The First-order isomorphism theorem

For any class C und closed under NC1 reductions, it is shown that all sets complete for C under first-order (equivalently, Dlogtimeuniform AC0) reductions are isomorphic under first-order computable isomorphisms

AND and/or OR: Uniform Polynomial-Size Circuits

We investigate the complexity of uniform OR circuits and AND circuits of polynomial-size and depth. As their name suggests, OR circuits have OR gates as their computation gates, as well as the usual input, output and constant (0/1) gates. As is the norm for Boolean circuits, our circuits have multiple sink gates, which implies that an OR circuit computes an OR function on some subset of its input variables. Determining that subset amounts to solving a number of reachability questions on a polynomial-size directed graph (which input gates are connected to the output gate?), taken from a very sparse set of graphs. However, it is not obvious whether or not this (restricted) reachability problem can be solved, by say, uniform AC^0 circuits (constant depth, polynomial-size, AND, OR, NOT gates). This is one reason why characterizing the power of these simple-looking circuits in terms of uniform classes turns out to be intriguing. Another is that the model itself seems particularly natural and worthy of study. Our goal is the systematic characterization of uniform polynomial-size OR circuits, and AND circuits, in terms of known uniform machine-based complexity classes. In particular, we consider the languages reducible to such uniform families of OR circuits, and AND circuits, under a variety of reduction types. We give upper and lower bounds on the computational power of these language classes. We find that these complexity classes are closely related to tallyNL, the set of unary languages within NL, and to sets reducible to tallyNL. Specifically, for a variety of types of reductions (many-one, conjunctive truth table, disjunctive truth table, truth table, Turing) we give characterizations of languages reducible to OR circuit classes in terms of languages reducible to tallyNL classes. Then, some of these OR classes are shown to coincide, and some are proven to be distinct. We give analogous results for AND circuits. Finally, for many of our OR circuit classes, and analogous AND circuit classes, we prove whether or not the two classes coincide, although we leave one such inclusion open.Comment: In Proceedings MCU 2013, arXiv:1309.104

Pseudo-random generators and structure of complete degrees

It is shown that, if there exist sets in E (the exponential complexity class) that require 2Ω(n)-sized circuits, then sets that are hard for class P (the polynomial complexity class) and above, under 1-1 reductions, are also hard under 1-1 size-increasing reductions. Under the assumption of the hardness of solving the RSA (Rivest-Shamir-Adleman, 1978) problem or the discrete log problem, it is shown that sets that are hard for class NP (nondeterministic polynomial) and above, under many-1 reductions, are also hard under (non-uniform) 1-1 and size-increasing reductions

The isomorphism conjecture for constant depth reductions

For any class C closed under TC0 reductions, and for any measure u of uniformity containing Dlogtime, it is shown that all sets complete for C under u-uniform AC0 reductions are isomorphic under u-uniform AC0-computable isomorphisms