Lower Bounds and PIT for Non-commutative Arithmetic Circuits with Restricted Parse Trees

Guillaume Lagarde,Srikanth Srinivasan,Nutan Limaye

doi:10.1007/s00037-018-0171-9

Guillaume Lagarde, Srikanth Srinivasan + Show 1 more

Open Access

https://doi.org/10.1007/s00037-018-0171-9

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Abstract

We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring $${\mathbb{F}\langle{x_1,\ldots,x_N\rangle}}$$ , where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions capture essentially all non-commutative circuit models for which lower bounds are known. We prove several results about such circuits.

Highlights

In this paper, we study questions related to Arithmetic Circuits, which are computational devices that use arithmetic operations to compute multivariate polynomials over a field F
We make progress on a question of Nisan (STOC 1991) regarding separating the power of Algebraic Branching Programs (ABPs) and Formulas in the non-commutative setting by showing a tight lower bound of nΩ(log d) for any Unique Parse Tree (UPT) formula computing the product of d n × n matrices
Nisan [21] justified this by proving exponential lower bounds for non-commutative formulas, and more generally Algebraic Branching Programs (ABPs), computing the Determinant and Permanent

Summary

Introduction

We study questions related to Arithmetic Circuits, which are computational devices that use arithmetic operations (such as + and ×) to compute multivariate polynomials over a field F. The results of Kayal, Saha, and Saptharishi [15] and Fournier, Limaye, Malod, and Srinivasan [8] together prove a superpolynomial lower bound on the size of regular formulas (defined by [15]) computing the product of d n × n matrices in the commutative setting. While these formulas (in the non-commutative setting) are definitely UPT, the converse is not true. Extending this idea exactly as in [10], we can efficiently check if the sum of any small number of UPT circuits is 0

Non-commutative polynomials

The partial derivative matrix

Standard definitions related to non-commutative circuits

Non-commutative circuits with restricted parse trees

Lower bounds for k-PT circuits

Other results