Related papers: How incomputable is Kolmogorov complexity?
Abduction is one of the most important forms of reasoning; it has been successfully applied to several practical problems such as diagnosis. In this paper we investigate whether the computational complexity of abduction can be reduced by an…
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…
Complexity remains one of the central challenges in science and technology. Although several approaches at defining and/or quantifying complexity have been proposed, at some point each of them seems to run into intrinsic limitations or…
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…
We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A…
Implicit computational complexity, which aims at characterizing complexity classes by machine-independent means, has traditionally been based, on the one hand, on programs and deductive formalisms for free algebras, and on the other hand on…
The Coding Theorem of L.A. Levin connects unconditional prefix Kolmogorov complexity with the discrete universal distribution. There are conditional versions referred to in several publications but as yet there exist no written proofs in…
This is an expository paper about the Borel complexity of structure and classification theorems. It sorts several classical problems relative to known benchmarks of complexity. As a corollary various problems proposed by people such as von…
We discuss a classical complexity of finite-dimensional unitary transformations, which can been seen as a computable approximation of classical descriptional complexity of a unitary transformation acting on a set of qubits.
We develop a notion of computability and complexity of functions over the reals, which seems to be very natural when one tries to determine just how "difficult" a certain function is. This notion can be viewed as an extension of both BSS…
The integer complexity $f(n)$ of a positive integer $n$ is defined as the minimum number of 1's needed to represent $n$, using additions, multiplications and parentheses. We present two simple and faster algorithms for computing the integer…
Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…
Metaquerying is a datamining technology by which hidden dependencies among several database relations can be discovered. This tool has already been successfully applied to several real-world applications. Recent papers provide only…
Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is…
We construct a universal decompressor $U$ for plain Kolmogorov complexity $\mathrm{C}_U$ such that the Halting Problem cannot be decided by any polynomial-time oracle machine with access to the set of random strings $R_{\mathrm{C}_U} = \{x…
In contrast to entropy, which increases monotonically, the "complexity" or "interestingness" of closed systems seems intuitively to increase at first and then decrease as equilibrium is approached. For example, our universe lacked complex…
Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
The aim of this work is to give an introduction to the theoretical background and computational complexity of Markov chain Monte Carlo methods. Most of the mathematical results related to the convergence are not found in most of the…
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,…