Related papers: On the Conditional Complexity of Sets of Strings
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…
Kolmogorov complexity is often used as a convenient language for counting and/or probabilistic existence proofs. However, there are some applications where Kolmogorov complexity is used in a more subtle way. We provide one (somehow)…
Information distance can be defined not only between two strings but also in a finite multiset of strings of cardinality greater than two. We give an elementary proof for expressing the information distance in terms of plain Kolmogorov…
While Kolmogorov complexity is the accepted absolute measure of information content of an individual finite object, a similarly absolute notion is needed for the relation between an individual data sample and an individual model summarizing…
It is not obvious what fraction of all the potential information residing in the molecules and structures of living systems is significant or meaningful to the system. Sets of random sequences or identically repeated sequences, for example,…
Loveland complexity is a variant of Kolmogorov complexity, where it is asked to output separately the bits of the desired string, instead of the string itself. Similarly to the resource-bounded Kolmogorov sets we define Loveland sets. We…
A drawback of Kolmogorov-Chaitin complexity (K) as a function from s to the shortest program producing s is its noncomputability which limits its range of applicability. Moreover, when strings are short, the dependence of K on a particular…
In this paper we give a definition for quantum Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a string is the length of the shortest program that can produce this string as its output. It is a measure of the…
The famous G\"odel incompleteness theorem states that for every consistent sufficiently rich formal theory T there exist true statements that are unprovable in T. Such statements would be natural candidates for being added as axioms, but…
Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes-no voting system is a set of rules that specifies exactly which collections of ``yea''…
If no optimal propositional proof system exists, we (and independently Pudl\'ak) prove that ruling out length $t$ proofs of any unprovable sentence is hard. This mapping from unprovable to hard-to-prove sentences powerfully translates facts…
Assume that for some $\alpha<1$ and for all nutural $n$ a set $F_n$ of at most $2^{\alpha n}$ "forbidden" binary strings of length $n$ is fixed. Then there exists an infinite binary sequence $\omega$ that does not have (long) forbidden…
In this note a very crude but simple approximation to the set of separable states in an arbitrary simplex of commutative states is given using the fact that on the lines connecting the maximally mixed state and an arbitrary pure state the…
We systematically investigate the complexity of model checking the existential positive fragment of first-order logic. In particular, for a set of existential positive sentences, we consider model checking where the sentence is restricted…
We study the possibility of scaling down algorithmic information quantities in tuples of correlated strings. In particular, we address a question raised by Alexander Shen: whether, for any triple of strings $(a, b, c)$, there exists a…
We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the…
Let $K$ denote prefix-free Kolmogorov Complexity, and $K^A$ denote it relative to an oracle $A$. We show that for any $n$, $K^{\emptyset^{(n)}}$ is definable purely in terms of the unrelativized notion $K$. It was already known that…
Randomness extraction is the process of constructing a source of randomness of high quality from one or several sources of randomness of lower quality. The problem can be modeled using probability distributions and min-entropy to measure…
The word "complexity" is most often used as a meta--linguistic expression referring to certain intuitive characteristics of a natural system and/or its scientific description. These characteristics may include: sheer amount of data that…
In this paper, we revisit a central concept in Kolmogorov complexity in which one would equate program-size complexity with information content. Despite the fact that Kolmogorov complexity has been widely accepted as an objective measure of…