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Comp 330 (Fall 2024): Assignment 2
1. Let A be the automaton depicted below.
(a) (10 points) Compute a minimal deterministic finite automata (DFA) from A.
(b) (10 points) Using the minimal DFA to determine a regular expression representing L(A)
2. (20 points) Let Σ be a (non-empty) alphabet and let w ∈ Σ* be a string. We say that x ∈ Σ* is a prefix of the string w if there exists a string u ∈ Σ* such that w = xu.
Consider the following language
L = {w ∈ {a, b}* |for every prefixx of w nb (x) ≥ na (x)}
Prove that L is a context-free language. Your proof should rely on mathematical induction.
3. (10 points) Compute the Chomsky Normal Form of the following context-free grammar (CFG).
S → aAa|bBb|∈
A → C|a
B → C|b
C → CDE|∈
D → A|B|ab
4. (20 points) State whether the following claim is true or false and prove your answer.
Claim: For any (non-empty) alphabet Σ, there is no language L ≤ Σ* which is both regular and inherently ambiguous.
Hint: Use a context-free grammar recognizing L to show a contradiction if it were ambiguous.
5. (20 points) Build a Pushdown Automaton (PDA) recognizing the language L = {aibj |i = 2 · j}. The PDA must recognize the words by an empty stack. Briefly explain how your PDA works. Then, describe an execution of your PDA on aaaabb and show it accepts it.
6. (10 points) Use the pumping lemma to show that the language L = {an bn an bn jn ≥ 0} is not context-free.