Width Hierarchy for k-OBDD of Small Width

In this paper was explored well known model k-OBDD. There are proven width based hierarchy of classes of boolean functions which computed by k-OBDD. The proof of hierarchy is based on sufficient condition of Boolean function's non representation as k-OBDD and complexity properties of Boolean function SAF. This function is modification of known Pointer Jumping (PJ) and Indirect Storage Access (ISA) functions.Comment: 8 page

A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $\Delta+1$ to one of product-depth $\Delta$ in the multilinear setting. We show that for every positive $\Delta = \Delta(n) = o(\log n/\log \log n),$ there is an explicit multilinear polynomial $P^{(\Delta)}$ on $n$ variables that can be computed by a multilinear formula of product-depth $\Delta+1$ and size $O(n)$, but not by any multilinear circuit of product-depth $\Delta$ and size less than $\exp(n^{\Omega(1/\Delta)})$. This result is tight up to the constant implicit in the double exponent for all $\Delta = o(\log n/\log \log n).$ This strengthens a result of Raz and Yehudayoff (Computational Complexity 2009) who prove a quasipolynomial separation for constant-depth multilinear circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an exponential separation in the case $\Delta = 1.$ Our separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits

Satisfiability Algorithm for Syntactic Read-kk-times Branching Programs

The satisfiability of a given branching program is to determine whether there exists a consistent path from the root to 1-sink. In a syntactic read-k-times branching program, each variable appears at most k times in any path from the root to a sink. We provide a satisfiability algorithm for syntactic read-k-times branching programs with n variables and m edges that runs in time Oleft(poly(n, m^{k^2})cdot 2^{(1-mu(k))n}right), where mu(k) = frac{1}{4^{k+1}}. Our algorithm is based on the decomposition technique shown by Borodin, Razborov and Smolensky [Computational Complexity, 1993]