ABSTRACT
An arithmetic formula is multi-linear if the polynomial computed by each of its sub-formulas is multi-linear. We prove that any multi-linear arithmetic formula for the permanent or the determinant of an n x n matrix is of size super-polynomial in n.Previously, super-polynomial lower bounds were not known (for any explicit function) even for the special case of multi-linear formulas of constant depth.
- S. Aaronson. Multilinear Formulas and Skepticism of Quantum Computing. STOC 2004 Google ScholarDigital Library
- N. Alon, J.H. Spencer, P.Erdos. The Probabiliatic Method. John Wiley and Sons, Inc., (1992)Google Scholar
- P. Burgisser, M. Clausen, M. A. Shokrollahi. Algebraic Complexity Theory. Springer-Verlag New York, Inc., (1997)Google Scholar
- J. von zur Gathen. Feasible Arithmetic Computations: Valiant's Hypothesis. J. Symbolic Computation 4(2): 137--172 (1987) Google ScholarDigital Library
- J. von zur Gathen. Algebraic Complexity Theory. Ann. Rev. Computer Science 3: 317--347 (1988) Google ScholarDigital Library
- D. Grigoriev, M. Karpinski. An Exponential Lower Bound for Depth 3 Arithmetic Circuits. STOC 1998: 577--582 Google ScholarDigital Library
- D. Grigoriev, A. A. Razborov. Exponential Lower Bounds for Depth 3 Arithmetic Circuits in Algebras of Functions over Finite Fields. Applicable Algebra in Engineering, Communication and Computing 10(6): 465--487 (2000) (preliminary version in FOCS 1998) Google ScholarDigital Library
- R. Impagliazzo, V. Kabanets. Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds. STOC 2003: 355--364 Google ScholarDigital Library
- A. Kalorkoti. The Formula Size of the Determinant. SIAM Journal of Computing 14: 678--687 (1995)Google ScholarCross Ref
- N. Nisan. Lower Bounds for Non-Commutative Computation. STOC 1991: 410--418 Google ScholarDigital Library
- N. Nisan, A. Wigderson. Lower Bounds on Arithmetic Circuits Via Partial Derivatives. Computational Complexity 6(3): 217--234 (1996) (preliminary version in FOCS 1995) Google ScholarDigital Library
- R. Raz, A. Shpilka. Deterministic Polynomial Identity Testing in Non Commutative Models. Conference on Computational Complexity 2004 (to appear) Google ScholarDigital Library
- C. P. Schnorr. A Lower Bound on the Number of Additions in Monotone Computations. Theoretical Computer Science 2(3): 305--315 (1976)Google ScholarCross Ref
- E. Shamir, M. Snir. On the Depth Complexity of Formulas. Mathematical Systems Theory 13: 301--322 (1980)Google ScholarCross Ref
- A. Shpilka, A. Wigderson. Depth-3 Arithmetic Circuits Over Fields of Characteristic Zero. Computational Complexity 10(1): 1--27 (2001) (preliminary version in Conference on Computational Complexity 1999) Google ScholarDigital Library
- L. G. Valiant. Negation can be Exponentially Powerful. Theoretical Computer Science 12: 303--314 (1980)Google ScholarCross Ref
- L. G. Valiant. Why is Boolean Complexity Theory Difficult? In Boolean Function Complexity (M. S. Paterson, ed.) Lond. Math. Soc. Lecture Note Ser. Vol. 169, Cambridge Univ. Press 84--94 (1992) Google ScholarDigital Library
Index Terms
- Multi-linear formulas for permanent and determinant are of super-polynomial size
Recommendations
Tensor-Rank and Lower Bounds for Arithmetic Formulas
We show that any explicit example for a tensor A : [n]r → F with tensor-rank ≥ nrċ(1−o(1)), where r = r(n) ≤ log n/log log n is super-constant, implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over F. This ...
Multi-linear formulas for permanent and determinant are of super-polynomial size
An arithmetic formula is multilinear if the polynomial computed by each of its subformulas is multilinear. We prove that any multilinear arithmetic formula for the permanent or the determinant of an n × n matrix is of size super-polynomial in n. ...
Non-commutative circuits and the sum-of-squares problem
STOC '10: Proceedings of the forty-second ACM symposium on Theory of computingWe initiate a direction for proving lower bounds on the size of non-commutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of non-commutative arithmetic circuits and a problem about commutative degree ...
Comments