The motivating question for this work is a long standing open problem, posed by Nisan [Noam Nisan, 1991], 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 [Hrubeš et al., 2011]) to define abecedarian polynomials and models that naturally compute them. Our main contribution is a possible new approach towards resolving the VF_{nc} vs VBP_{nc} question, via lower bounds against abecedarian formulas. In particular, we show the following. There is an explicit n²-variate degree d abecedarian polynomial f_{n,d}(𝐱) such that - f_{n, d}(𝐱) can be computed by an abecedarian ABP of size O(nd); - any abecedarian formula computing f_{n, log n}(𝐱) must have size at least n^{Ω(log log n)}. We also show that a super-polynomial lower bound against abecedarian formulas for f_{log n, n}(𝐱) would separate the powers of formulas and ABPs in the non-commutative setting.
@InProceedings{chatterjee:LIPIcs.CCC.2021.7, author = {Chatterjee, Prerona}, title = {{Separating ABPs and Some Structured Formulas in the Non-Commutative Setting}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {7:1--7:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.7}, URN = {urn:nbn:de:0030-drops-142812}, doi = {10.4230/LIPIcs.CCC.2021.7}, annote = {Keywords: Non-Commutative Formulas, Lower Bound, Separating ABPs and Formulas} }
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