We survey a variety of experimental paradigms for studying animal abilities to count, to understand numerical information and to perform simple arithmetic operations. There is a huge body of evidence that different forms and elements of quantitative judgement and numerical competence are spread across a wide range of species, both vertebrate and invertebrate. Here we pay particular attention to the display of numerical competence in ants. The reason is that most of the existing experimental schemes for studying numerical processing in animals, although often elegant, are restricted by studying subjects at the individual level, or by the use of artificial communicative systems. In contrast, the information-theoretic approach that was elaborated for studying number-related skills in ants employs their own communicative means and, thus, does not require the subjects to solve any artificial learning problems, such as learning intermediary languages, or even learning to solve multiple choice problems. Using this approach, it was discovered that members of highly social ant species possessed numerical competence. They were shown to be able to pass information about numbers and to perform simple arithmetic operations with small numbers. We suggest that applying ideas of information theory and using the natural communication systems of highly social animals can open new horizons in studying numerical cognition.

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We suggest a new approach to hypothesis testing for ergodic and stationary processes. In contrast to standard methods, the suggested approach gives a possibility to make tests, based on any lossless data compression method even if the distribution law of the codeword lengths is not known. We apply this approach to the following four problems: goodness-of-fit testing (or identity testing), testing for independence, testing of serial independence and homogeneity testing and suggest nonparametric statistical tests for these problems. It is important to note that practically used so-called archivers can be used for suggested testing.

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We address the problem of detecting deviations of binary sequence from randomness,which is very important for random number (RNG) and pseudorandom number generators (PRNG). Namely, we consider a null hypothesis H 0 that a given bit sequence is generated by Bernoulli source with equal probabilities of 0 and 1 and the alternative hypothesis H 1 that the sequence is generated by a stationary and ergodic source which differs from the source under H 0 . We show that data compression methods can be used as a basis for such testing and describe two new tests for randomness, which are based on ideas of universal coding. Known statistical tests and suggested ones are applied for testing PRNGs. Those experiments show that the power of the new tests is greater than of many known algorithms.

We show that Kolmogorov complexity and such its estimators as universal codes (or data compression methods) can be applied for hypotheses testing in a framework of classical mathematical statistics. The methods for identity testing and nonparametric testing of serial independence for time series are suggested.

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