Finite automata are abstract machines used to recognize patterns in input sequences, forming the basis for understanding regular languages in computer science.
- Consist of states, transitions, and input symbols, processing each symbol step-by-step.
- If ends in an accepting state after processing the input, then the input is accepted; otherwise, rejected.
- Finite automata come in deterministic (DFA) and non-deterministic (NFA), both of which can recognize the same set of regular languages.
- Widely used in text processing, compilers, and network protocols.
Features of Finite Automata
- Input: Set of symbols or characters provided to the machine.
- Output: Accept or reject based on the input pattern.
- States of Automata: The conditions or configurations of the machine.
- State Relation: The transitions between states.
- Output Relation: Based on the final state, the output decision is made.
Formal Definition of Finite Automata
A finite automaton can be defined as a tuple:
{ Q, Σ, q, F, δ }, where:
- Q: Finite set of states
- Σ: Set of input symbols
- q: Initial state
- F: Set of final states
- δ: Transition function
Types of Finite Automata
There are two types of finite automata:
- Deterministic Finite Automata (DFA)
- Non-Deterministic Finite Automata (NFA)
1. Deterministic Finite Automata (DFA)
A DFA is represented as {Q, Σ, q, F, δ}. In DFA, for each input symbol, the machine transitions to one and only one state. DFA does not allow any null transitions, meaning every state must have a transition defined for every input symbol.
DFA consists of 5 tuples {Q, Σ, q, F, δ}.
Q : set of all states.
Σ : set of input symbols. ( Symbols which machine takes as input )
q : Initial state. ( Starting state of a machine )
F : set of final state.
δ : Transition Function, defined as δ : Q X Σ --> Q.
Example: Construct a DFA that accepts all strings ending with 'a'.
Given:
Σ = {a, b},
Q = {q0, q1},
F = {q1}
| State\Symbol | a | b |
|---|---|---|
| q0 | q1 | q0 |
| q1 | q1 | q0 |
In this example, if the string ends in 'a', the machine reaches state q1, which is an accepting state.
2. Non-Deterministic Finite Automata (NFA)
NFA is similar to DFA but includes the following features:
- It can transition to multiple states for the same input.
- It allows null (ϵ) moves, where the machine can change states without consuming any input.
Example: Construct an NFA that accepts strings ending in 'a'.
Given:
Σ = {a, b},
Q = {q0, q1},
F = {q1}
State Transition Table for above Automaton,
| State\Symbol | a | b |
|---|---|---|
| q0 | {q0,q1} | q0 |
| q1 | φ | φ |
In an NFA, if any transition leads to an accepting state, the string is accepted.
NFA vs DFA
DFA | NFA |
|---|---|
Deterministic: exactly one transition per input symbol | Non-deterministic: multiple transitions per input symbol allowed |
No ε (null) moves | Allows ε (null) moves |
Simpler but sometimes larger in design | More flexible and easier to design |
Next state is uniquely determined | Next state may be multiple possibilities |
Recognizes regular languages; NFAs can be converted to DFA | Recognizes regular languages; equivalent in power to DFA |