A very simple function that requires exponential size nondeterministic graph-driven read-once branching programs

Branching programs are a well-established computation model for Boolean functions, especially read-once branching programs (BP1s) have been studied intensively. A very simple function f in n² variables is exhibited such that both the function f and its negation ¬f can be computed by Σ³_p-circuits, the function f has nondeterministic BP1s (with one nondeterministic node) of linear size and ¬f has size O(n⁴) for oblivious nondeterministic BP1s but f requires nondeterministic graph-driven BP1s of size . This answers an open question stated by Jukna, Razborov, Savický, and Wegener [Comput. Complexity 8 (1999) 357-370].