### Approximating Boolean functions by OBDDs

Andre Gronemeier
2007 Discrete Applied Mathematics
In learning theory and genetic programming, OBDDs are used to represent approximations of Boolean functions. This motivates the investigation of the OBDD complexity of approximating Boolean functions with respect to given distributions on the inputs. We present a new type of reduction for one-round communication problems that is suitable for approximations. Using this new type of reduction, we prove the following results on OBDD approximations of Boolean functions: 1. We show that OBDDs
more » ... ating the well-known hidden weighted bit function for uniformly distributed inputs with constant error have size 2 Θ(n 1/4 ) , improving a previously known result. 2. We prove that for every variable order π the approximation of some output bits of integer multiplication with constant error requires π-OBDDs of exponential size. 1. Alice computes a message m := P A ( x, r ) ∈ M and sends m to Bob. Alice computes her oracle input q 4. Alice and Bob query the oracle g for the input (q A , q B ). Bob gets the oracle's output z := g(q A , q B ) ∈ Z . 5. Bob computes the output P [g](x, y, r) := P B ( m, y, r, z ) ∈ Z. The protocol P [g] computes a function f : X×Y −→ Z with error ε if Prob ρ ( P [g](x, y, r) = f (x, y) ) ≤ ε for all inputs (x, y) ∈ X×Y . The cost of the protocol is c(P [g]) := log 2 |M | . If the output of the protocol does not depend on the oracle, then the corresponding part of the notation is omitted. A deterministic 1-round communication protocol is a randomized protocol, as defined above, where the output of the protocol does not depend on the random input r. Given a probability distribution on the input set X×Y of a communication problem f : X×Y −→ Z, one can define approximations of functions by deterministic 1-round communication protocols. This is done analogously to approximations by OBDDs. Note that the cost of a protocol for a given Boolean function may strongly depend on how the input is distributed among the players. Definition 6. Let f be a Boolean function on the variable set X = {x 1 , . . . , x n }. Each partition Π of the set X into two sets X A and X B defines a corresponding communication problem Π-f where Alice receives the variables from X A while Bob receives the variables from X B . Given a probability distribution µ on the inputs of f , let D A→B ε (Π-f µ ) denote the cost of a cheapest deterministic 1-round communication protocol P that approximates Π-f with error ε with respect to µ. Communication complexity has been used by many authors to prove lower bounds on the size of BPs computing Boolean functions (see  ). Bollig, Sauerhoff and Wegener  observed that the same proof method can be applied to approximations of Boolean functions by BPs.