Catalytic branching programs from groups and general protocols

CCCatalytic branching programs (catalytic bps) compute the same n-bit boolean function f at multiple entry points that need to be remembered at the exit nodes of the bp. When a doubly exponential number of entry points is allowed, linear amortized catalytic bp size is known to be achievable for any f. Here a method is introduced that produces a catalytic bp out of a reversible bp and a deterministic dag-like communication protocol. In a multiplicity range as low as linear, approximating a threshold is shown possible at linear amortized cost. In the same low range, computing \(\texttt {Maj} \) and \(\texttt {Mod} \) are shown possible at a cost that beats the brute force repetition of the best known bp for these functions by a polylog factor. In the exponential range, the method yields O(nlog n) amortized cost for any symmetric function.