Final answer:
The NPDA recognizes a language consisting of strings with any number of 'a's followed by groups of 'b's sandwiched between more 'a's which can repeat. The regular expression for this language is a*ba*ba*b(a*ba*ba*b)*a*.
Step-by-step explanation:
The language that the given NPDA (Nondeterministic Pushdown Automaton) recognizes can be determined by examining the transition functions provided. The first transition δ(q₀,a,$) = {(q₀, A$)} says that when the machine reads 'a' and the stack top is '$' (the bottom of the stack symbol), it pushes 'A' onto the stack. The second transition δ(q₀,b,A) = {(q₀, AA)} implies that for input 'b' with 'A' on the stack top, the machine will push another 'A' onto the stack. The last transition δ(q₀,a,A) = {(q₁, λ)} indicates that when 'a' is read and 'A' is on the stack, the machine moves to state q₁ and pops 'A' from the stack without pushing anything back.
Basically, the automaton can recognize strings that start with a series of 'a's, push an equal number of 'A's onto the stack, and then for every subsequent 'b', two more 'A's are pushed onto the stack. When reading 'a' again, the process of popping 'A's begins, which requires an equal number of 'a's as there were 'b's to return to the initial stack configuration with just $ at the end. So, the regular expression representing this language would be a*ba*ba*b(a*ba*ba*b)*a*. This regular expression signifies any number of 'a's followed by groups of 'b's sandwiched between more 'a's, and can repeat any number of times including zero (effectively, b can be seen as bb due to its stack effect).