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- W1606475279 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 ⊗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's theorem expresses its determinant. Second, restricted tensor formulas are shown to capture the classes LOGCFL and NL, their parity counterparts ⊗LOGCFL and ⊗L, and several other counting classes. Finally, the known inclusions NP/poly ⊆ ⊗P/poly, LOGCFL/poly ⊆ ⊗LOGCFL/poly, and NL/poly ⊆ ⊗L/poly, which have scattered proofs in the literature (Valiant & Vazirani 1986; Gal & 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 S." @default.
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- W1606475279 date "2000-01-01" @default.
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- W1606475279 title "The Complexity of Tensor Calculus" @default.
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