal depth, very small size circuits for symmetric functions in AC0
Information and Computation
The complexity of iterated multiplication
Information and Computation
Inductive counting for width-restricted branching programs
Information and Computation
Space-Efficient Deterministic Simulation of Probabilistic Automata
SIAM Journal on Computing
Time, hardware, and uniformity
Complexity theory retrospective II
Computing with sublogarithmic space
Complexity theory retrospective II
Threshold circuits of small majority-depth
Information and Computation
Nondeterministic NC1 computation
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
Efficient threshold circuits for power series
Information and Computation
Isolation, matching and counting uniform and nonuniform upper bounds
Journal of Computer and System Sciences
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
On TC0, AC0, and arithmetic circuits
Journal of Computer and System Sciences - Eleventh annual conference on computational learning theory&slash;Twelfth Annual IEEE conference on computational complexity
Reducing the complexity of reductions
Computational Complexity
Introduction to Circuit Complexity: A Uniform Approach
Introduction to Circuit Complexity: A Uniform Approach
Turing Machines with Sublogarithmic Space
Turing Machines with Sublogarithmic Space
Space Bounded Computations: Review And New Separation Results
MFCS '89 Proceedings on Mathematical Foundations of Computer Science 1989
On Counting AC0 Circuits with Negative Constants
MFCS '98 Proceedings of the 23rd International Symposium on Mathematical Foundations of Computer Science
Weak Bounded Arithmetic, the Diffie-Hellman Problem and Constable's Class K
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Type two computational complexity
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
Uniform Circuits for Division: Consequences and Problems
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
The computational complexity of nash equilibria in concisely represented games
EC '06 Proceedings of the 7th ACM conference on Electronic commerce
A survey of lower bounds for satisfiability and related problems
Foundations and Trends® in Theoretical Computer Science
The Orbit Problem Is in the GapL Hierarchy
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Amplifying lower bounds by means of self-reducibility
Journal of the ACM (JACM)
Planarity, Determinants, Permanents, and (Unique) Matchings
ACM Transactions on Computation Theory (TOCT)
On defining integers in the counting hierarchy and proving arithmetic circuit lower bounds
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Algorithms and theory of computation handbook
Uniform derandomization from pathetic lower bounds
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory
Journal of Computer and System Sciences
Extensions of MSO and the monadic counting hierarchy
Information and Computation
The orbit problem is in the GapL hierarchy
Journal of Combinatorial Optimization
On the complexity of regular-grammars with integer attributes
Journal of Computer and System Sciences
On the complexity of powering in finite fields
Proceedings of the forty-third annual ACM symposium on Theory of computing
Characterizing definability of second-order generalized quantifiers
WoLLIC'11 Proceedings of the 18th international conference on Logic, language, information and computation
Permanent does not have succinct polynomial size arithmetic circuits of constant depth
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Marginal hitting sets imply super-polynomial lower bounds for permanent
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Constant-Depth circuits for arithmetic in finite fields of characteristic two
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
On the complexity of parallel hardness amplification for one-way functions
TCC'06 Proceedings of the Third conference on Theory of Cryptography
The Computational Complexity of Nash Equilibria in Concisely Represented Games
ACM Transactions on Computation Theory (TOCT)
Green's theorem and isolation in planar graphs
Information and Computation
Computing bits of algebraic numbers
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
Information Processing Letters
CRN Elimination and Substitution Bases for Complexity Classes
Fundamenta Informaticae
Root finding with threshold circuits
Theoretical Computer Science
Permanent does not have succinct polynomial size arithmetic circuits of constant depth
Information and Computation
Decomposition of threshold functions into bounded fan-in threshold functions
Information and Computation
Log-Space Algorithms for Paths and Matchings in k-Trees
Theory of Computing Systems
Journal of Computer and System Sciences
Monomials, multilinearity and identity testing in simple read-restricted circuits
Theoretical Computer Science
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It has been known since the mid-1980s (SIAM J. Comput. 15 (1986) 994; SIAM J. Comput. 21 (1992) 896) that integer division can be performed by poly-time uniform constant-depth circuits of MAJORITY gates; equivalently, the division problem lies in P-uniform TC0. Recently, this was improved to L-uniform TC0 (RAIRO Theoret. Inform. Appl. 35 (2001) 259), but it remained unknown whether division can be performed by DLOGTIME-uniform TC0 circuits. The DLOGTIME uniformity condition is regarded by many as being the most natural notion of uniformity to apply to small circuit complexity classes such as TC0; DLOGTIME-uniform TC0 is also known as FOM, because it corresponds to first-order logic with MAJORITY quantifiers, in the setting of finite model theory. Integer division has been the outstanding example of a natural problem known to be in a P-uniform circuit complexity class, but not known to be in its DLOGTIME-uniform version.We show that indeed division is in DLOGTIME-uniform TC0. First we show that division lies in the complexity class FOM + POW obtained by augmenting FOM with a predicate for powering modulo small primes. Then we show that the predicate POW itself lies in FOM. (In fact, it lies in FO, or DLOGTIME-uniform AC0.)The essential idea in the fast parallel computation of division and related problems is that of Chinese remainder representation (CRR)--storing a number in the form of its residues modulo many small primes. The fact that CRR operations can be carried out in log space has interesting implications for small space classes. We define two versions of s(n) space for s(n) = o(log n): dspace(s(n)) as the traditional version where the worktape begins blank, and DSPACE(s(n)) where the space bound is established by endmarkers before the computation starts. We present a new translational lemma characterizing the unary languages in the DSPACE classes. It is known (Theoret. Comput. Sci. 3 (1976) 213) that {0n : n is prime} ∉ dspace(log logn). We show that if this can be improved to {0n:n is prime} ∉ DSPACE(log log n), it follows that L ≠ NP.