On the Normality of Numbers to Different Bases

We prove independence of normality to different bases We show that the set of real numbers that are normal to some base is Sigma^0_4 complete in the Borel hierarchy of subsets of real numbers. This was an open problem, initiated by Alexander Kechris, and conjectured by Ditzen 20 years ago

Normal Numbers and the Borel Hierarchy

We show that the set of absolutely normal numbers is $\mathbf \Pi^0_3$-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is $\Pi^0_3$-complete in the effective Borel hierarchy

Kolmogorov complexity for possibly infinite computations

In this paper we study a variant of the Kolmogorov complexity for non-effective computations, that is, either halting or non-halting computations on Turing machines. This complexity function is defined as the length of the shortest inputs that produce a desired output via a possibly non-halting computation. Clearly this function gives a lower bound of the classical Kolmogorov complexity. In particular, if the machine is allowed to overwrite its output, this complexity coincides with the classical Kolmogorov complexity for halting computations relative to the first jump of the halting problem. However, on machines that cannot erase their output –called monotone machines–, we prove that our complexity for non effective computations and the classical Kolmogorov complexity separate as much as we want. We also consider the prefix-free complexity for possibly infinite computations. We study several properties of the graph of these complexity functions and specially their oscillations with respect to the complexities for effective computations.Eje: Teoría (TEOR)Red de Universidades con Carreras en Informática (RedUNCI

Perfect Necklaces

We introduce a variant of de Bruijn words that we call perfect necklaces. Fix a finite alphabet. Recall that a word is a finite sequence of symbols in the alphabet and a circular word, or necklace, is the equivalence class of a word under rotations. For positive integers k and n, we call a necklace (k,n)-perfect if each word of length k occurs exactly n times at positions which are different modulo n for any convention on the starting point. We call a necklace perfect if it is (k,k)-perfect for some k. We prove that every arithmetic sequence with difference coprime with the alphabet size induces a perfect necklace. In particular, the concatenation of all words of the same length in lexicographic order yields a perfect necklace. For each k and n, we give a closed formula for the number of (k,n)-perfect necklaces. Finally, we prove that every infinite periodic sequence whose period coincides with some (k,n)-perfect necklace for any n, passes all statistical tests of size up to k, but not all larger tests. This last theorem motivated this work

A linearly computable measure of string complexity

AbstractWe present a measure of string complexity, called I-complexity, computable in linear time and space. It counts the number of different substrings in a given string. The least complex strings are the runs of a single symbol, the most complex are the de Bruijn strings. Although the I-complexity of a string is not the length of any minimal description of the string, it satisfies many basic properties of classical description complexity. In particular, the number of strings with I-complexity up to a given value is bounded, and most strings of each length have high I-complexity

A construction of a λ\lambda- Poisson generic sequence

Years ago Zeev Rudnick defined the $\lambda$-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter $\lambda$. Although it has long been known that almost all sequences, with respect to Lebesgue measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit $\lambda$-Poisson generic sequence over an alphabet of at least three symbols, for any fixed positive real number $\lambda$. Since $\lambda$-Poisson genericity implies Borel normality, the constructed sequence is Borel normal. The same construction provides explicit instances of Borel normal sequences that are not $\lambda$-Poisson generic.Comment: 14 pages. Accepted in Mathematics of Computatio

On the logic for utopia

In this study we propose the standard modal logic KD43 as the logle governing expressions about Utopia. We define a formal construction corresponding to Utopian expressions In ordlnary language that we name utopian eonditionals. They possess the singular properties of admitting Strengthening of the Antecedent whlle possibly defeating the rule of Modus Ponens. Perhaps the most interesting aspect of this work is that, as far as the authors know, this is the first time a category of expressions in the ordinary language corresponding to these two singular properties is provided.Eje: 2do. Workshop sobre aspectos teóricos de la inteligencia artificialRed de Universidades con Carreras en Informática (RedUNCI