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Tomislav Petrović

Publications and source records attributed to Tomislav Petrović.

4 recordsLinked to original sources
arXiv2025

Kolmogorov-Loveland betting strategies lose the Betting game on open sets

Whether Kolmogorov-Loveland randomness (KLR) is the same as Martin-Löf randomness (MLR) is a major open problem in the study of algorithmic randomness. More general classes of betting strategies than Kolmogorov-Loveland ones have been studied in \cite{MMS, Rute, TP}. In each case it was proven that the class induces a notion of randomness equivalent to MLR. In all of those proofs, it was shown that the class contains a finite set of betting strategies such that for any given bound, when betting on a binary sequence contained in an effective open set of small enough measure, at least one of the betting strategies in the set earns capital larger than the bound. We show that the class of Kolmogorov-Loveland betting strategies does not have this property.

cs.IT↗
arXiv2022

Betting strategies with bounded splits

We show that a pair of Kolmogorov-Loveland betting strategies cannot win on every non-Martin-Löf random sequence if either of the two following conditions is true: (I) There is an unbounded computable function $g$ such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all $\ell$, bet on at most $\ell-g(\ell)$ positions among the first $\ell$ positions of the sequence. (II) There is a sublinear function $g$ such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all $\ell$, bet on at least $\ell-g(\ell)$ positions among the first $\ell$ positions of the sequence.

cs.IT↗
arXiv2015

A pair of universal sequence-set betting strategies

We introduce the sequence-set betting game, a generalization of An. A. Muchnik's non-monotonic betting game. Instead of successively partitioning the infinite binary strings by their value of a bit at a chosen position, as in the non-monotonic game, the player is allowed to partition the strings into any two clopen sets with equal measure. We show that, while there is no single computable sequence-set betting strategy that predicts all non-Martin-Löf random strings, we can construct two strategies such that every non-Martin-Löf random string is predicted by at least one of them.

cs.DS↗