### Working paper

## Randomized communication complexity of appropximating Kolmogorov complexity

We present a new structural lemma for deterministic con- text free languages. From the first sight, it looks like a pumping lemma, because it is also based on iteration properties, but it has significant distinctions that makes it much easier to apply. The structural lemma is a combinatorial analogue of KC-DCF-Lemma (based on Kolmogorov complexity), presented by Li and Vit ́anyi in 1995 and corrected by Glier in 2003. The structural lemma allows not only to prove that a language is not a DCFL, but discloses the structure of DCFLs Myhill-Nerode classes.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

Joseph Miller [16] and independently Andre Nies, Frank Stephan and Sebastiaan Terwijn [18] gave a complexity characterization of 2-random sequences in terms of plain Kolmogorov complexity C: they are sequences that have infinitely many initial segments with O(1)-maximal plain complexity (among the strings of the same length). Later Miller [17] showed that prefix complexity K can also be used in a similar way: a sequence is 2-random if and only if it has infinitely many initial segments with O(1)-maximal prefix complexity (which is n + K (n) for strings of length n). The known proofs of these results are quite involved; in this paper we provide simple direct proofs for both of them. In [16] Miller also gave a quantitative version of the first result: the 0'-randomness deficiency of a sequence {\omega} equals lim inf [n - C ({\omega}1 . . . {\omega}n)] + O(1). (Our simplified proof can also be used to prove this.) We show (and this seems to be a new result) that a similar quantitative result is also true for prefix complexity: 0'-randomness deficiency equals lim inf [n + K (n) -- K ({\omega}1 . . . {\omega}n)] + O(1). Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs (PDF Download Available). Available from: http://www.researchgate.net/publication/258082399_Prefix_and_plain_Kolmogorov_complexity_characterizations_of_2-randomness_simple_proofs [accessed Oct 20, 2015].

The paper [Harry Buhrman, Michal Kouck ́, Nikolay Vereshcha- y gin. Randomized Individual Communication Complexity. IEEE Con- ference on Computational Complexity 2008: 321-331] considered com- munication complexity of the following problem. Alice has a bi- nary string x and Bob a binary string y, both of length n, and they want to compute or approximate Kolmogorov complexity C(x|y) of x conditional to y. It is easy to show that deterministic communica- tion complexity of approximating C(x|y) with precision α is at least n − 2α − O(1). The above referenced paper asks what is random- ized communication complexity of this problem and shows that for r- round randomized protocols its communication complexity is at least Ω((n/α)1/r ). In this paper, for some positive ε, we show the lower bound 0.99n for (worst case) communication length of any random- ized protocol that with probability at least 0.01 approximates C(x|y) with precision εn for all input pairs.

This book constitutes the refereed proceedings of the First International Conference on Data Compression, Communications and Processing held in Palinuro, Italy, in June 2011.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.