The Complexity of Tensor Circuit Evaluation
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
The complexity of tensor calculus
Computational Complexity
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Tensor calculus over semirings is shown relevant to complexity theory in unexpected ways. First, evaluating well-formed tensor formulas with explicit tensor entries is shown complete for \mathP, for NP, and for #P as the semiring varies. Indeed the permanent of a matrix is shown expressible as the value of a tensor formula in much the same way that Berkowitz' theorem expresses its determinant. Second, restricted tensor formulas are shown to capture the classes LOGCFL and NL, their parity counterparts place \mathLOGCFL and place \mathL, and several other counting classes. Finally, the known inclusions NP/poly \math\mathP/poly, LOGCFL/poly \math\mathLOGCFL/poly, and NL/poly \math\mathL/poly, which have scattered proofs in the literature [21, 39], are shown to follow from the new characterizations in a single blow.