The complexity of tensor calculus

Abstract

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 $\bigoplusP$, 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’s theorem expresses its determinant. Second, restricted tensor formulas are shown to capture the classes LOGCFL and NL, their parity counterparts $\bigoplusLOGCFL$ and $\bigoplusL$, and several other counting classes. Finally, the known inclusions $\NP/\poly \subseteq \bigoplusP/\poly$, $\LOGCFL/\poly \subseteq \bigoplusLOGCFL/\poly$, and $\NL/\poly \subseteq \bigoplusL/\poly$, which have scattered proofs in the literature (Valiant & Vazirani 1986; Gál & Wigderson 1996), are shown to follow from the new characterizations in a single blow. As an intermediate tool, we define and make use of the natural notion of an algebraic Turing machine over a semiring $ \mathcal{S}$.