Final answer:
The question is about creating a finite automaton for the regular expression (a+b)*a, which accepts any string of 'a's and 'b's ending with an 'a'. The FA will have a repeat state for both symbols and a final state upon the last 'a'. This reflects the algebraic rule of exponents, (x^a)^b = x^(a.b), in the sequential processing of strings.
Step-by-step explanation:
The student is asking about the construction of a finite automaton (FA) for the regular expression (a+b)*a. In formal languages, finite automata are abstract machines that recognize patterns within input strings. In the given expression, (a+b)* signifies any combination, including the empty string, of 'a's and 'b's, and this is followed by an 'a'. Consequently, any string accepted by this FA must end with an 'a'.
To construct an FA for this pattern, you can create a state machine that accepts any number of 'a's and 'b's and then transitions to a final state upon receiving an 'a'. The FA will have at least two states: an initial state that is also a repeat state for both 'a' and 'b' input, painting both letters back into the same state, and a final state that is only reached when an 'a' is received from the repeat state.
This construction relies on our understanding of powers in algebra, where the rule (xa)b = xa.b can be taken as an analogous concept representing the sequential processing of string symbols in finite automata.