Abstract:
In unrestricted branching programs all variables may be tested arbitrarily often on each path. But exponential lower bounds are only known, if on each path the number of tests of each variable is bounded (Borodin, Razborov and Smolensky (1993)). We examine branching programs in which for each path the number of variables that are tested more than once is bounded by $k$, but we do not bound the number of tests of those variables. A new lower bound method admits to prove that we can enhance the expressive power of such branching programs by increasing $k$ only by $1$: For $k\leq(1-\varepsilon)(n/3)^{1/3}/\log^{2/3}n$, where $\varepsilon > 0$, we exhibit Boolean functions that can be represented in polynomial size, if $k$ variables may be tested more than once on each path, but only in exponential size, if $(k-1)$ variables may be tested more than once on each path. Therefore, we obtain a tight hierarchy.