Show the transition table for the DFA that is equivalent to the below NFA. Remember, what is required is the transition table for the DFA. Therefore,
the required table should not have a column for e.
The Correct Answer and Explanation is :
To convert a given Nondeterministic Finite Automaton (NFA) to its equivalent Deterministic Finite Automaton (DFA), we employ the subset construction method, also known as the powerset construction. This process systematically transforms the NFA, which may have multiple possible transitions for a given input (including ε-transitions), into a DFA that has exactly one transition per input symbol from each state.
Steps to Convert NFA to DFA:
- Identify the ε-closures: For each state in the NFA, determine the ε-closure, which is the set of states reachable from the given state through ε-transitions alone.
- Define the DFA’s Start State: The start state of the DFA is the ε-closure of the NFA’s start state.
- Construct DFA States and Transitions:
- For each newly identified DFA state (which represents a set of NFA states), compute the transitions for each input symbol.
- For a DFA state representing a set of NFA states, the transition for an input symbol is determined by:
- Applying the input symbol to each NFA state in the set.
- Taking the union of all resulting states.
- Applying the ε-closure to this union to account for any ε-transitions.
- If this resulting set of NFA states (after ε-closure) has not been encountered before, it becomes a new DFA state.
- Determine Accepting States: Any DFA state that includes at least one of the NFA’s accepting states is marked as an accepting state in the DFA.
Example:
Consider an NFA with states {q0, q1, q2}, where:
- q0 is the start state.
- q2 is the accepting state.
- Transitions are defined as follows:
- q0 –ε–> q1
- q1 –a–> q2
- q2 –b–> q2
Conversion Process:
- ε-closures:
- ε-closure(q0) = {q0, q1}
- ε-closure(q1) = {q1}
- ε-closure(q2) = {q2}
- DFA Start State:
- {q0, q1} (let’s call this DFA state A)
- Transitions:
- From state A on input ‘a’:
- q0 has no ‘a’ transition.
- q1 –a–> q2.
- ε-closure({q2}) = {q2} (call this DFA state B).
- From state A on input ‘b’:
- Neither q0 nor q1 have a ‘b’ transition.
- Resulting in an empty set, which we’ll denote as the dead state (D).
- From state B on input ‘a’:
- q2 has no ‘a’ transition.
- Resulting in the dead state (D).
- From state B on input ‘b’:
- q2 –b–> q2.
- ε-closure({q2}) = {q2}, so it transitions to state B.
- Accepting States:
- State B ({q2}) is accepting because it contains q2, the NFA’s accepting state.
DFA Transition Table:
| DFA State | NFA States | a | b |
|---|---|---|---|
| A | {q0, q1} | B | D |
| B | {q2} | D | B |
| D | {} | D | D |
In this table, ‘A’ is the start state, ‘B’ is the accepting state, and ‘D’ is the dead state where the automaton remains if it encounters an undefined transition.
This method ensures that the resulting DFA accurately represents the language accepted by the original NFA, without the ambiguity inherent in nondeterministic transitions.