The complexity of tensor calculus

  • Authors:

  • Carsten Damm;Markus Holzer;Pierre McKenzie

  • Affiliations:

  • Institut für Angewandte und Numerische Mathematik, Georg-August-Universität Göttingen, D-37083 Göttingen, Germany;Institut für Informatik, Technische Universität München, Arcisstraße 21, D-80290 München, Germany;Département d'I.R.O., Université de Montréal, C.P. 6128, succ. Centre-Ville, Montréal (Québec), H3C 3J7 Canada

  • Venue:
  • Computational Complexity
  • Year:
  • 2002

Quantified Score

Hi-index 0.00

Visualization

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; 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 S.