Collapsing Exact Arithmetic Hierarchies

  title={Collapsing Exact Arithmetic Hierarchies},
  author={Nikhil Balaji and Samir Datta},
We provide a uniform framework for proving the collapse of the hierarchy \({\sf NC}^1(\mathcal{C})\) for an exact arithmetic class \(\mathcal{C}\) of polynomial degree. These hierarchies collapse all the way down to the third level of the AC 0-hierarchy, \({\sf AC^0_3}(\mathcal{C})\). Our main collapsing exhibits are the classes 



Collapsing Oracle Hierarchies, Census Functions and Logarithmically Many Queries

Using a uniform method based on a technique by Kadin [5], many oracle and alternation hierarchies can be shown to collapse to their level Δ2. This improves in different respects earlier such collapse

Collapsing oracle-tape hierarchies

  • G. Gottlob
  • Computer Science, Mathematics
    Proceedings of Computational Complexity (Formerly Structure in Complexity Theory)
  • 1996
It turns out that for an extremely large number of central complexity classes C, the oracle tape hierarchy for C collapses totally, and any class C is smooth if it is closed under marked union and positive polynomial-time Turing reductions.

Counting classes and the fine structure between NC1 and L

The complexity of matrix rank and feasible systems of linear equations

It is shown that this natural complexity class for which the problems of determining if a system of linear equations is feasible and computing the rank of an integer matrix are complete under logspace reductions is closed under NC1-reducibility.

The PL hierarchy collapses

It is shown that the PL hierarchy PLH, defined in terms of the Ruzzo-Simon-Tompa relativization, collapses to PL and is closed under logspace-uniform AC^0-reductions.

PP is closed under intersection

It is shown that PP is closed under a variety of polynomial-time truth-table reductions and in complexity theory include the definite collapse and (assuming P ? PP) separation of certain query hierarchies over PP.

Nondeterministic NC 1 Computation.

It is proved that boolean circuits, algebraic circuits, programs over nondeterministic-nite automata, and programs over constant integer matrices yield equivalent deenitions of the latter three classes, and closure properties are investigated.

Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds

Verifying the determinant in parallel

It is proved that computing the rank is equivalent under AC0 reductions to verifying the determinant, and it is shown that for functions, there exists an NC1 checker even if they are hard to verify, and that they can be extended into functions whose verification is easy.

Generalized theorems on relationships among reducibility notions to certain complexity classes

It is proved that, for a classK, reducibility notions of sets toK under polynomial-time constant-round truth-table reducible, polynometric-time log-Turing reducibles, logspace constant- round truth- table reducibilities, and logspace Turing reducibly are all equivalent.