Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Non-commutative circuits and the sum-of-squares problem
Proceedings of the forty-second ACM symposium on Theory of computing
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Almost settling the hardness of noncommutative determinant
Proceedings of the forty-third annual ACM symposium on Theory of computing
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In 1991, Nisan proved an exponential lower bound on the size of an algebraic branching program (ABP) that computes the determinant of a matrix in the noncommutative “free algebra” setting, in which there are no nontrivial relationships between the matrix entries. By contrast, when the matrix entries commute there are polynomial size ABPs for the determinant. This paper extends Nisan’s result to a much wider class of noncommutative algebras, including all nontrivial matrix algebras over any field of characteristic 0, group algebras of all nonabelian finite groups over algebraically closed fields of characteristic 0, the quaternion algebra, and the Clifford algebras. As a result, we obtain more compelling evidence for the essential role played by commutativity in the efficient computation of the determinant. The key to our approach is a characterization of noncommutative algebras by means of the polynomial identities that they satisfy. Extending Nisan’s lower bound framework, we find that any reduction in complexity compared to the free algebra must arise from the ability of the identities to reduce the rank of certain naturally associated matrices. Using results from the theory of algebras with polynomial identities, we are able to show that none of the identities of the above classes of algebras is able to achieve such a rank reduction.