Algebraic Methods in Computational Complexity

From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 “Algebraic Methods in Computational Complexity “ was held in Schloss Dagstuhl-Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

09421 Abstracts Collection -- Algebraic Methods in Computational Complexity

From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 ``Algebraic Methods in Computational Complexity \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Levin–Kolmogorov Complexity is not in Linear Time

Understanding the computational hardness of Kolmogorov complexity is a central open question in complexity theory. An important notion is Levin\u27s version of Kolmogorov complexity, Kt, and its decisional variant, MKtP, due to its connections to universal search, derandomization, and oneway functions, among others. The question whether MKtP can be computed in polynomial time is particularly interesting because it is not subject to known technical barriers such as algebrization or natural proofs that would explain the lack of a proof for MKtP not in P. We take a significant step towards proving MKtP not in P by developing an algorithmic approach for showing unconditionally that MKtP not in DTIME[O(n)] cannot be decided in deterministic linear time in the worst-case. This allows us to partially affirm a conjecture by Ren and Santhanam [STACS:RS22] about a non-halting variant of Kt complexity. Additionally, we give conditional lower bounds for MKtP that tolerate either more runtime or one-sided error

Computational complexity theory and the philosophy of mathematics

Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the P≠NP problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof

An Axiomatic Approach to Algebrization

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: (1) Fortnow [12] 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. (2) Aaronson and Wigderson [1] give definitions of algebrizing separations an