  Under the auspices of the Computational Complexity Foundation (CCF)     REPORTS > KEYWORD > KOLMOGOROV COMPLEXITY:
Reports tagged with Kolmogorov complexity:
TR00-015 | 16th February 2000
Andrej Muchnik, Alexej Semenov

#### Multi-conditional Descriptions and Codes in Kolmogorov Complexity

TR00-016 | 29th February 2000
Mikhail V. Vyugin

#### Information Distance and Conditional Complexities

C.H.~Bennett, P.~G\'acs, M.~Li, P.M.B.~Vit\'anyi, and W.H.~Zurek
have defined information distance between two strings $x$, $y$
as
$$d(x,y)=\max\{ K(x|y), K(y|x) \}$$
where $K(x|y)$ is the conditional Kolmogorov complexity.
It is easy to see that for any string $x$ and any integer $n$
there is a string $y$ ... more >>>

TR00-026 | 11th April 2000
Andrei Romashchenko, Alexander Shen, Nikolay Vereshchagin

#### Combinatorial Interpretation of Kolmogorov Complexity

The very first Kolmogorov's paper on algorithmic
information theory was entitled Three approaches to the
definition of the quantity of information'. These three
approaches were called combinatorial, probabilistic and
algorithmic. Trying to establish formal connections
between combinatorial and algorithmic approaches, we
prove that any ... more >>>

TR00-035 | 6th June 2000
Nikolay Vereshchagin, Mikhail V. Vyugin

#### Independent minimum length programs to translate between given strings

A string $p$ is called a program to compute $y$ given $x$
if $U(p,x)=y$, where $U$ denotes universal programming language.
Kolmogorov complexity $K(y|x)$ of $y$ relative to $x$
is defined as minimum length of
a program to compute $y$ given $x$.
Let $K(x)$ denote $K(x|\text{empty string})$
(Kolmogorov complexity of $x$) ... more >>>

TR01-052 | 26th April 2001
Mikhail V. Vyugin, Vladimir Vyugin

#### Non-linear Inequalities between Predictive and Kolmogorov Complexity

Predictive complexity is a generalization of Kolmogorov complexity
which gives a lower bound to ability of any algorithm to predict
elements of a sequence of outcomes. A variety of types of loss
functions makes it interesting to study relations between corresponding
predictive complexities.

Non-linear inequalities between predictive complexity of ... more >>>

TR01-059 | 20th July 2001
Elvira Mayordomo

#### A Kolmogorov complexity characterization of constructive Hausdorff dimension

Revisions: 3

We obtain the following full characterization of constructive dimension
in terms of algorithmic information content. For every sequence A,
cdim(A)=liminf_n (K(A[0..n-1])/n.

more >>>

TR01-083 | 29th October 2001
Nikolay Vereshchagin

#### An enumerable undecidable set with low prefix complexity: a simplified proof

Revisions: 1

We present a simplified proof of Solovay-Calude-Coles theorem
stating that there is an enumerable undecidable set with the
following property: prefix
complexity of its initial segment of length n is bounded by prefix
complexity of n (up to an additive constant).

more >>>

TR01-086 | 29th October 2001
Nikolay Vereshchagin

#### Kolmogorov Complexity Conditional to Large Integers

Assume that for almost all n Kolmogorov complexity
of a string x conditional to n is less than m.
We prove that in this case
there is a program of size m+O(1) that given any sufficiently large
n outputs x.

more >>>

TR01-087 | 29th October 2001
Bruno Durand, Alexander Shen, Nikolay Vereshchagin

#### Descriptive complexity of computable sequences

We study different notions of descriptive complexity of
computable sequences. Our main result states that if for almost all
n the Kolmogorov complexity of the n-prefix of an infinite
binary sequence x conditional to n
is less than m then there is a program of length
m^2+O(1) that for ... more >>>

TR01-088 | 29th October 2001
Alexander Shen, Nikolay Vereshchagin

#### Logical operations and Kolmogorov complexity

We define Kolmogorov complexity of a set of strings as the minimal
Kolmogorov complexity of its element. Consider three logical
operations on sets going back to Kolmogorov
and Kleene:
(A OR B) is the direct sum of A,B,
(A AND B) is the cartesian product of A,B,
more >>>

TR01-089 | 29th October 2001
Andrej Muchnik, Nikolay Vereshchagin

#### Logical operations and Kolmogorov complexity. II

We invistigate what is the minimal length of
a program mapping A to B and at the same time
mapping C to D (where A,B,C,D are binary strings). We prove that
it cannot be expressed
in terms of Kolmogorv complexity of A,B,C,D, their pairs (A,B), (A,C),
..., their ... more >>>

TR02-011 | 14th October 2001
Boris Ryabko

#### The nonprobabilistic approach to learning the best prediction.

The problem of predicting a sequence $x_1, x_2,....$ where each $x_i$ belongs
to a finite alphabet $A$ is considered. Each letter $x_{t+1}$ is predicted
using information on the word $x_1, x_2, ...., x_t$ only. We use the game
theoretical interpretation which can be traced to Laplace where there ... more >>>

TR02-028 | 15th May 2002
Eric Allender, Harry Buhrman, Michal Koucky, Detlef Ronneburger, Dieter van Melkebeek

#### Power from Random Strings

Revisions: 1

We consider sets of strings with high Kolmogorov complexity, mainly
in resource-bounded settings but also in the traditional
recursion-theoretic sense. We present efficient reductions, showing
that these sets are hard and complete for various complexity classes.

In particular, in addition to the usual Kolmogorov complexity measure
K, ... more >>>

TR04-002 | 8th January 2004
Troy Lee, Dieter van Melkebeek, Harry Buhrman

#### Language Compression and Pseudorandom Generators

The language compression problem asks for succinct descriptions of
the strings in a language A such that the strings can be efficiently
recovered from their description when given a membership oracle for
A. We study randomized and nondeterministic decompression schemes
and investigate how close we can get to the information ... more >>>

TR04-015 | 24th February 2004
Richard Beigel, Harry Buhrman, Peter Fejer, Lance Fortnow, Piotr Grabowski, Luc Longpré, Andrej Muchnik, Frank Stephan, Leen Torenvliet

#### Enumerations of the Kolmogorov Function

A recursive enumerator for a function $h$ is an algorithm $f$ which
enumerates for an input $x$ finitely many elements including $h(x)$.
$f$ is an $k(n)$-enumerator if for every input $x$ of length $n$, $h(x)$
is among the first $k(n)$ elements enumerated by $f$.
If there is a $k(n)$-enumerator for ... more >>>

TR04-030 | 9th March 2004
Nikolay Vereshchagin

#### Kolmogorov complexity of enumerating finite sets

Solovay has proven that
the minimal length of a program enumerating a set A
is upper bounded by 3 times the absolute value of the
logarithm of the
probability that a random program will enumerate A.
It is unknown whether one can replace the constant
3 by a smaller constant.
more >>>

TR04-031 | 22nd March 2004
Troy Lee, Andrei Romashchenko

#### On Polynomially Time Bounded Symmetry of Information

The information contained in a string $x$ about a string $y$
is defined as the difference between the Kolmogorov complexity
of $y$ and the conditional Kolmogorov complexity of $y$ given $x$,
i.e., $I(x:y)=\C(y)-\C(y|x)$. From the well-known Kolmogorov--Levin Theorem it follows that $I(x:y)$ is symmetric up to a small ... more >>>

TR04-044 | 1st June 2004
Eric Allender, Harry Buhrman, Michal Koucky

#### What Can be Efficiently Reduced to the Kolmogorov-Random Strings?

We investigate the question of whether one can characterize complexity
classes (such as PSPACE or NEXP) in terms of efficient
reducibility to the set of Kolmogorov-random strings R_C.
We show that this question cannot be posed without explicitly dealing
with issues raised by the choice of universal
machine in the ... more >>>

TR04-054 | 5th June 2004
Andrej Muchnik, Alexander Shen, Nikolay Vereshchagin, Mikhail V. Vyugin

#### Non-reducible descriptions for conditional Kolmogorov complexity

Let a program p on input a output b. We are looking for a
shorter program p' having the same property (p'(a)=b). In
addition, we want p' to be simple conditional to p (this
means that the conditional Kolmogorov complexity K(p'|p) is
negligible). In the present paper, we prove that ... more >>>

TR04-080 | 7th September 2004
Lance Fortnow, Troy Lee, Nikolay Vereshchagin

#### Kolmogorov Complexity with Error

We introduce the study of Kolmogorov complexity with error. For a
metric d, we define C_a(x) to be the length of a shortest
program p which prints a string y such that d(x,y) \le a. We
also study a conditional version of this measure C_{a,b}(x|y)
where the task is, given ... more >>>

TR04-081 | 9th September 2004
Harry Buhrman, Lance Fortnow, Ilan Newman, Nikolay Vereshchagin

#### Increasing Kolmogorov Complexity

How much do we have to change a string to increase its Kolmogorov complexity. We show that we can
increase the complexity of any non-random string of length n by flipping O(sqrt(n)) bits and some strings
require
Omega(sqrt(n)) bit flips. For a given m, we also give bounds for ... more >>>

TR05-105 | 24th September 2005
Lance Fortnow, John Hitchcock, A. Pavan, N. V. Vinodchandran, Fengming Wang

#### Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws

We apply recent results on extracting randomness from independent
sources to extract'' Kolmogorov complexity. For any $\alpha, \epsilon > 0$, given a string $x$ with $K(x) > \alpha|x|$, we show
how to use a constant number of advice bits to efficiently
compute another string $y$, $|y|=\Omega(|x|)$, with $K(y) > (1-\epsilon)|y|$. ... more >>>

TR05-144 | 5th December 2005
Lance Fortnow, Luis Antunes

#### Time-Bounded Universal Distributions

We show that under a reasonable hardness assumptions, the time-bounded Kolmogorov distribution is a universal samplable distribution. Under the same assumption we exactly characterize the worst-case running time of languages that are in average polynomial-time over all P-samplable distributions.

more >>>

TR06-006 | 16th December 2005
Alexander Shen

#### Multisource algorithmic information theory

Multisource information theory in Shannon setting is well known. In this article we try to develop its algorithmic information theory counterpart and use it as the general framework for many interesting questions about Kolmogorov complexity.

more >>>

TR06-047 | 11th February 2006
Maria Lopez-Valdes

#### Scaled Dimension of Individual Strings

We define a new discrete version of scaled dimension and we find
connections between the scaled dimension of a string and its Kolmogorov
complexity and predictability. We give a new characterization
of constructive scaled dimension by Kolmogorov complexity, and prove
a new result about scaled dimension and prediction.

more >>>

TR06-070 | 23rd May 2006
Ludwig Staiger

#### The Kolmogorov complexity of infinite words

We present a brief survey of results on relations between the Kolmogorov
complexity of infinite strings and several measures of information content
(dimensions) known from dimension theory, information theory or fractal
geometry.

Special emphasis is laid on bounds on the complexity of strings in
more >>>

TR06-080 | 16th June 2006
David Doty

#### Dimension Extractors

A dimension extractor is an algorithm designed to increase the effective dimension -- i.e., the computational information density -- of an infinite sequence. A constructive dimension extractor is exhibited by showing that every sequence of positive constructive dimension is Turing equivalent to a sequence of constructive strong dimension arbitrarily ... more >>>

TR06-125 | 20th September 2006
Luis Antunes, Lance Fortnow, Alexandre Pinto, Andre Souto

#### Low-Depth Witnesses are Easy to Find

Antunes, Fortnow, van Melkebeek and Vinodchandran captured the
notion of non-random information by computational depth, the
difference between the polynomial-time-bounded Kolmogorov
complexity and traditional Kolmogorov complexity We show how to
find satisfying assignments for formulas that have at least one
assignment of logarithmic depth. The converse holds under a
standard ... more >>>

TR08-109 | 10th November 2008
Marc Kaplan, Sophie Laplante

#### Kolmogorov complexity and combinatorial methods in communication complexity

A very important problem in quantum communication complexity is to show that there is, or isn?t, an exponential gap between randomized and quantum complexity for a total function. There are currently no clear candidate functions for such a separation; and there are fewer and fewer randomized lower bound techniques that ... more >>>

TR09-051 | 2nd July 2009
Eric Allender, Michal Koucky, Detlef Ronneburger, Sambuddha Roy

#### The Pervasive Reach of Resource-Bounded Kolmogorov Complexity in Computational Complexity Theory

We continue an investigation into resource-bounded Kolmogorov complexity \cite{abkmr}, which highlights the close connections between circuit complexity and Levin's time-bounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity.
The Kolmogorov measures that have been ... more >>>

TR09-071 | 1st September 2009
John Hitchcock, A. Pavan, N. V. Vinodchandran

#### Kolmogorov Complexity in Randomness Extraction

We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity
extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction
more >>>

TR10-055 | 31st March 2010
Eric Allender

#### Avoiding Simplicity is Complex

Revisions: 2

It is a trivial observation that every decidable set has strings of length $n$ with Kolmogorov complexity $\log n + O(1)$ if it has any strings of length $n$ at all. Things become much more interesting when one asks whether a similar property holds when one
considers *resource-bounded* Kolmogorov complexity. ... more >>>

TR10-128 | 15th August 2010
Scott Aaronson

#### The Equivalence of Sampling and Searching

Revisions: 1

In a sampling problem, we are given an input $x\in\left\{0,1\right\} ^{n}$, and asked to sample approximately from a probability
distribution $D_{x}$ over poly(n)-bit strings. In a search problem, we are given an input
$x\in\left\{ 0,1\right\} ^{n}$, and asked to find a member of a nonempty set
$A_{x}$ with high probability. ... more >>>

TR10-138 | 17th September 2010
Eric Allender, Luke Friedman, William Gasarch

#### Exposition of the Muchnik-Positselsky Construction of a Prefix Free Entropy Function that is not Complete under Truth-Table Reductions

In this paper we give an exposition of a theorem by Muchnik and Positselsky, showing that there is a universal prefix Turing machine U, with the property that there is no truth-table reduction from the halting problem to the set {(x,i) : there is a description d of length at ... more >>>

TR10-139 | 17th September 2010
Eric Allender, Luke Friedman, William Gasarch

#### Limits on the Computational Power of Random Strings

Revisions: 1

Let C(x) and K(x) denote plain and prefix Kolmogorov complexity, respectively, and let R_C and R_K denote the sets of strings that are `random'' according to these measures; both R_K and R_C are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both R_K ... more >>>

TR11-014 | 3rd February 2011
Antoine Taveneaux

#### Towards an axiomatic system for Kolmogorov complexity

Revisions: 1

In \cite{shenpapier82}, it is shown that four basic functional properties are enough to characterize plain Kolmogorov complexity, hence obtaining an axiomatic characterization of this notion. In this paper, we try to extend this work, both by looking at alternative axiomatic systems for plain complexity and by considering potential axiomatic systems ... more >>>

TR11-080 | 11th May 2011
mohammad iftekhar husain, steve ko, Atri Rudra, steve uurtamo

#### Storage Enforcement with Kolmogorov Complexity and List Decoding

We consider the following problem that arises in outsourced storage: a user stores her data $x$ on a remote server but wants to audit the server at some later point to make sure it actually did store $x$. The goal is to design a (randomized) verification protocol that has the ... more >>>

TR12-028 | 30th March 2012
Eric Allender, George Davie, Luke Friedman, Samuel Hopkins, Iddo Tzameret

#### Kolmogorov Complexity, Circuits, and the Strength of Formal Theories of Arithmetic

Revisions: 1

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems $C$ defined ... more >>>

TR12-054 | 2nd May 2012
Eric Allender, Harry Buhrman, Luke Friedman, Bruno Loff

#### Reductions to the set of random strings:the resource-bounded case

Revisions: 1

This paper is motivated by a conjecture that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out by [Allender et al] to settle this conjecture cannot succeed without significant alteration, but that it ... more >>>

TR13-007 | 8th January 2013
Bruno Bauwens, Anton Makhlin, Nikolay Vereshchagin, Marius Zimand

#### Short lists with short programs in short time

Revisions: 1

Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any universal machine, it is possible to compute in polynomial time on ... more >>>

TR13-161 | 23rd October 2013
Jack H. Lutz

#### The Frequent Paucity of Trivial Strings

Revisions: 1

A 1976 theorem of Chaitin, strengthening a 1969 theorem of Meyer,says that infinitely many lengths n have a paucity of trivial strings (only a bounded number of strings of length n having trivially low plain Kolmogorov complexities). We use the probabilistic method to give a new proof of this fact. ... more >>>

TR14-133 | 15th October 2014
Adam Case, Jack H. Lutz

#### Mutual Dimension

We define the lower and upper mutual dimensions $mdim(x:y)$ and $Mdim(x:y)$ between any two points $x$ and $y$ in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by $x$ and $y$. We show that these quantities satisfy the main desiderata for a satisfactory ... more >>>

TR14-176 | 16th December 2014
Eric Allender, Dhiraj Holden, Valentine Kabanets

#### The Minimum Oracle Circuit Size Problem

We consider variants of the Minimum Circuit Size Problem MCSP, where the goal is to minimize the size of oracle circuits computing a given function. When the oracle is QBF, the resulting problem MCSP$^{QBF}$ is known to be complete for PSPACE under ZPP reductions. We show that it is not ... more >>>

TR15-159 | 28th September 2015
Juraj Hromkovic

#### Why the Concept of Computational Complexity is Hard for Verifiable Mathematics

Mathematics was developed as a strong research instrument with fully verifiable argumentations. We call any formal theory based on syntactic rules that enables to algorithmically verify for any given text whether it is a proof or not algorithmically verifiable mathematics (AV-mathematics for short). We say that a decision problem L ... more >>>

TR17-043 | 3rd March 2017
Alexey Milovanov, Nikolay Vereshchagin

#### Stochasticity in Algorithmic Statistics for Polynomial Time

A fundamental notion in Algorithmic Statistics is that of a stochastic object, i.e., an object having a simple plausible explanation. Informally, a probability distribution is a plausible explanation for $x$ if it looks likely that $x$ was drawn at random with respect to that distribution. In this paper, we ... more >>>

TR17-073 | 28th April 2017
Eric Allender, Shuichi Hirahara

#### New Insights on the (Non)-Hardness of Circuit Minimization and Related Problems

The Minimum Circuit Size Problem (MCSP) and a related problem (MKTP) that deals with time-bounded Kolmogorov complexity are prominent candidates for NP-intermediate status. We show that, under very modest cryptographic assumptions (such as the existence of one-way functions), the problem of approximating the minimum circuit size (or time-bounded Kolmogorov complexity) ... more >>>

TR18-043 | 22nd February 2018
Andrei Romashchenko, Marius Zimand

#### An operational characterization of mutual information in algorithmic information theory

Revisions: 2

We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings
$x$ and $y$ is equal, up to logarithmic precision, to the length of the longest shared secret key that
two parties, one having $x$ and the complexity profile of the pair and the ... more >>>

TR19-025 | 28th February 2019
Shuichi Hirahara, Osamu Watanabe

#### On Nonadaptive Reductions to the Set of Random Strings and Its Dense Subsets

Revisions: 1

We investigate the computational power of an arbitrary distinguisher for (not necessarily computable) hitting set generators as well as the set of Kolmogorov-random strings. This work contributes to (at least) two lines of research. One line of research is the study of the limits of black-box reductions to some distributional ... more >>>

TR19-064 | 23rd April 2019
Igor Carboni Oliveira

#### Randomness and Intractability in Kolmogorov Complexity

We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin's notion of Kolmogorov complexity from 1984. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability.

This complexity measure gives rise to a ... more >>>

TR20-050 | 18th April 2020
Shuichi Hirahara

#### Unexpected Hardness Results for Kolmogorov Complexity Under Uniform Reductions

Hardness of computing the Kolmogorov complexity of a given string is closely tied to a security proof of hitting set generators, and thus understanding hardness of Kolmogorov complexity is one of the central questions in complexity theory. In this paper, we develop new proof techniques for showing hardness of computing ... more >>>

TR20-052 | 14th April 2020
Yanyi Liu, Rafael Pass

#### On One-way Functions and Kolmogorov Complexity

Revisions: 2

We prove the equivalence of two fundamental problems in the theory of computation:

- Existence of one-way functions: the existence of one-way functions (which in turn are equivalent to pseudorandom generators, pseudorandom functions, private-key encryption schemes, digital signatures, commitment schemes, and more).

- Mild average-case hardness of $K^{poly}$-complexity: ... more >>>

TR20-078 | 21st May 2020
Eric Allender

#### The New Complexity Landscape around Circuit Minimization

We survey recent developments related to the Minimum Circuit Size Problem

more >>>

TR20-143 | 16th September 2020
Shuichi Hirahara

#### Characterizing Average-Case Complexity of PH by Worst-Case Meta-Complexity

We exactly characterize the average-case complexity of the polynomial-time hierarchy (PH) by the worst-case (meta-)complexity of GapMINKT(PH), i.e., an approximation version of the problem of determining if a given string can be compressed to a short PH-oracle efficient program. Specifically, we establish the following equivalence:

DistPH is contained in ... more >>>

TR21-010 | 11th February 2021
Eric Allender, John Gouwar, Shuichi Hirahara, Caleb Robelle

#### Cryptographic Hardness under Projections for Time-Bounded Kolmogorov Complexity

A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT ... more >>>

TR21-039 | 15th March 2021
Zhenjian Lu, Igor Carboni Oliveira, Rahul Santhanam

#### Pseudodeterministic Algorithms and the Structure of Probabilistic Time

We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $BPTIME$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows.

1. A new pseudorandom generator and its consequences: We build on techniques developed to ... more >>>

TR21-041 | 15th March 2021
Zhenjian Lu, Igor Carboni Oliveira

#### An Efficient Coding Theorem via Probabilistic Representations and its Applications

A probabilistic representation of a string $x \in \{0,1\}^n$ is given by the code of a randomized algorithm that outputs $x$ with high probability [Oliveira, ICALP 2019]. We employ probabilistic representations to establish the first unconditional Coding Theorem in time-bounded Kolmogorov complexity. More precisely, we show that if a distribution ... more >>>

TR21-049 | 1st April 2021
Juraj Hromkovic

#### Kolmogorov complexity and nondeterminism versus determinism for polynomial time computations

We call any consistent and sufficiently powerful formal theory that enables to algorithmically in polynomial time verify whether a text is a proof \textbf{efficiently verifiable mathematics} (ev-mathematics). We study the question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of ev-mathematics. Our main results ... more >>>

TR21-053 | 13th April 2021
Jan Krajicek

#### Information in propositional proofs and algorithmic proof search

We study from the proof complexity perspective the (informal) proof search problem:
Is there an optimal way to search for propositional proofs?
We note that for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal ... more >>>

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