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Separating ABPs and Some Structured Formulas in the Non-Commutative Setting

Author Prerona Chatterjee

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Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Non-Commutative Formulas
  • Lower Bound
  • Separating ABPs and Formulas

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Abstract

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.

Cite As Get BibTex

Prerona Chatterjee. Separating ABPs and Some Structured Formulas in the Non-Commutative Setting. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.7

Author Details

Prerona Chatterjee
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

Research supported by the Department of Atomic Energy, Government of India, under project number RTI4001.

Acknowledgements

We are thankful to Ramprasad Saptharishi, Mrinal Kumar, C. Ramya and especially Anamay Tengse for the discussions at various stages of this work. We would also like to thank Ramprasad Saptharishi, Anamay Tengse and Kshitij Gajjar for helping with the presentation of the paper. Finally, we would like to thank the anonymous reviewers for their valuable comments that have helped in improving the paper.

References

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