What is a discrete system?
by Stephen M. Walker II, CoFounder / CEO
What is a discrete system?
A discrete system is a system with a countable number of states. It is characterized by state changes that occur abruptly at specific, discrete points in time. This is in contrast to continuous systems, where state variables change continuously over time.
In the context of theoretical computer science, a computer is a prime example of a discrete system. It's a finitestate machine with a countable number of states, and it can be modeled with a directed graph. Computers are often used to model not only other discrete systems but also continuous systems. Methods have been developed to represent realworld continuous systems as discrete systems, such as sampling a continuous signal at discrete time intervals.
In discrete systems, both input and output signals are discrete signals, and the variables in these systems vary with time. The changes in these systems are predominantly discontinuous, meaning the state of variables changes only at a discrete set of points in time.
Discrete mathematics is the study of techniques, ideas, and modes of reasoning that are indispensable in applied disciplines such as computer science or information technology. It focuses on the study of discrete systems. For example, the set of integers is a discrete mathematical system because even though there are infinitely many integers, any one integer can be expressed by typing a finite sequence of digits.
How are discrete systems different from continuous systems?
Discrete systems differ from continuous systems in the nature of their state variables and the manner in which these variables change over time:

State Variables — In discrete systems, state variables change only at a discrete set of points in time. For example, the number of customers in a bank changes only when a customer arrives or departs. In contrast, continuous systems have state variables that change continuously over time, such as the water level behind a dam, which can change with inflow, outflow, and evaporation.

Signal Types — Discrete systems use discrete signals for both input and output, meaning the signals are defined only at specific, separate intervals. Continuous systems, however, have input and output signals that are continuous, varying smoothly over time without abrupt changes.

System Response — The response of a discrete system to an input signal is characterized by abrupt changes at particular times, whereas a continuous system responds with outputs that vary smoothly and continuously in response to the input signal.

Modeling and Control — Discrete systems are often modeled and controlled using digital systems due to their inherent discrete nature, which aligns with the binary processing of computers. Continuous systems may require analog or digital control, with digital control systems needing to sample the continuous signals at discrete intervals.

Complexity and Accuracy — Discrete systems can be easier to model and solve because they deal with a finite number of states, but they may be less accurate compared to continuous systems, which can represent an infinite number of states and are often more reflective of natural phenomena.
Properties of Discrete Systems
Discrete systems are characterized by a finite state space, ensuring predictability and finality in their behavior. They exhibit determinism, meaning the evolution of the system is consistent and repeatable under identical initial conditions. Completeness is another property, where systems inevitably reach a conclusive state. Lastly, consistency is maintained, as the same input always results in the same output.
Common Types of Discrete Systems in AI
Discrete systems are categorized based on their structure and function. Finite state machines model sequential processes, decision trees handle complex decisionmaking, and Bayesian networks represent probabilistic relationships. Each system offers a unique approach to problemsolving, from the simplicity of state transitions in finite state machines to the intricate conditional probabilities in Bayesian networks.
Common Applications of Discrete Systems in AI
Discrete systems, characterized by a finite set of states, are pivotal in artificial intelligence for managing large and complex state spaces that are impractical for continuous representation. They enable efficient exploration and problemsolving across various domains.
Search algorithms leverage discrete systems to navigate through state spaces, enhancing the efficiency of finding solutions. Planning algorithms utilize these systems to streamline the process of devising strategic plans. Additionally, reasoning systems employ discrete systems to facilitate more effective problem analysis and decisionmaking processes.
Challenges in Working with Discrete Systems
Discrete systems, characterized by a finite set of states, offer advantages in modeling and simulation over continuous systems, which have an infinite number of states. However, they present unique challenges. Selecting an appropriate set of discrete values is critical; too few values may lead to inaccurate representations, while too many can complicate the model unnecessarily.
Additionally, optimizing discrete systems can be complex due to the vast number of potential states, making it challenging to identify the optimal configuration.
Despite these difficulties, discrete systems often provide a more practical and comprehensible approach to problemsolving in various applications, necessitating a careful evaluation of tradeoffs when choosing between discrete and continuous systems.