Hierarchy theorems for kOBDDs and kIBDDs

Restricted Nondeterministic Read-Once Branching Programs and an Exponential Lower Bound for Integer Multiplication

“…Furthermore, it is not difficult to see that P (k-OBDD), k constant, is a proper subclass of P (OR-OBDD) which has been already stated implicitly in [7]. Proving that there are efficient algorithms for the manipulation of k-OBDDs Bollig, Sauerhoff, Sieling, and Wegener [5] have decomposed a k-OBDD representing a function f n into an OR-OBDD of size O(|G| 2k−1 ) for f n . Therefore, all functions representable by polynomial size k-OBDDs, k constant, can also be represented by OR-T 0 -BP1s of polynomial size.…”

Nondeterministic ordered binary decision diagrams with repeated tests and various modes of acceptance

“…• f (1) j (g (1) j ) is the function represented by the parity (existential) OBDD spanned between s as source, u 2,j as 1-target, and a 0-target that results from merging all nodes u 2,ι (ι = j ); • f (λ) i,j (g (λ) i,j ) is the function represented by the parity (existential) OBDD spanned between u λ,i as source, u λ+1,j as 1-target, and a 0-target that results from merging all nodes u λ+1,ι (ι = j );…”

“…In the k-OBDD-case all layers are guided by the same variable ordering. Though the equivalence test for 2-IBDD is NP-complete (see [1]), Bitner et al [9] present a lot of successful experiments with IBDDs.…”

“…Let us outline the hierarchy result presented in [1]. Pointer jumping functions PJ k,n are defined, where n is the number of Boolean variables, such that the size of each deterministic (k −1)-IBDD for PJ k,n is not polynomially bounded, if k (1 − ε) log log n, for some constant ε > 0, whereas there is a deterministic k-OBDD of size O(kn 2 ) for PJ k,n .…”

“…. , x n } of size n O(1) and width w, where the number k of layers is a constant greater or equal to two, and s and t 1 are the source and the 1-target. Moreover, let B represent the Boolean function f and g under the existential and the parity acceptance mode, respectively.…”