Laurent Bienvenu

Reconciling data compression and kolmogorov complexity

While data compression andKolmogorov complexity are both about effective coding of words, the two settings differ in the following respect. A compression algorithm or compressor, for short, has to map a word to a unique code for this word in one shot, whereas with the standard notions of Kolmogorov complexity a word has many different codes and the minimum code for a given word cannot be found effectively. This gap is bridged by introducing decidable Turing machines and a corresponding notion of Kolmogorov complexity, where compressors and suitably normalized decidable machines are essentially the same concept. Kolmogorov complexity defined via decidable machines yields characterizations in terms of the intial segment complexity of sequences of the concepts of Martin-Lof randomness, Schnorr randomness, Kurtz randomness, and computable dimension. These results can also be reformulated in terms of time-bounded Kolmogorov complexity. Other applications of decidable machines are presented, such as a simplified proof of the Miller-Yu theorem (characterizing Martin-Lof randomness by the plain complexity of the initial segments) and a new characterization of computably traceable sequences via a natural lowness notion for decidable machines.

Coherent randomness tests and computing the

We show that a Martin-Lof random set for which the effective version of the Lebesgue density theorem fails computes every K-trivial set. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem (Question 4.6 in Randomness and computability: Open questions. Bull. Symbolic Logic, 12(3):390–410, 2006). On the other hand, we settle stronger variants of the covering problem in the negative. We show that any witness for the solution of the covering problem, namely an incomplete random set which computes all K-trivial sets, must be very close to being Turing complete. For example, such a random set must be LR-hard. Similarly, not every K-trivial set is computed by the two halves of a random set. The work passes through a notion of randomness which characterises computing K-trivial sets by random sets. This gives a “smart” K-trivial set, all randoms from whom this set is computed have to compute all K-trivial sets.

K

We study generalizations of Demuths Theorem, which states that the image of a Martin-Lof random real under a tt-reduction is either computable or Turing equivalent to a Martin-Lof random real. We show that Demuths Theorem holds for Schnorr randomness and computable randomness (answering a question of Franklin), but that it cannot be strengthened by replacing the Turing equivalence in the statement of the theorem with wtt-equivalence. We also provide some additional results about the Turing and tt-degrees of reals that are random with respect to some computable measure.

-trivial sets

This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim H(S) and constructive packing dimension dim P(S) is Turing equivalent to a sequence R with dim H(R)≥(dim H(S)/dim P(S))−ε, for arbitrary ε>0. Furthermore, if dim P(S)>0, then dim P(R)≥1−ε. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension.A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dim H(S)/dim P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim H(S)=dim P(S)) such that dim H(S)>0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1.Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.

Strong reductions in effective randomness

The Denjoy-Young-Saks Theorem from classical analysis states that for an arbitrary real-valued function f, the Denjoy alternative holds outside a null set. This means that for almost every real x, either the derivative of f exists at x, or the derivative fails to exist in the worst possible way: the slopes of f around x take arbitrarily large positive values and arbitrarily large negative values. Demuth studied effective versions of this theorem, in particular the effective version when the function f is Markov computable. He then looked at the set DA of reals x such that any Markov computable function satisfies the Denjoy alternative at x. He introduced a notion of algorithmic randomness (now known as Demuth randomness) which he proved to be sufficient for a point to belong to DA. In this paper, we show that DA is in fact strictly contained in the set of Demuth random points. We also obtain interesting side results about effective versions of Lebesgues density theorem.

Constructive Dimension and Turing Degrees

A theorem of Kucera states that given a Martin-Lof random infinite binary sequence ω and an effectively open set A of measure less than 1, some tail of ω is not in A. We show that this result can be seen as an effective version of Birkhoffs ergodic theorem (in a special case). We prove several results in the same spirit and generalize them via an effective ergodic theorem for bijective ergodic maps.

The Denjoy alternative for computable functions

This paper examines the constructive Hausdorff and packingdimensions of weak truth-table degrees. The main result is thatevery infinite sequence Swith constructive Hausdorffdimension dimH(S) and constructive packingdimension dimP(S) is weak truth-tableequivalent to a sequence Rwith

Ergodic-type characterizations of algorithmic randomness

{\rm dim_H}({\it R}) \geq{\rm dim_H}(S) / {\rm dim_P}(S) - \epsilon

Constructive Dimension and Weak Truth-Table Degrees

, for arbitrarye> 0. Furthermore, ifdimP(S) > 0, thendimP(R) ≥ 1 - e. Thereduction thus serves as a randomness extractorthatincreases the algorithmic randomness of S, as measured byconstructive dimension. A number of applications of this result shed new light on theconstructive dimensions of wtt degrees (and, by extension, Turingdegrees). A lower bound of dimH(S) /dimP(S) is shown to hold for the wtt degree ofany sequence S. A new proof is given of a previously-knownzero-one law for the constructive packing dimension of wtt degrees.It is also shown that, for any regularsequence S(that is, dimH(S) =dimP(S)) such that dimH(S)> 0, the wtt degree of Shas constructive Hausdorff andpacking dimension equal to 1. Finally, it is shown that no single Turing reduction can be auniversalconstructive Hausdorff dimension extractor.

Effective Randomness for Computable Probability Measures

Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of random elements coincide. Elaborating on work of Bienvenu [Bienvenu, L., Constructive equivalence relations on computable probability measures, International Computer Science Symposium in Russia, Lecture Notes in Computer Science 3967 (2006), 92-103], we determine all the implications that hold between the equivalence relations induced by Martin-Lof randomness, computable randomness, Schnorr randomness, and weak randomness, and the equivalence and consistency relations of probability measures, except that we do not know whether two computable probability measures need to be equivalent in case their respective concepts of weakly randomness coincide.