8,096 research outputs found
Stratification and enumeration of Boolean functions by canalizing depth
Boolean network models have gained popularity in computational systems
biology over the last dozen years. Many of these networks use canalizing
Boolean functions, which has led to increased interest in the study of these
functions. The canalizing depth of a function describes how many canalizing
variables can be recursively picked off, until a non-canalizing function
remains. In this paper, we show how every Boolean function has a unique
algebraic form involving extended monomial layers and a well-defined core
polynomial. This generalizes recent work on the algebraic structure of nested
canalizing functions, and it yields a stratification of all Boolean functions
by their canalizing depth. As a result, we obtain closed formulas for the
number of n-variable Boolean functions with depth k, which simultaneously
generalizes enumeration formulas for canalizing, and nested canalizing
functions
The Partial Visibility Representation Extension Problem
For a graph , a function is called a \emph{bar visibility
representation} of when for each vertex , is a
horizontal line segment (\emph{bar}) and iff there is an
unobstructed, vertical, -wide line of sight between and
. Graphs admitting such representations are well understood (via
simple characterizations) and recognizable in linear time. For a directed graph
, a bar visibility representation of , additionally, puts the bar
strictly below the bar for each directed edge of
. We study a generalization of the recognition problem where a function
defined on a subset of is given and the question is whether
there is a bar visibility representation of with for every . We show that for undirected graphs this problem
together with closely related problems are \NP-complete, but for certain cases
involving directed graphs it is solvable in polynomial time.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Some Applications of Coding Theory in Computational Complexity
Error-correcting codes and related combinatorial constructs play an important
role in several recent (and old) results in computational complexity theory. In
this paper we survey results on locally-testable and locally-decodable
error-correcting codes, and their applications to complexity theory and to
cryptography.
Locally decodable codes are error-correcting codes with sub-linear time
error-correcting algorithms. They are related to private information retrieval
(a type of cryptographic protocol), and they are used in average-case
complexity and to construct ``hard-core predicates'' for one-way permutations.
Locally testable codes are error-correcting codes with sub-linear time
error-detection algorithms, and they are the combinatorial core of
probabilistically checkable proofs
Efficient Parallel Path Checking for Linear-Time Temporal Logic With Past and Bounds
Path checking, the special case of the model checking problem where the model
under consideration is a single path, plays an important role in monitoring,
testing, and verification. We prove that for linear-time temporal logic (LTL),
path checking can be efficiently parallelized. In addition to the core logic,
we consider the extensions of LTL with bounded-future (BLTL) and past-time
(LTL+Past) operators. Even though both extensions improve the succinctness of
the logic exponentially, path checking remains efficiently parallelizable: Our
algorithm for LTL, LTL+Past, and BLTL+Past is in AC^1(logDCFL) \subseteq NC
Envelopes of conditional probabilities extending a strategy and a prior probability
Any strategy and prior probability together are a coherent conditional
probability that can be extended, generally not in a unique way, to a full
conditional probability. The corresponding class of extensions is studied and a
closed form expression for its envelopes is provided. Then a topological
characterization of the subclasses of extensions satisfying the further
properties of full disintegrability and full strong conglomerability is given
and their envelopes are studied.Comment: 2
Involutive Categories and Monoids, with a GNS-correspondence
This paper develops the basics of the theory of involutive categories and
shows that such categories provide the natural setting in which to describe
involutive monoids. It is shown how categories of Eilenberg-Moore algebras of
involutive monads are involutive, with conjugation for modules and vector
spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS)
construction is identified as a bijective correspondence between states on
involutive monoids and inner products. This correspondence exists in arbritrary
involutive categories
Searchable Sky Coverage of Astronomical Observations: Footprints and Exposures
Sky coverage is one of the most important pieces of information about
astronomical observations. We discuss possible representations, and present
algorithms to create and manipulate shapes consisting of generalized spherical
polygons with arbitrary complexity and size on the celestial sphere. This shape
specification integrates well with our Hierarchical Triangular Mesh indexing
toolbox, whose performance and capabilities are enhanced by the advanced
features presented here. Our portable implementation of the relevant spherical
geometry routines comes with wrapper functions for database queries, which are
currently being used within several scientific catalog archives including the
Sloan Digital Sky Survey, the Galaxy Evolution Explorer and the Hubble Legacy
Archive projects as well as the Footprint Service of the Virtual Observatory.Comment: 11 pages, 7 figures, submitted to PAS
On the Complexity and Performance of Parsing with Derivatives
Current algorithms for context-free parsing inflict a trade-off between ease
of understanding, ease of implementation, theoretical complexity, and practical
performance. No algorithm achieves all of these properties simultaneously.
Might et al. (2011) introduced parsing with derivatives, which handles
arbitrary context-free grammars while being both easy to understand and simple
to implement. Despite much initial enthusiasm and a multitude of independent
implementations, its worst-case complexity has never been proven to be better
than exponential. In fact, high-level arguments claiming it is fundamentally
exponential have been advanced and even accepted as part of the folklore.
Performance ended up being sluggish in practice, and this sluggishness was
taken as informal evidence of exponentiality.
In this paper, we reexamine the performance of parsing with derivatives. We
have discovered that it is not exponential but, in fact, cubic. Moreover,
simple (though perhaps not obvious) modifications to the implementation by
Might et al. (2011) lead to an implementation that is not only easy to
understand but also highly performant in practice.Comment: 13 pages; 12 figures; implementation at
http://bitbucket.org/ucombinator/parsing-with-derivatives/ ; published in
PLDI '16, Proceedings of the 37th ACM SIGPLAN Conference on Programming
Language Design and Implementation, June 13 - 17, 2016, Santa Barbara, CA,
US
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