Final answer:
A minimal DFA can be constructed with five states to accept strings where all occurrences of 'c' precede any 'a', without containing two consecutive 'b's, with the addition of a dead state for invalid strings violating these rules.
Step-by-step explanation:
We need to design a minimal Deterministic Finite Automaton (DFA) that recognizes the set of all strings (including the empty string) such that all 'c' characters appear before any 'a' character, and the string does not contain two consecutive 'b's. To do this, we can use states to represent the conditions of our string as it is being processed.
Here's a description of the DFA states:
- q0: The start state, where we have seen no 'a's, and the last character was not 'b'.
- q1: We've seen at least one 'a', and the last character was not 'b'.
- q2: The last character we saw was 'b', and we have not yet seen an 'a'.
- q3: The last character we saw was 'b', following an 'a'.
- q4: A dead state, which we transition to if we see two consecutive 'b's or if we see a 'c' after an 'a'.
The transitions between these states reflect the above conditions. Since we can have an infinite number of 'd's at any point, all states loop on 'd'. The initial state q0 accepts 'c' to remain in q0. After seeing an 'a', we move to q1, which accepts all characters except 'c'. From q0 or q1, if a 'b' is encountered, we transition to an intermediate state q2 or q3 respectively, to check for another 'b'. Any transition violating our 'c' before 'a' or two consecutive 'b's conditions moves to the dead state q4. Since empty strings are accepted, q0 is also an accept state.