The Complexity of Tensor Circuit Evaluation

• Authors: Martin Beaudry;Markus Holzer
• Venue: Computational Complexity
• Year: 2007

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 驴 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.