A deterministic finite automaton (DFA) is a mathematical model used in computer science and formal language theory to recognize and accept strings of symbols. It falls under the category of finite automata, which are abstract machines with a finite set of states, input symbols, transition rules, and an initial state.
Here are the key components of a deterministic finite automaton:
- Alphabet (Σ): The finite set of input symbols.
- States (Q): A finite set of states, including an initial state and one or more accepting (or final) states.
- Transition Function (δ):
- A function that defines the state transitions based on the current state and input symbol.
- For a DFA, the transition is deterministic, meaning that for each combination of a state and an input symbol, there is only one possible next state.
- Mathematically, δ: Q × Σ → Q
- Initial State (q₀): The starting state before any input is processed.
- Accepting States (F): A subset of states that are considered accepting or final states. If the DFA is in an accepting state after processing the entire input string, the string is accepted; otherwise, it is rejected.
The language recognized by a DFA is the set of all strings that, when input to the automaton, lead it to an accepting state. DFAs are particularly useful for recognizing regular languages, which are a class of languages defined by regular expressions.
The processing of a string by a DFA involves starting in the initial state, reading each symbol from the input string one at a time, and transitioning between states according to the transition function. After processing the entire string, if the DFA is in an accepting state, the string is accepted; otherwise, it is rejected.
DFAs are simpler than non-deterministic finite automata (NFAs) because they have a unique transition for each combination of state and input symbol, making their behavior predictable and deterministic.
