Language of Finite Automata
The language of a finite automaton refers to the set of all strings that the automaton can accept. A finite automaton is a mathematical model used in computer science and formal language theory to recognize patterns in strings. It consists of a finite set of states, an input alphabet, a transition function, an initial state, and a set of accepting states.
The language accepted by a finite automaton is the set of all strings for which the automaton reaches an accepting state when it processes the input string. Here’s a brief explanation with an example:
Finite Automaton Components:
- States (Q): A finite set of states, such as {q0, q1, q2}.
- Alphabet (Σ): A finite set of input symbols, like {0, 1}.
- Transition Function (δ): Describes how the automaton transitions between states based on input symbols.
- Initial State (q0): The starting state.
- Accepting States (F): A set of states where the automaton accepts the input.
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Example.
Let’s consider a simple finite automaton that recognizes strings with an odd number of 1s. The components are:Â
Let’s consider a simple finite automaton that recognizes strings with an odd number of 1s. The components are:
- States (Q): {q1, q2}
- Alphabet (Σ): {0, 1}
- Transition Function (δ):
- δ(q1, 0) = q1
- δ(q1, 1) = q2
- δ(q2, 0) = q1
- δ(q2, 1) = q2
- Initial State (q1): The starting state is q0.
- Accepting States (F): {q2}
Explanation:
- The automaton starts in the initial state q1.
- For each input symbol, it transitions between states according to the transition function.
- If the automaton ends up in an accepting state after processing the entire input string, it accepts the string.
Computation:
Let’s consider the input string “1101”:
- q0 (initial state) -> δ(q1, 1) = q2
- q2 -> δ(q2, 1) = q2
- q2 -> δ(q2, 0) = q1
- q1 -> δ(q1, 1) = q2 (final state)
Since the automaton reaches the accepting state q2, the input string “1101” is accepted. The language accepted by this automaton consists of all strings with an odd number of 1s.
