Relational queries computable in polynomial time
Information and Control
Log depth circuits for division and related problems
SIAM Journal on Computing
The Boolean formula value problem is in ALOGTIME
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
The complexity of Boolean functions
The complexity of Boolean functions
Expressibility and parallel complexity
SIAM Journal on Computing
Journal of Computer and System Sciences - 3rd Annual Conference on Structure in Complexity Theory, June 14–17, 1988
A Uniform Circuit Lower Bound For the Permanent
SIAM Journal on Computing
The complexity of iterated multiplication
Information and Computation
The expressive power of finitely many generalized quantifiers
Information and Computation
Time, hardware, and uniformity
Complexity theory retrospective II
Journal of the ACM (JACM)
First Order Logic, Fixed Point Logic and Linear Order
CSL '95 Selected Papers from the9th International Workshop on Computer Science Logic
Invariant Definability and P/poly
Proceedings of the 12th International Workshop on Computer Science Logic
First-Order Logic vs. Fixed-Point Logic in Finite Set Theory
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Borel sets and circuit complexity
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Elementary induction on abstract structures (Studies in logic and the foundations of mathematics)
Elementary induction on abstract structures (Studies in logic and the foundations of mathematics)
Space-bounded reducibility among combinatorial problems
Journal of Computer and System Sciences
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We investigate the complexity of the fixed-points of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a built-in BIT predicate, or equivalently, with a built-in membership relation between hereditarily finite sets (input relations are allowed). We show that the iteration of a positive bounded formula converges in polylogarithmically many steps in the cardinality of the structure. This extends a previously known much weaker result. We obtain a number of connections with the rudimentary languages and deterministic polynomial-time. Moreover, our results provide a natural characterization of the complexity class consisting of all languages computable by bounded-depth, polynomial-size circuits, and polylogarithmic-time uniformity. As a byproduct, we see that this class coincides with LH(P), the logarithmic-time hierarchy with an oracle to deterministic polynomial-time. Finally, we discuss the connection of this result with the well-studied algorithms for integer division.