Related papers: On the Running Time of the Shortest Programs
The logical depth with significance $b$ of a finite binary string $x$ is the shortest running time of a binary program for $x$ that can be compressed by at most $b$ bits. There is another definition of logical depth. We give two theorems…
We consider the notion of information distance between two objects $x$ and $y$ introduced by Bennett, G\'acs, Li, Vit\'anyi, and Zurek in 1998 as the minimal length of a program that computes $x$ from $y$ as well as computing $y$ from $x$.…
The LZ-End parsing [Kreft & Navarro, 2011] of an input string yields compression competitive with the popular Lempel-Ziv 77 scheme, but also allows for efficient random access. Kempa and Kosolobov showed that the parsing can be computed in…
All strings with low mutual information with the halting sequence will have flat Kolmogorov Structure Functions, in the context of Algorithmic Statistics. Assuming the Independence Postulate, strings with non-negligible information with the…
The information in an individual finite object (like a binary string) is commonly measured by its Kolmogorov complexity. One can divide that information into two parts: the information accounting for the useful regularity present in the…
We show that there are infinitely many binary strings z, such that the sum of the on-line decision complexity of predicting the even bits of z given the previous uneven bits, and the decision complexity of predicting the uneven bits given…
We study the classical problem of minimizing the total weighted completion time on a fixed set of $m$ identical machines working in parallel, the $Pm||\sum w_jC_j$ problem in the standard three field notation for scheduling problems. This…
We revisit parallel-innermost term rewriting as a model of parallel computation on inductive data structures and provide a corresponding notion of runtime complexity parametric in the size of the start term. We propose automatic techniques…
A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new ``normalized information distance'', based on the noncomputable notion of…
TThe problem is to identify a probability associated with a set of natural numbers, given an infinite data sequence of elements from the set. If the given sequence is drawn i.i.d. and the probability mass function involved (the target)…
The last theme of Kolmogorov's mathematics research was algorithmic theory of information, now often called Kolmogorov complexity theory. There are only two main publications of Kolmogorov (1965 and 1968-1969) on this topic. So Kolmogorov's…
Let $F$ be a free group. We present for arbitrary $g\in\mathbb{N}$ a LogSpace (and thus polynomial time) algorithm that determines whether a given $w\in F$ is a product of at most $g$ commutators; and more generally an algorithm that…
The nascent field of compressed sensing is founded on the fact that high-dimensional signals with "simple structure" can be recovered accurately from just a small number of randomized samples. Several specific kinds of structures have been…
We consider the following problem: given that a finite automaton $M$ of $N$ states accepts at least one $k$-power-free (resp., overlap-free) word, what is the length of the shortest such word accepted? We give upper and lower bounds which,…
Word Break is a prototypical factorization problem in string processing: Given a word $w$ of length $N$ and a dictionary $\mathcal{D} = \{d_1, d_2, \ldots, d_{K}\}$ of $K$ strings, determine whether we can partition $w$ into words from…
The use of algorithmic information theory (Kolmogorov complexity theory) to explain the relation between mathematical probability theory and `real world' is discussed.
In this paper, we consider two versions of the Text Assembling problem. We are given a sequence of strings $s^1,\dots,s^n$ of total length $L$ that is a dictionary, and a string $t$ of length $m$ that is texts. The first version of the…
Through a straightforward Bayesian approach we show that under some general conditions a maximum running time, namely the number of discrete steps performed by a computer program during its execution, can be defined such that the…
One of the greatest algorithms of all time is Quicksort. Its average running time is famously O(nlog(n)), and its variance, less famously, is O(n^2) (hence its standard deviation is O(n)). But what about higher moments? Here we find…
We develop conjectures and theorems expressing the idea that the prime sequence exhibits computational irreducibility in the transition from one prime to its successor. Informally, given a prime pp p, no general algorithm can compute the…