The number of states required to automate the last question i.e. {a,b}*{aba}{a,b}* using finite automata:a)4b)3c)5d)6Correct answer is option 'A'. Can you explain this answer?

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The finite automata can be represented as:

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The finite automata can be represented as:

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The finite automata can be represented as:

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Number of states required to automate the given regular expression:

The regular expression {a,b}*{aba}{a,b}* represents the language that consists of all strings that have the substring 'aba' in them. To automate this regular expression using a finite automaton, we need to determine the minimum number of states required.

Step 1: Construction of DFA

To construct the DFA for the given regular expression, we can follow the below steps:

1. Start with an initial state.
2. For each input symbol (a and b), create a transition from the current state to a new state.
3. If the input symbol is 'a' and we are in the initial state, transition to a new state. If the input symbol is 'a' and we are in the state representing the substring 'ab', transition to another new state. If the input symbol is 'b' and we are in the state representing the substring 'aba', transition to a new state.
4. Repeat step 3 for each input symbol and each state.

Step 2: Determining the minimum number of states

To determine the minimum number of states required, we need to eliminate any redundant states. Two states are considered redundant if they have the same set of outgoing transitions for each input symbol.

1. Initially, we start with the initial state.
2. For each input symbol (a and b), we check the set of outgoing transitions from the current state.
3. If there is a state with the same set of outgoing transitions for each input symbol, we can merge these two states into one.
4. Repeat step 3 for each input symbol and each state.

Step 3: Analysis of DFA

After constructing the DFA and eliminating redundant states, we can analyze the resulting DFA to determine the minimum number of states required.

In the given regular expression, there are three important parts:
1. {a,b}* before the substring 'aba'
2. The substring 'aba' itself
3. {a,b}* after the substring 'aba'

Part 1: {a,b}* before the substring 'aba'

This part can accept any combination of 'a' and 'b' before encountering the substring 'aba'. Therefore, it can be represented by a single state that loops back to itself for each input symbol.

Part 2: The substring 'aba'

This part can be represented by three states: one state for each character in the substring 'aba'. The first state represents the first 'a' in 'aba', the second state represents 'b', and the third state represents the second 'a'.

Part 3: {a,b}* after the substring 'aba'

Similar to part 1, this part can accept any combination of 'a' and 'b' after encountering the substring 'aba'. Therefore, it can be represented by a single state that loops back to itself for each input symbol.

Total number of states:

Considering the analysis of the DFA, we can determine that there are a total of 4 states required to automate the given regular expression: one state for part 1, three states for part 2, and one state for part 3.

Therefore, the correct answer is option 'A' - 4 states.

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