We initiate the study of the complexity of arithmetic circuits with division gates over non-commuting variables. Such circuits and formulas compute <i>non-commutative</i> rational functions, which, despite their name, can no longer be expressed as ratios of polynomials. We prove some lower and upper bounds, completeness and simulation results, as follows.
If <i>X</i> is <i>n</i> x <i>n</i> matrix consisting of <i>n</i><sup>2</sup> distinct mutually non-commuting variables, we show that:
(<i>i… CONTINUE READING