40 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Graph Parameters, Universal Obstructions, and WQO
We introduce the notion of universal obstruction of a graph parameter, with
respect to some quasi-ordering relation. Universal obstructions may serve as
compact characterizations of the asymptotic behavior of graph parameters. We
provide order-theoretic conditions which imply that such a characterization is
finite and, when this is the case, we present some algorithmic implications on
the existence of fixed-parameter algorithms
Kolmogorov Last Discovery? (Kolmogorov and Algorithmic Statictics)
The last theme of Kolmogorov's mathematics research was algorithmic theory of
information, now often called Kolmogorov complexity theory. There are only two
main publications of Kolmogorov (1965 and 1968-1969) on this topic. So
Kolmogorov's ideas that did not appear as proven (and published) theorems can
be reconstructed only partially based on work of his students and
collaborators, short abstracts of his talks and the recollections of people who
were present at these talks.
In this survey we try to reconstruct the development of Kolmogorov's ideas
related to algorithmic statistics (resource-bounded complexity, structure
function and stochastic objects).Comment: [version 2: typos and minor errors corrected
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Real patterns and indispensability
While scientific inquiry crucially relies on the extraction of patterns from data, we still have a far from perfect understanding of the metaphysics of patterns—and, in particular, of what makes a pattern real. In this paper we derive a criterion of real-patternhood from the notion of conditional Kolmogorov complexity. The resulting account belongs to the philosophical tradition, initiated by Dennett (J Philos 88(1):27–51, 1991), that links real-patternhood to data compressibility, but is simpler and formally more perspicuous than other proposals previously defended in the literature. It also successfully enforces a non-redundancy principle, suggested by Ladyman and Ross (Every thing must go: metaphysics naturalized, Oxford University Press, Oxford, 2007), that aims to exclude from real-patternhood those patterns that can be ignored without loss of information about the target dataset, and which their own account fails to enforce.Manolo MartĂnez would like to acknowledge research funding awarded by the Spanish Ministry of Economy, Industry and Competitiveness, in the form of grants PGC2018-101425-B-I00 and RYC-2016-20642
Stashing And Parallelization Pentagons
Parallelization is an algebraic operation that lifts problems to sequences in
a natural way. Given a sequence as an instance of the parallelized problem,
another sequence is a solution of this problem if every component is
instance-wise a solution of the original problem. In the Weihrauch lattice
parallelization is a closure operator. Here we introduce a dual operation that
we call stashing and that also lifts problems to sequences, but such that only
some component has to be an instance-wise solution. In this case the solution
is stashed away in the sequence. This operation, if properly defined, induces
an interior operator in the Weihrauch lattice. We also study the action of the
monoid induced by stashing and parallelization on the Weihrauch lattice, and we
prove that it leads to at most five distinct degrees, which (in the maximal
case) are always organized in pentagons. We also introduce another closely
related interior operator in the Weihrauch lattice that replaces solutions of
problems by upper Turing cones that are strong enough to compute solutions. It
turns out that on parallelizable degrees this interior operator corresponds to
stashing. This implies that, somewhat surprisingly, all problems which are
simultaneously parallelizable and stashable have computability-theoretic
characterizations. Finally, we apply all these results in order to study the
recently introduced discontinuity problem, which appears as the bottom of a
number of natural stashing-parallelization pentagons. The discontinuity problem
is not only the stashing of several variants of the lesser limited principle of
omniscience, but it also parallelizes to the non-computability problem. This
supports the slogan that "non-computability is the parallelization of
discontinuity"