The class TFNP, defined by Megiddo and Papadimitriou, consists of
multivalued functions with values that are polynomially verifiable
and guaranteed to exist. Do we have evidence that such functions are
hard, for example, if TFNP is computable in polynomial-time does
this imply the polynomial-time hierarchy collapses?

We give a relativized ... more >>>

Our main result implies the following easily formulated statement. The set of edges E of every finite bipartite graph can be split into poly(log |E|) subsets so the all the resulting bipartite graphcs are almost regular. The latter means that the ratio between the maximal and minimal non-zero degree of ... more >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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.

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).

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 >>>

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 >>>

Assume that A, B are finite families of n-element sets.
We prove that there is an element that simultaneously
belongs to at least |A|/2n sets
in A and to at least |B|/2n sets in B. We use this result to prove
that for any inconsistent DNF's F,G with OR ... more >>>

It is well known that probabilistic boolean decision trees
cannot be much more powerful than deterministic ones (N.~Nisan, SIAM
Journal on Computing, 20(6):999--1007, 1991). Motivated by a question
if randomization can significantly speed up a nondeterministic
computation via a boolean decision tree, we address structural
properties of Arthur-Merlin games in ... more >>>