Question

For σ = {a, b}, construct DFA's that accept the sets consisting of:

a) All strings with at least two a's
b) All strings with at most two a's
c) All strings with exactly two a's
d) All strings with no a's

Answer 1

To address the student's request, four separate DFA's are constructed to accept strings with varying occurrences of the letter 'a': at least two, at most two, exactly two, and none, adhering to a defined transition logic for each.

Step-by-step explanation:

Constructing DFA's for Specific String PatternsTo construct Deterministic Finite Automata (DFA) for the given string patterns over the alphabet σ = {a, b}, we will create separate automata for each condition:All strings with at least two a's: Start with a state Q0 that transitions to itself on 'b' and to Q1 on 'a'. Q1 transitions to itself on 'b' and to Q2 on 'a'. Q2 is the accepting state and transitions to itself on both 'a' and 'b'All strings with at most two a's: Start with an accepting state Q0 that transitions to Q1 on 'a' and stays on itself on 'b'. Q1 is also accepting and transitions to Q2 on 'a', stays on itself on 'b'. Q2 transitions to a non-accepting dead state Q3 on 'a' and stays on itself on 'b'All strings with exactly two a's: Starting at Q0, on 'a' transition to Q1, remain in Q0 on 'b'.

From Q1, on 'a' go to accepting state Q2, remain in Q1 on 'b'. Q2 transitions to a non-accepting dead state Q3 on 'a' and stays in Q2 on 'bAll strings with no a's: Start with an accepting state Q0 that transitions to a non-accepting state Q1 on 'a' and stays on itself on 'b'. Q1 is a dead state and stays on itself for all inputsEach DFA is based on the number of 'a' characters present in the string, with transitions reflecting this count and accepting states determined by the specified conditions.To construct the DFA that accepts strings with at least two a's: Start with the initial state s0.. If the input is 'a,' transition to state s1If the input is 'b,' stay in state s0.From state s1, if the input is 'a,' transition to the accepting state s2. Otherwise, stay in state s1. The accepting state s2 accepts any further 'a' inputs while staying in the state s2.To construct the DFAs for the other sets of strings, the approach is similar, but with different transition conditions.