Improved construction for universality of determinant and permanent. (English) Zbl 1185.68016
Summary: L. Valiant [in: Proc. 11th Annual ACM Symposium on the Theory of Computing, Atlanta, GA, 249–261 (1979)] proved that every polynomial of formula size \(e\) is a projection of the \((e+2)\times (e+2)\) determinant polynomial. We improve “\(e+2\)” to “\(e+1\)”, also for a definition of formula size that does not count multiplications by constants as gates. Our proof imitates the “\(2e+2\)” proof of J. von zur Gathen [J. Symb. Comput. 4, 137–172 (1987; Zbl 0662.68033)], but uses different invariants and a tighter set of base cases.
MSC:
| 68M07 | Mathematical problems of computer architecture |
Citations:
Zbl 0662.68033References:
| [1] | K. Mulmuley, M. Sohoni, Geometric complexity theory, P vs. NP, and explicit obstructions, in: Proceedings, International Conference on Algebra and Geometry, Hyderabad, 2001; K. Mulmuley, M. Sohoni, Geometric complexity theory, P vs. NP, and explicit obstructions, in: Proceedings, International Conference on Algebra and Geometry, Hyderabad, 2001 · Zbl 1046.68057 |
| [2] | L. Valiant, Completeness classes in algebra, in: Proc. 11th Annual ACM Symposium on the Theory of Computing, Atlanta, GA, 1979, pp. 249-261; L. Valiant, Completeness classes in algebra, in: Proc. 11th Annual ACM Symposium on the Theory of Computing, Atlanta, GA, 1979, pp. 249-261 |
| [3] | von zur Gathen, J., Feasible arithmetic computations: Valiant’s hypothesis, Journal of Symbolic Computation, 4, 137-172 (1987) · Zbl 0662.68033 |
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