On the Hierarchies for Deterministic, Nondeterministic and Probabilistic Ordered Read-k-times Branching Programs

The paper examines hierarchies for nondeterministic and deterministic ordered read-$k$-times Branching programs. The currently known hierarchies for deterministic $k$-OBDD models of Branching programs for $ k=o(n^{{1/2}/\log{3/2}n)$} are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic $k$-OBDD it is known that, if $k$ is constant then polynomial size $k$-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read $k$-times Branching programs for $k=o(\sqrt{\log{n}}/\log\log{n})$ are proved by Okolnishnikova in 1997, and for probabilistic read $k$-times Branching programs for $k\leq \log n/3$ are proved by Hromkovic and Saurhoff in 2003. We show that increasing $k$ for polynomial size nodeterministic $k$-OBDD makes model more powerful if $k$ is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic $k$-OBDDs for $ k=o(n/ \log n)$. These results extends hierarchies for read $k$-times Branching programs, but $k$-OBDD has more regular structure. The lower bound techniques we propose are a "functional description" of Boolean function presented by nondeterministic $k$-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic $k$-OBDDs. Additionally we expand the hierarchies for deterministic $k$-OBDDs using our lower bounds for $ k=o(n/ \log n)$. We also analyze similar hierarchies for superpolynomial and subexponential width $k$-OBDDs.