Lower Bounds for the Sum of Graph--driven Read--Once Parity Branching Programs

We prove the first lower bound for restricted read–once parity branching programs with unlimited parity nondeterminism where for each input the variables may be tested according to several orderings. Proving a superpolynomial lower bound for read–once parity branching programs is still a challenging open problem. The following variant of read–once parity branching programs is well–motivated. Let k be a fixed integer. For each input a there are k orderings σ1(a), . . . , σk(a) of the variables such that for each computation path activated by a the bits are queried according to σi(a) for some i, 1 ≤ i ≤ k. This model that we call k–⊕BP1s for convenience strictly generalizes all restricted variants of read–once parity branching programs for that lower bounds are known. We consider a slightly more restricted version, i.e. the sum of k graph–driven ⊕BP1s with polynomial size graph– orderings. We prove lower bounds for linear codes and show that the considered variant strictly generalizes well–structured graph–driven ⊕BP1s as well as (⊕, k)-BPs examined by Savický and Sieling in [24].