An axiomatic approach to algebrization
Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova
Non-relativization of complexity issues can be interpreted as giving some evidence that these issues cannot be resolved by ``black-box'' techniques. In the early 1990's, a sequence of important non-relativizing results was proved, mainly using algebraic techniques. Two approaches have been proposed to understand the power and limitations of these algebraic techniques:
- Fortnow gives a construction of a class of oracles which have a similar algebraic and logical structure, although they are arbitrarily powerful. He shows that many of the non-relativizing results proved using algebraic techniques hold for all such oracles, but he does not show, e.g., that the outcome of the ``P vs. NP'' question differs between different oracles in that class.
- Aaronson and Wigderson give definitions of algebrizing separations and collapses of complexity classes, by comparing classes relative to one oracle to classes relative to an algebraic extension of that oracle. Using these definitions, they show both that the standard collapses and separations ``algebrize'' and that many of the open questions in complexity fail to ``algebrize'', suggesting that the arithmetization technique is close to its limits. However, it is unclear how to formalize algebrization of more complicated complexity statements than collapses or separations, and whether the algebrizing statements are, e.g., closed under modus ponens; so it is conceivable that several algebrizing premises could imply (in a relativizing way) a non-algebrizing conclusion.
In this paper, building on the work of Arora, Impagliazzo, and Vazirani, we propose an axiomatic approach to ``algebrization'', which complements and clarifies the approaches of Fortnow and Aaronson&Wigderson. We present logical theories formalizing the notion of algebrizing techniques in the following sense: most known complexity results proved using arithmetization are provable within our theories, while many open questions are independent of the theories. So provability in the proposed theories can serve as a surrogate for provability using the arithmetization technique.
Our theories extend the Arora et al.'s theory with a new axiom, Arithmetic Checkability which intuitively says that all NP languages have verifiers that are efficiently computable low-degree polynomials (over the integers). We show the following:
- Arithmetic checkability holds relative to arbitrarily powerful oracles (since Fortnow's algebraic oracles all satisfy the Arithmetic Checkability axiom).
- Most of the algebrizing collapses and separations from Aaronson&Wigderson, such as IP=PSPACE, NP ⊂ ZKIP if one-way functions exist, MA-EXP ⊄ P/poly, etc., are provable from Arithmetic Checkability.
- Many of the open complexity questions (including most of those shown to require non-algebrizing techniques in Aaronson&Wigderson), such as ``P vs. NP'', ``NP vs. BPP'', etc., cannot be proved from Arithmetic Checkability.
- Arithmetic Checkability is also insufficient to prove one known result, NEXP=MIP (although relative to an oracle satisfying Arithmetic Checkability, NEXPO restricted to poly-length queries is contained in MIPO, mirroring a similar result from Aaronson&Wigderson).
- extended abstract in Proceedings of the Forty-First ACM Symposium on Theory of Computing (STOC'09), pages 695-705, 2009.
- full version (draft).