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Bruno Bauwens

Publications and source records attributed to Bruno Bauwens.

31 records · Page 2Linked to original sources
arXiv2016

Conditional probabilities and van Lambalgen theorem revisited

The definition of conditional probability in case of continuous distributions was an important step in the development of mathematical theory of probabilities. How can we define this notion in algorithmic probability theory? In this survey we discuss the developments in this direction trying to explain what are the difficulties and what can be done to avoid them. Almost all the results discussed in this paper have been published (and we provide the references), but we tried to put them into perspective and to explain the proofs in a more intuitive way. We assume that the reader is familiar with basic notions of measure theory and algorithmic randomness.

math.LO↗
arXiv2016

Van Lambalgen's theorem fails for some computable measure

Van Lambalgen's theorem states that a pair $(α,β)$ of bitsequences is Martin-Löf random if and only if $α$ is Martin-Löf random and $β$ is Martin-Löf random relative to $α$. In [Information and Computation 209.2 (2011): 183-197, Theorem 3.3], Hayato Takahashi generalized van Lambalgen's theorem for computable measures $P$ on a product of two Cantor spaces; he showed that the equivalence holds for each $β$ for which the conditional probability $P(\cdot | β)$ is computable. He asked whether this computability condition is necessary. We give a positive answer by providing a computable measure for which van Lambalgen's theorem fails. We also present a simple construction of a measure for which conditional measure is not computable. Such measures were first constructed by N. Ackerman, C. Freer and D. Roy in [Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 107-116. IEEE (2011)].

math.LO↗
arXiv2015

Linear list-approximation for short programs (or the power of a few random bits)

A $c$-short program for a string $x$ is a description of $x$ of length at most $C(x) + c$, where $C(x)$ is the Kolmogorov complexity of $x$. We show that there exists a randomized algorithm that constructs a list of $n$ elements that contains a $O(\log n)$-short program for $x$. We also show a polynomial-time randomized construction that achieves the same list size for $O(\log^2 n)$-short programs. These results beat the lower bounds shown by Bauwens et al. \cite{bmvz:c:shortlist} for deterministic constructions of such lists. We also prove tight lower bounds for the main parameters of our result. The constructions use only $O(\log n)$ ($O(\log^2 n)$ for the polynomial-time result) random bits . Thus using only few random bits it is possible to do tasks that cannot be done by any deterministic algorithm regardless of its running time.

cs.CC↗
arXiv2014

Relating and contrasting plain and prefix Kolmogorov complexity

In [3] a short proof is given that some strings have maximal plain Kolmogorov complexity but not maximal prefix-free complexity. The proof uses Levin's symmetry of information, Levin's formula relating plain and prefix complexity and Gacs' theorem that complexity of complexity given the string can be high. We argue that the proof technique and results mentioned above are useful to simplify existing proofs and to solve open questions. We present a short proof of Solovay's result [21] relating plain and prefix complexity: $K (x) = C (x) + CC (x) + O(CCC (x))$ and $C (x) = K (x) - KK (x) + O(KKK (x))$, (here $CC(x)$ denotes $C(C(x))$, etc.). We show that there exist $ω$ such that $\liminf C(ω_1\dots ω_n) - C(n)$ is infinite and $\liminf K(ω_1\dots ω_n) - K(n)$ is finite, i.e. the infinitely often C-trivial reals are not the same as the infinitely often K-trivial reals (i.e. [1,Question 1]). Solovay showed that for infinitely many $x$ we have $|x| - C (x) \le O(1)$ and $|x| + K (|x|) - K (x) \ge \log^{(2)} |x| - O(\log^{(3)} |x|)$, (here $|x|$ denotes the length of $x$ and $\log^{(2)} = \log\log$, etc.). We show that this result holds for prefixes of some 2-random sequences. Finally, we generalize our proof technique and show that no monotone relation exists between expectation and probability bounded randomness deficiency (i.e. [6, Question 1]).

cs.CC↗
arXiv2014

Asymmetry of the Kolmogorov complexity of online predicting odd and even bits

Symmetry of information states that $C(x) + C(y|x) = C(x,y) + O(\log C(x))$. We show that a similar relation for online Kolmogorov complexity does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring $x_1x_2... x_n$ be the length of a shortest program that computes $x_2$ on input $x_1$, computes $x_4$ on input $x_1x_2x_3$, etc; and similar for odd complexity. We show that for all n there exist an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence $x_2x_1x_4x_3\ldots$, decreases the sum of odd and even complexity to $C(x)$.

cs.IT↗
arXiv2013

Upper semicomputable sumtests for lower semicomputable semimeasures

A sumtest for a discrete semimeasure $P$ is a function $f$ mapping bitstrings to non-negative rational numbers such that \[ \sum P(x)f(x) \le 1 \,. \] Sumtests are the discrete analogue of Martin-Löf tests. The behavior of sumtests for computable $P$ seems well understood, but for some applications lower semicomputable $P$ seem more appropriate. In the case of tests for independence, it is natural to consider upper semicomputable tests (see [B.Bauwens and S.Terwijn, Theory of Computing Systems 48.2 (2011): 247-268]). In this paper, we characterize upper semicomputable sumtests relative to any lower semicomputable semimeasures using Kolmogorov complexity. It is studied to what extend such tests are pathological: can upper semicomputable sumtests for $m(x)$ be large? It is shown that the logarithm of such tests does not exceed $\log |x| + O(\log^{(2)} |x|)$ (where $|x|$ denotes the length of $x$ and $\log^{(2)} = \log\log$) and that this bound is tight, i.e. there is a test whose logarithm exceeds $\log |x| - O(\log^{(2)} |x|$) infinitely often. Finally, it is shown that for each such test $e$ the mutual information of a string with the Halting problem is at least $\log e(x)-O(1)$; thus $e$ can only be large for ``exotic'' strings.

cs.CC↗
arXiv2013

Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs

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 ω equals lim inf [n - C (ω1 . . . ωn)] + 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 (ω1 . . . ωn)] + O(1).

cs.IT↗
arXiv2013

Complexity of complexity and strings with maximal plain and prefix Kolmogorov complexity

Peter Gacs showed (Gacs 1974) that for every n there exists a bit string x of length n whose plain complexity C(x) has almost maximal conditional complexity relative to x, i.e., C(C(x)|x) > log n - log^(2) n - O(1). (Here log^(2) i = log log i.) Following Elena Kalinina (Kalinina 2011), we provide a simple game-based proof of this result; modifying her argument, we get a better (and tight) bound log n - O(1). We also show the same bound for prefix-free complexity. Robert Solovay showed (Solovay 1975) that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length): for some c there exist infinitely many x such that |x| - C(x) < c and |x| + K(|x|) - K(x) > log^(2) |x| - c log^(3) |x|. In fact, the results of Solovay and Gacs are closely related. Using the result above, we provide a short proof for Solovay's result. We also generalize it by showing that for some c and for all n there are strings x of length n with n - C (x) < c and n + K(n) - K(x) > K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. We also prove a close upper bound K(K(n)|n) + O(1). Finally, we provide a direct game proof for Joseph Miller's generalization (Miller 2006) of the same Solovay's theorem: if a co-enumerable set (a set with c.e. complement) contains for every length a string of this length, then it contains infinitely many strings x such that |x| + K(|x|) - K(x) > log^(2) |x| + O(log^(3) |x|).

cs.CC↗
arXiv2010

m-sophistication

The m-sophistication of a finite binary string x is introduced as a generalization of some parameter in the proof that complexity of complexity is rare. A probabilistic near sufficient statistic of x is given which length is upper bounded by the m-sophistication of x within small additive terms. This shows that m-sophistication is lower bounded by coarse sophistication and upper bounded by sophistication within small additive terms. It is also shown that m-sophistication and coarse sophistication can not be approximated by an upper or lower semicomputable function, not even within very large error.

cs.CC↗
arXiv2009

Influence tests I: ideal composite hypothesis tests, and causal semimeasures

Ratios of universal enumerable semimeasures corresponding to hypotheses are investigated as a solution for statistical composite hypotheses testing if an unbounded amount of computation time can be assumed. Influence testing for discrete time series is defined using generalized structural equations. Several ideal tests are introduced, and it is argued that when Halting information is transmitted, in some cases, instantaneous cause and consequence can be inferred where this is not possible classically. The approach is contrasted with Bayesian definitions of influence, where it is left open whether all Bayesian causal associations of universal semimeasures are equal within a constant. Finally the approach is also contrasted with existing engineering procedures for influence and theoretical definitions of causation.

math.ST↗
arXiv2009

On the equivalence between minimal sufficient statistics, minimal typical models and initial segments of the Halting sequence

It is shown that the length of the algorithmic minimal sufficient statistic of a binary string x, either in a representation of a finite set, computable semimeasure, or a computable function, has a length larger than the computational depth of x, and can solve the Halting problem for all programs with length shorter than the m-depth of x. It is also shown that there are strings for which the algorithmic minimal sufficient statistics can contain a substantial amount of information that is not Halting information. The weak sufficient statistic is introduced, and it is shown that a minimal weak sufficient statistic for x is equivalent to a minimal typical model of x, and to the Halting problem for all strings shorter than the BB-depth of x.

cs.CC↗