The Complexity of Tensor Circuit Evaluation

Abstract

The study of tensor calculus over semirings in terms of complexity theory was initiated by Damm et al. in [8]. Here we first look at tensor circuits, a natural generalization of tensor formulas; we show that the problem of asking whether the output of such circuits is non-zero is complete for the class NE = NTIME(2^o(n)) for circuits over the boolean semiring, ⊕E for the field $\mathbb{F}_2$, and analogous results for other semirings. Common sense restrictions such as imposing a logarithmic upper bound on circuit depth are also discussed. Second, we analyze other natural problems concerning tensor formulas and circuits over various semirings, such as asking whether the output matrix is diagonal or a null matrix.

Supported by the Québec FCAR, by the NSERC of Canada, and by the DFG of Germany.