Related papers: Kolmogorov Complexity of Categories
Descriptional complexity is the study of the conciseness of the various models representing formal languages. The state complexity of a regular language is the size, measured by the number of states of the smallest, either deterministic or…
Algorithmic information theory translates statements about classes of objects into statements about individual objects; it defines individual random sequences, effective Hausdorff dimension of individual points, amount of information in…
In this paper, we present some results on information, complexity and entropy as defined below and we discuss their relations with the Kolmogorov-Sinai entropy which is the most important invariant of a dynamical system. These results have…
Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a…
We prove the formula C(a,b) = K(a|C(a,b)) + C(b|a,C(a,b)) + O(1) that expresses the plain complexity of a pair in terms of prefix and plain conditional complexities of its components.
Peter Gacs showed (Gacs 1974) that for every n there exists a bit string x of length n whose plain complexity C(x) has almost maximal conditional complexity relative to x, i.e., C(C(x)|x) > log n - log^(2) n - O(1). (Here log^(2) i = log…
By Kolmogorov Complexity,two number-theoretic problems are solved in different way than before,one problem is Maxim Kontsevich and Don Bernard Zagier's Problem 3 \emph{Exhibit at least one number which does not belong to} $ \mathcal{P}$…
Any positive word comprised of random sequence of tokens form a finite alphabet can be reduced (without change of length) using an appropriate size Braid group relationships. Surprisingly the Braid relations dramatically reduce the…
This work concerns a comparison of SVM kernel methods in text categorization tasks. In particular I define a kernel function that estimates the similarity between two objects computing by their compressed lengths. In fact, compression…
We extend algorithmic information theory to quantum mechanics, taking a universal semicomputable density matrix (``universal probability'') as a starting point, and define complexity (an operator) as its negative logarithm. A number of…
It is well known that normality can be described as incompressibility via finite automata. Still the statement and the proof of this result as given by Becher and Heiber (2013) in terms of "lossless finite-state compressors" do not follow…
The provably asymptotically fastest algorithm within a factor of 5 for formally described problems will be constructed. The main idea is to enumerate all programs provably equivalent to the original problem by enumerating all proofs. The…
The piecewise complexity $h(u)$ of a word is the minimal length of subwords needed to exactly characterise $u$. Its piecewise minimality index $\rho(u)$ is the smallest length $k$ such that $u$ is minimal among its order-$k$ class $[u]_k$…
We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea…
The paper proposes open problems in classical Kolmogorov complexity. Each problem is presented with background information and thus the article also surveys some recent studies in the area.
Given a countable mathematical structure, its Scott sentence is a sentence of the infinitary logic $\mathcal{L}_{\omega_1 \omega}$ that characterizes it among all countable structures. We can measure the complexity of a structure by the…
The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary…
`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program…
Semantics of logic programs has been given by proof theory, model theory and by fixpoint of the immediate-consequence operator. If clausal logic is a programming language, then it should also have a compositional semantics. Compositional…
Cognitive complexity measures quantify human difficulty in understanding the source code based on cognitive informatics foundation. The discipline derives cognitive complexity on a basis of fundamental software factors i.e, inputs, outputs,…