Automata theory studies computational systems (automata) that operate on input sequences. These systems often follow a structured progression, typically divided into four key stages or classes, each with increasing computational power and complexity:

Finite Automata (FA):

• The simplest class, finite automata, can recognize regular languages.
• These automata operate with a finite number of states and are often represented as deterministic (DFA) or nondeterministic (NFA).
• They are limited in power and cannot handle problems that require memory beyond their state transitions.

Pushdown Automata (PDA):

• PDAs extend finite automata with a stack, giving them memory and enabling them to recognize context-free languages.
• They can handle nested structures, like parentheses matching, making them useful for parsing expressions in programming languages.
• However, they still lack the power to recognize context-sensitive languages.

Linear Bounded Automata (LBA):

• LBAs have a finite tape (bounded linearly by the input size), unlike Turing machines with an infinite tape.
• These automata can recognize context-sensitive languages, which are more complex than context-free languages.
• They are commonly used in scenarios where resource limitations are a constraint.

Turing Machines (TM):

• The most powerful class, Turing machines, have an infinite tape, allowing them to recognize recursively enumerable languages.
• Turing machines can perform any computation that a computer can, representing the theoretical foundation of modern computing.
• They are used to understand the limits of computation and decide problems solvable by algorithmic processes.