CS138: Automata and Formal Languages Lecturer: Stefano Tessaro UC Santa Barbara Fall 2014 Homework 2 Posted: Friday, Oct 10, 2014 – 11:59pm Due: Tuesday, Oct 21, 2014 – 12pm (HFH 2108) or 12:30pm (in class) Task 1 – DFAs (6 points) Consider the DFA M defined by the following transition graph: a) Represent the DFA as a tuple M = (Q, Σ, δ, q0 , F ). b) What is δ ∗ (q0 , ab)? What is δ ∗ (q0 , abaabbb)? What is δ ∗ (q0 , abbbba)? c) Describe the language L(M ) accepted by M . Hint: It will be easier to represent L(M ) as L1 ∪ L2 , for appropriate L1 and L2 . Task 2 – Regular Languages (9 points) Show that each of the following languages with alphabet Σ = {a, b} is regular. Concretely, give corresponding DFAs accepting them, and justify (in words, but as precisely as possible) why your DFAs accept the corresponding languages. a) L1 := {a2 b3 ai : i ≥ 2}. b) L2 := {w ∈ Σ∗ : |w| is even}. c) L3 := {abw : w ∈ Σ∗ , w does not contain two consecutive a’s} 1 Task 3 – Two-state DFAs (7 points) Consider the DFA with alphabet Σ = {0, 1} defined by the following transition graph: a) Find a language L ⊆ {0, 1}∗ such that L(M ) = L∗ . b) How many different DFAs with states Q = {q0 , q1 }, initial state q0 , and alphabet Σ = {0, 1} are there? Hint: Here, different means with different structure, i.e., two different DFAs may well accept the same language and be equivalent. Task 4 – Finite Languages (8 points) The goal of this task is to convince yourself that finite languages (i.e., languages L that only contain a finite number of sentences), are regular. a) Let L1 = {010} be a language with one single sentence, and alphabet Σ = {0, 1}. Show that L1 is regular by giving a corresponding DFA. b) Let L2 = {010, 011, 100}. Show that L2 is regular by giving a corresponding DFA. c) Show that every finite language L over an alphabet Σ is regular by giving a method to construct the corresponding DFA for any finite language 2