20,641 research outputs found
Relating and contrasting plain and prefix Kolmogorov complexity
In [3] a short proof is given that some strings have maximal plain Kolmogorov
complexity but not maximal prefix-free complexity. The proof uses Levin's
symmetry of information, Levin's formula relating plain and prefix complexity
and Gacs' theorem that complexity of complexity given the string can be high.
We argue that the proof technique and results mentioned above are useful to
simplify existing proofs and to solve open questions.
We present a short proof of Solovay's result [21] relating plain and prefix
complexity: and , (here denotes , etc.).
We show that there exist such that is infinite and is
finite, i.e. the infinitely often C-trivial reals are not the same as the
infinitely often K-trivial reals (i.e. [1,Question 1]).
Solovay showed that for infinitely many we have
and , (here
denotes the length of and , etc.). We show that this
result holds for prefixes of some 2-random sequences.
Finally, we generalize our proof technique and show that no monotone relation
exists between expectation and probability bounded randomness deficiency (i.e.
[6, Question 1]).Comment: 20 pages, 1 figur
NP-Hardness of Approximately Solving Linear Equations Over Reals
URL lists article on conference siteIn this paper, we consider the problem of approximately solving a system of homogeneous
linear equations over reals, where each equation contains at most three variables.
Since the all-zero assignment always satisfies all the equations exactly, we restrict the
assignments to be “non-trivial”. Here is an informal statement of our result: it is NP-hard
to distinguish whether there is a non-trivial assignment that satisfies fraction of the
equations or every non-trivial assignment fails to satisfy a constant fraction of the equations
with a ``margin" of .
We develop linearity and dictatorship testing procedures for functions f : Rn 7--> R over
a Gaussian space, which could be of independent interest.
We believe that studying the complexity of linear equations over reals, apart from being
a natural pursuit, can lead to progress on the Unique Games Conjecture.National Science Foundation (U.S.) (NSF CAREER grant CCF-0833228)National Science Foundation (U.S.) (Expeditions grant CCF-0832795)U.S.-Israel Binational Science Foundation (BSF grant 2008059
- …