### Summary

A discrete system, within the context of AI, is one that operates in distinct, separate states as opposed to continuous ones. It’s like a digital clock showing time in chunks, rather than an analog one where the time flow seems fluid.

### ELI5

Imagine you are collecting seashells from a beach. You can’t collect half a shell or three-and-a-half shells, right? You have to collect whole numbers, say, 1, 2, 3, and so on. This is a type of discrete system because you’re dealing with complete seashells, and there are clear gaps between each number of shells you collect. So in AI, a discrete system means the AI can only be in distinct or separate states.

### In-depth explanation

Discrete systems occur in many parts of artificial intelligence, most notably in machine learning, control theory, optimization, and game theory. In this context, ‘discrete’ refers to systems that evolve in distinct time intervals and have a finite set of state values. This is contrasted with continuous systems which evolve over continuous time and have an infinite number of possible state values.

A simple example of a discrete system is a chess game which moves from one discrete state to another. Each state of the game is a specific arrangement of the chess pieces on the board. After a player makes a move, the game transitions to a new state. Despite the move duration, the game isn’t in a state of continuous change during the move itself.

In machine learning, discrete systems often come into play in classification tasks, where a model must determine which of a set of distinct class labels an instance belongs to. Furthermore, reinforcement learning algorithms often operate in discrete action spaces where the agent selects its next action from a finite set.

The discrete nature of these systems means that they are often easier to model digitally compared to continuous systems. However, they may require more computation to work with, as each distinct state and transition needs to be modeled explicitly.

The modeling and manipulation of discrete systems often involve techniques and structures that are distinct from those used for continuous ones. This includes combinatorial algorithms for searching and optimising over discrete spaces, Markov Decision Processes for defining state transitions, and discrete mathematics for analyzing system properties.

While discrete systems have many advantages, they are inherently approximations to the real, continuous world. The quality of a discrete system’s approximation to reality may vary, depending on the granularity at which the system’s state and time are discretized. This can bring about questions of precision, efficiency, and risk of error during performance.

Markov Decision Processes, Reinforcement Learning (RL),, Classification, Optimization, Game Theory, Control Theory, State Space