discretestate

Discretestate is a term sometimes used to describe a system whose state is discrete. In formal usage the standard expression is discrete state or discrete-state, referring to a model in which the state variables can take only a finite or countably infinite set of distinct values. Such systems evolve in time through transitions among states, either in fixed time steps or as events occur, and may be deterministic or stochastic.

State space and transitions: The collection of all possible states is called the state space. A deterministic discrete-state system is described by a transition function x_{t+1} = f(x_t, u_t), with x_t in a finite or countable set S. A stochastic or probabilistic discrete-state model uses transition probabilities P(x_{t+1} | x_t, u_t) or a transition matrix, as in Markov chains, hidden Markov models, or probabilistic automata.

Examples: A digital circuit with flip-flops has two on/off states per component; a traffic-light controller cycles through red, green, amber; a finite-state machine implements lexical analysis; cellular automata like the Game of Life use discrete cell states and update rules.

Relation to other modeling approaches: Discrete-state models contrast with continuous-state models in which state variables take values in a continuum such as real numbers; many systems are hybrid, combining discrete and continuous dynamics. Discretization and state enumeration can simplify analysis and controller synthesis but may suffer from state explosion as the number of distinct states grows.

Applications: computer science (protocol design, formal verification), operations research (queueing networks), robotics (discrete planners), and systems biology (genetic regulatory networks modeled with discrete states).