We prove exponential size lower bounds for nondeterministic and randomized read-k BPs as well as a time-space tradeoff lower bound for unrestricted, deterministic multi-way BPs computing the middle bit of integer multiplication. The lower bound for randomized read-k BPs is superpolynomial as long as the error probability is superpolynomially small. For polynomially small error, we have a polynomial upper bound on the size of approximating read once BPs for this function. The lower bounds follow from a more general result for the graphs of universal hash classes that is applicable to the graphs of arithmetic functions such as integer multiplication, convolution, and finite field multiplication.

Branching problems are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e.g., verification), exact representations are required. For learning boolean functions f on the basis of classified examples, it is sufficient to produce the representation of a function g approximating f. This motivates the investigation of the size of the smallest branching program approximating f. Although several non-approximability results are contained in the papers on randomized branching programs, these results often do not work for the uniform distribution (which is the most important one in applications). Here, the following non-approximability results are presented. (1) It is proven that a simple function from the branching program literature requires exponential size to be approximated with respect to the uniform distribution by OBDDs, which are the most important type of branching programs in applications. (2) The first truly exponential lower bound on the size of approximating syntactic read-k-times branching programs with respect to the uniform distribution and error probability 1/2-2/sup -/spl Omega/(n)/, n the input size, is shown. In order to improve upon the so far best results for error probabilities smaller than 1/3, a strong combinatorial lemma from a recent paper of Ajtai on linear-length branching programs is exploited.

Ordered functional decision diagrams (OFDD's) are a data structure for representation and manipulation of Boolean functions. A polynomial time algorithm for the test whether f/spl les/g for functions f and g given by OFDD's is presented. This is the last basic operation on OFDD's whose complexity was unknown. The result also solves an open question on the complexity of minimizing OFDD's for incompletely specified functions.

The problem of constructing an OBDD cover of minimal size for an incompletely specified Boolean function arises in several applications in the CAD domain, e.g., the verification of sequential machines and the construction of OBDD's for incompletely specified circuits. The complexity of this problem is determined. The decision problem is NP-complete. Efficient approximation algorithms exist only if NP=P.