JVe consider algebraic computations which are not allowed to rely on the commut,at,ivity of multiplication. We obtain various lower bounds for algebraic formula size in this model: (1) Computing the determinant is as hard as computing the permanent and tight exponential upper and lower bounds are given. (2) Computation cannot be parallelized, as opposed to in the commutative case - this solves in the negative an open problem of hliller et al . (3) The question of the power of negation in this model is shown to be closely related to a well known open problem relating communication complexity and rank. We then take modest steps towards extending our results to general, commutative algebraic computation, and prove exponential lower bounds for monotone algebraic circuit size, as well as for the size of certain types of constant depth algebraic circuits.
|Title of host publication
|Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, STOC 1991
|Association for Computing Machinery
|Number of pages
|Published - 3 Jan 1991
|23rd Annual ACM Symposium on Theory of Computing, STOC 1991 - New Orleans, United States
Duration: 5 May 1991 → 8 May 1991
|Proceedings of the Annual ACM Symposium on Theory of Computing
|23rd Annual ACM Symposium on Theory of Computing, STOC 1991
|5/05/91 → 8/05/91
Bibliographical notePublisher Copyright:
© 1991 ACM.