Abstract
Nisan (STOC 1991) exhibited a polynomial which is computable by linear sized noncommutative circuits but requires exponential sized non-commutative algebraic branching programs. Nisan’s hard polynomial is in fact computable by linear sized skew circuits (skew circuits are circuits where every multiplication gate has the property that all but one of its children is an input variable or a scalar). We prove that any non-commutative skew circuit which computes the square of the polynomial defined by Nisan must have exponential size. As a further step towards proving exponential lower bounds for general non-commutative circuits, we also extend our techniques to prove an exponential lower bound for a class of circuits which is a restriction of general non-commutative circuits and a generalization of non-commutative skew circuits. More precisely, we consider non-commutative circuits of small non-skew depth, which denotes the maximum number of non-skew gates on any path from the output gate to an input gate. We show that for any k < d, there is an explicit polynomial of degree d over n variables that has non-commutative circuits of polynomial size but such that any circuit with non-skew depth k must have size at least n. It is not hard to see that any polynomial of degree d that has polynomial size circuits has a polynomial-sized circuit with non-skew depth d. Hence, our results should be interpreted as proving lower bounds for the class of circuits with non-trivially small non-skew depth. As far as we know, this is the strongest model of non-commutative computation for which we have superpolynomial lower bounds.
Highlights
The second reason is that proving explicit lower bounds for non-commutative arithmetic circuits is formally an easier problem than that of proving lower bounds for arithmetic circuits described in the previous paragraph, and it is hoped that techniques discovered in the course proving non-commutative lower bounds will be useful in the commutative setting as well
Note that a superpolynomial lower bound for non-commutative skew circuits was claimed by Allender et al [1], but, the proof of this particular result in the paper (Theorem 7.12) seems to fail because it did not take into account possible cancellations
Our lower bound shows that skew circuits are exponentially less powerful than circuits with just one non-skew gate. This is because the explicit polynomial for which we prove a lower bound is just the square of a polynomial considered by Nisan, and this polynomial in turn has skew circuits of linear size
Summary
If we want to design an efficient algorithm for a computational problem that is naturally stated as a polynomial—such as the determinant or the permanent, matrix multiplication, Fast Fourier Transform, etc.— arithmetic circuits capture most natural candidate algorithms that we might consider. Proving explicit superpolynomial lower bounds for general arithmetic circuits is a celebrated open question in complexity theory and one of the possible approaches to the P versus NP question (The formal definition of an ABP is given in Definition 3.1.) This might have led one to think that a superpolynomial lower bound for general (non-commutative) arithmetic circuits was close at hand. In a more recent result, Hrubeš, Wigderson, and Yehudayoff [9] suggested a new line of attack on the general arithmetic circuit lower bound question. Their result introduces a “product lemma” for general arithmetic circuits that generalizes a decomposition of ABPs due to Nisan [15]. The strongest known computational model for which we have superpolynomial lower bounds remains the ABPs from the paper of Nisan [15]
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