A Theory for Discrete-time Boolean Finite Dynamical Systems with Uncertainty

by   Mitsunori Ogihara, et al.

Dynamical Systems is a field that studies the collective behavior of objects that update their states according to some rules. Discrete-time Boolean Finite Dynamical System (DT-BFDS) is a subfield where the systems have some finite number of objects whose states are Boolean values, and the state updates occur in discrete time. In the subfield of DT-BFDS, researchers aim to (i) design models for capturing real-world phenomena and using the models to make predictions and (ii) develop simulation techniques for acquiring insights about the systems' behavior. Useful for both aims is understanding the system dynamics mathematically before executing the systems. Obtaining a mathematical understanding of BFDS is quite challenging, even for simple systems, because the state space of a system grows exponentially in the number of objects. Researchers have used computational complexity to circumvent the challenge. The complexity theoretic research in DT-BFDS has successfully produced complete characterizations for many dynamical problems. The DT-BFDS studies have mainly dealt with deterministic models, where the update at each time step is deterministic, so the system dynamics are completely determinable from the initial setting. However, natural systems have uncertainty. Models having uncertainty may lead to far-better understandings of nature. Although a few attempts have explored DT-BFDS with uncertainty, including stochastic initialization and tie-breaking, they have scratched only a tiny surface of models with uncertainty. The introduction of uncertainty can be through two schemes. One is the introduction of alternate update functions. The other is the introduction of alternate update schedules. 37This paper establishes a theory of models with uncertainty and proves some fundamental results.


page 1

page 2

page 3

page 4


Finite-Time Error Analysis of Asynchronous Q-Learning with Discrete-Time Switching System Models

This paper develops a novel framework to analyze the convergence of Q-le...

On the Complexity of the Escape Problem for Linear Dynamical Systems over Compact Semialgebraic Sets

We study the computational complexity of the Escape Problem for discrete...

On countings and enumerations of block-parallel automata networks

When we focus on finite dynamical systems from both the computability/co...

Moment-Based Exact Uncertainty Propagation Through Nonlinear Stochastic Autonomous Systems

In this paper, we address the problem of uncertainty propagation through...

Attractor Stability in Finite Asynchronous Biological System Models

We present mathematical techniques for exhaustive studies of long-term d...

Factorisation in the semiring of finite dynamical systems

Finite dynamical systems (FDSs) are commonly used to model systems with ...

Conley's fundamental theorem for a class of hybrid systems

We establish versions of Conley's (i) fundamental theorem and (ii) decom...

Please sign up or login with your details

Forgot password? Click here to reset