88,429 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
Conflict Detection for Edits on Extended Feature Models using Symbolic Graph Transformation
Feature models are used to specify variability of user-configurable systems
as appearing, e.g., in software product lines. Software product lines are
supposed to be long-living and, therefore, have to continuously evolve over
time to meet ever-changing requirements. Evolution imposes changes to feature
models in terms of edit operations. Ensuring consistency of concurrent edits
requires appropriate conflict detection techniques. However, recent approaches
fail to handle crucial subtleties of extended feature models, namely
constraints mixing feature-tree patterns with first-order logic formulas over
non-Boolean feature attributes with potentially infinite value domains. In this
paper, we propose a novel conflict detection approach based on symbolic graph
transformation to facilitate concurrent edits on extended feature models. We
describe extended feature models formally with symbolic graphs and edit
operations with symbolic graph transformation rules combining graph patterns
with first-order logic formulas. The approach is implemented by combining
eMoflon with an SMT solver, and evaluated with respect to applicability.Comment: In Proceedings FMSPLE 2016, arXiv:1603.0857
Recursive analysis and estimation for the discrete Boolean random set model
Random sets provide a powerful class of models for images containing randomly placed objects of random shapes and orientation. Those pixels within the foreground are members of a random set realization. The discrete Boolean model is the simplest general random set model in which a Bernoulli point process (called a germ process) is coupled with an independent shape or grain process. A typical realization consists of many overlapping shapes. Estimation in these models is difficult owing to the fact that many outcomes of the process obscure other outcomes. The directional one-dimensional (ID) model, in which random- length line segments emanate to the right from germs on the line, is analyzed via recursive expressions to provide a complete characterization of these discrete models in terms of the distributions of their black and white runlengths. An analytic representation is given for the optimal windowed filter for the signalunion- noise process, where both signal and noise are Boolean models. Several of these results are extended to the nondirectional case where segments can emanate to the left and right. Sufficient conditions are presented for a two-dimensional (2D) discrete Boolean model to induce a one dimensional Boolean model on an intersecting line. When inducement holds, the likelihood of runlength observations of the two-dimensional model is used to provide maximum-likelihood estimation of parameters of the 2D model. The ID directional discrete Boolean model is equivalent to the discrete-time infinite-server queue. Analysis for the Boolean model is extended to provide densities for many random variables of interest in queueing theory
A saturation property of structures obtained by forcing with a compact family of random variables
A method how to construct Boolean-valued models of some fragments of
arithmetic was developed in Krajicek (2011), with the intended applications in
bounded arithmetic and proof complexity. Such a model is formed by a family of
random variables defined on a pseudo-finite sample space. We show that under a
fairly natural condition on the family (called compactness in K.(2011)) the
resulting structure has a property that is naturally interpreted as saturation
for existential types. We also give an example showing that this cannot be
extended to universal types.Comment: preprint February 201
NP-hardness of circuit minimization for multi-output functions
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive.
In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators.
Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions
Stable and unstable attractors in Boolean networks
Boolean networks at the critical point have been a matter of debate for many
years as, e.g., scaling of number of attractor with system size. Recently it
was found that this number scales superpolynomially with system size, contrary
to a common earlier expectation of sublinear scaling. We here point to the fact
that these results are obtained using deterministic parallel update, where a
large fraction of attractors in fact are an artifact of the updating scheme.
This limits the significance of these results for biological systems where
noise is omnipresent. We here take a fresh look at attractors in Boolean
networks with the original motivation of simplified models for biological
systems in mind. We test stability of attractors w.r.t. infinitesimal
deviations from synchronous update and find that most attractors found under
parallel update are artifacts arising from the synchronous clocking mode. The
remaining fraction of attractors are stable against fluctuating response
delays. For this subset of stable attractors we observe sublinear scaling of
the number of attractors with system size.Comment: extended version, additional figur
- …