Separating ABPs and some structured formulas in the non-commutative setting

The motivating question for this work is a long standing open problem, posed by Nisan [20], regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question remains open, we make some progress towards its resolution. To that effect, we generalise the notion of ordered polynomials in the non-commutative setting (defined by Hrubeš, Wigderson and Yehudayoff [11]) to define abecedarian polynomials and models that naturally compute them. Our main contribution is a possible new approach towards resolving the VFnc vs VBPnc question, via lower bounds against abecedarian formulas. In particular, we show the following. There is an explicit n2-variate degree d abecedarian polynomial fn,d(x) such that • fn,d(X) can be computed by an abecedarian ABP of size O(nd); • any abecedarian formula computing fn,log n(X) must have size at least nΩ(log log n). We also show that a super-polynomial lower bound against abecedarian formulas for flog n, n(X) would separate the powers of formulas and ABPs in the non-commutative setting.