The Complexity of Tensor Circuit Evaluation

Abstract.

We continue the study of tensor calculus over semirings in terms of complexity theory initiated by Damm et al. (2003). First, we look at tensor circuits, a natural generalization of tensor formulas; we show that the problem to determine whether the circuit output over a certain semiring is non-zero is complete for NE = NTime(2^O(n)) over the Boolean semiring, for $\oplus$ E over the field $\mathbb{F}_{2}$, and for analogous classes over other semirings. Moreover, common-sense restrictions such as imposing bounds on circuit and/or tensor depth, are shown to elegantly capture the classes P, NTime $(2^{O(log^{k} n)})$, NTimeSpace($2^{O(log^{k} n)}$, n^O(1)), for k ≥ 1, PSPACE, and their counting counterparts. The proofs of these results use a model of algebraic Turing machines over a semiring together with a predicate-based approach on counting, which is similar to that of Toda (1991). This allows characterizations of the classes $\oplus$ P, NP, co-NP, co-DP, C₌P, SPP, USP, and UP, and their exponential time counterparts, in a single framework. Finally, we show that a number of natural problems concerning tensor formulas and circuits, such as asking whether the output of a formula/circuit is a diagonal matrix, or the identity matrix, or a permutation matrix, capture the classes $\prod^{p}_{2}$ for formulas and $\prod^{e}_{2}$ for circuits over the Boolean semiring; other semirings are also discussed.