56,084 research outputs found
On the robustness of temporal properties for stochastic models
Stochastic models such as Continuous-Time Markov Chains (CTMC) and Stochastic Hybrid Automata (SHA) are powerful formalisms to model and to reason about the dynamics of biological systems, due to their ability to capture the stochasticity inherent in biological processes. A classical question in formal modelling with clear relevance to biological modelling is the model checking problem. i.e. calculate the probability that a behaviour, expressed for instance in terms of a certain temporal logic formula, may occur in a given stochastic process. However, one may not only be interested in the notion of satisfiability, but also in the capacity of a system to mantain a particular emergent behaviour unaffected by the perturbations, caused e.g. from extrinsic noise, or by possible small changes in the model parameters. To address this issue, researchers from the verification community have recently proposed several notions of robustness for temporal logic providing suitable definitions of distance between a trajectory of a (deterministic) dynamical system and the boundaries of the set of trajectories satisfying the property of interest. The contributions of this paper are twofold. First, we extend the notion of robustness to stochastic systems, showing that this naturally leads to a distribution of robustness scores. By discussing two examples, we show how to approximate the distribution of the robustness score and its key indicators: the average robustness and the conditional average robustness. Secondly, we show how to combine these indicators with the satisfaction probability to address the system design problem, where the goal is to optimize some control parameters of a stochastic model in order to best maximize robustness of the desired specifications
Adversarial Robustness Verification and Attack Synthesis in Stochastic Systems
Probabilistic model checking is a useful technique for specifying and
verifying properties of stochastic systems including randomized protocols and
reinforcement learning models. Existing methods rely on the assumed structure
and probabilities of certain system transitions. These assumptions may be
incorrect, and may even be violated by an adversary who gains control of system
components.
In this paper, we develop a formal framework for adversarial robustness in
systems modeled as discrete time Markov chains (DTMCs). We base our framework
on existing methods for verifying probabilistic temporal logic properties and
extend it to include deterministic, memoryless policies acting in Markov
decision processes (MDPs). Our framework includes a flexible approach for
specifying structure-preserving and non structure-preserving adversarial
models. We outline a class of threat models under which adversaries can perturb
system transitions, constrained by an ball around the original
transition probabilities.
We define three main DTMC adversarial robustness problems: adversarial
robustness verification, maximal synthesis, and worst case attack
synthesis. We present two optimization-based solutions to these three problems,
leveraging traditional and parametric probabilistic model checking techniques.
We then evaluate our solutions on two stochastic protocols and a collection of
Grid World case studies, which model an agent acting in an environment
described as an MDP. We find that the parametric solution results in fast
computation for small parameter spaces. In the case of less restrictive
(stronger) adversaries, the number of parameters increases, and directly
computing property satisfaction probabilities is more scalable. We demonstrate
the usefulness of our definitions and solutions by comparing system outcomes
over various properties, threat models, and case studies.Comment: To Appear, 35th IEEE Computer Security Foundations Symposium (2022
Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka-Volterra Models
We study the general properties of stochastic two-species models for
predator-prey competition and coexistence with Lotka-Volterra type interactions
defined on a -dimensional lattice. Introducing spatial degrees of freedom
and allowing for stochastic fluctuations generically invalidates the classical,
deterministic mean-field picture. Already within mean-field theory, however,
spatial constraints, modeling locally limited resources, lead to the emergence
of a continuous active-to-absorbing state phase transition. Field-theoretic
arguments, supported by Monte Carlo simulation results, indicate that this
transition, which represents an extinction threshold for the predator
population, is governed by the directed percolation universality class. In the
active state, where predators and prey coexist, the classical center
singularities with associated population cycles are replaced by either nodes or
foci. In the vicinity of the stable nodes, the system is characterized by
essentially stationary localized clusters of predators in a sea of prey. Near
the stable foci, however, the stochastic lattice Lotka-Volterra system displays
complex, correlated spatio-temporal patterns of competing activity fronts.
Correspondingly, the population densities in our numerical simulations turn out
to oscillate irregularly in time, with amplitudes that tend to zero in the
thermodynamic limit. Yet in finite systems these oscillatory fluctuations are
quite persistent, and their features are determined by the intrinsic
interaction rates rather than the initial conditions. We emphasize the
robustness of this scenario with respect to various model perturbations.Comment: 19 pages, 11 figures, 2-column revtex4 format. Minor modifications.
Accepted in the Journal of Statistical Physics. Movies corresponding to
Figures 2 and 3 are available at
http://www.phys.vt.edu/~tauber/PredatorPrey/movies
Turing's model for biological pattern formation and the robustness problem
One of the fundamental questions in developmental biology is how the vast range of pattern and structure we observe in nature emerges from an almost uniformly homogeneous fertilized egg. In particular, the mechanisms by which biological systems maintain robustness, despite being subject to numerous sources of noise, are shrouded in mystery. Postulating plausible theoretical models of biological heterogeneity is not only difficult, but it is also further complicated by the problem of generating robustness, i.e. once we can generate a pattern, how do we ensure that this pattern is consistently reproducible in the face of perturbations to the domain, reaction time scale, boundary conditions and so forth. In this paper, not only do we review the basic properties of Turing's theory, we highlight the successes and pitfalls of using it as a model for biological systems, and discuss emerging developments in the area
Partial differential equations for self-organization in cellular and developmental biology
Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field
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