The complexity of tensor calculus

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 /spl oplus/P, 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 /spl oplus/LOGCFL and /spl oplus/L, and several other counting classes. Finally, the known inclusions NP/poly /spl sube//spl oplus/P/poly, LOGCFL/poly /spl sube//spl oplus/LOGCFL/poly, and NL/poly/spl sube//spl oplus//poly, which have scattered proofs in the literature, are shown to follow from the new characterizations in a single blow.