automata problems
- 1. Homework One Solution– CSE 355 Due: 31 January 2011 Please note that there is more than one way to answer most of these questions. The following only represents a sample solution. Problem 1: Linz 2.1.7(b)(c)(g), 2.2.7. and 2.2.11 2.1.7: Find dfa’s for the following languages on Σ = {a, b} (b): L = {w : |w| mod 5 = 0} A dfa for L is given by the following transition graph: a,b a,b a,b a,ba,b (c): L = {w : na(w) mod 3 > 1} A dfa for L is given by the following transition graph: a a a b b b 1
- 2. (g): L = {w : |w| mod 3 = 0, |w| = 6} A dfa for L is given by the following transition graph: a,b a,b a,b a,b a,b a,ba,ba,b a,b a,b 2.2.7: Design an nfa with no more than five states for the set {ababn : n ≥ 0} ∪ {aban : n ≥ 0}. An nfa for the set is given by the following transition graph: a b b b a a λ 2.2.11: Find an nfa with foour states for L = {an : n ≥ 0} ∪ {bn a : n ≥ 1}. An nfa for L is given by the following transition graph: b b aλ a 2
- 3. Problem 2: Linz 2.39 and 2.3.12 2.39: Let L be a regular language that does not contain λ. Show that there exists an nfa without λ-transitions and with a single final state that accept L. Since L is regular there exists a dfa, D = (Q, Σ, δ, q0, F), with an associated transition graph, GD, such that L(D) = L. We will construct an nfa N = (Q ∪ {qf }, Σ, δ , q0, {qf }) where qf /∈ Q by giving its transition graph GN as follows: 1. From GD, remove the final label from every final state (making them nonfinal states). 2. Add a new state qf and label it as a final state. 3. For every state qi, if there is a transition from qi to a state in F on input a ∈ Σ, then add a transition from qi to qf on input a. Clearly, N has a single accept state, qf , and no λ-transitions (since D is a dfa and we did not add any λ-transitions in our construction of N). We will now show that L(N) = L. First note that since λ /∈ L, every w ∈ L can be written as w = va for some v ∈ Σ∗ and an a ∈ Σ. Now, w = va ∈ L iff there is a walk on GD labeled with w from q0 to qi with qi ∈ F iff there is a walk on GD labeled with v from q0 to qj and a transition from qj to qi on input a iff there is a walk on GN labeled with v from q0 to qj and a transition from qj to qf on input a (since every transition in GD is a transition in GN and from step (3) in the construction of GN ) iff there is a walk on GN labeled with w from q0 to qf iff w ∈ L(N). Thus, w ∈ L iff w ∈ L(N). Therefore we conclude that L(N) = L and that for any regular language that does not contain λ, there exists an nfa without λ-transitions and with a single final state that accept L. 2.3.12: Show that if L is regular, so is LR . Since L is a regular language, we can construct a corresponding dfa, N, such that L(N) = L (For every regular language, there is a corresponding dfa, by definition, and for every dfa, there is an equivalent nfa). By definition, LR consists of all strings in language L in reverse order. We will construct a nfa, NR, representing LR such that L(NR) = LR. NR will contain an additional start state with λ-transitions to the final states of N. The direction of every transition in N is reversed. Also, the start state of N will be the final state of NR. The construction of nfa NR is as follows: Let N = (Q, Σ, δ, qn, F) NR = (Q ∪ {q0}, Σ, δr, qr, {qn}) Set of states of NR = set of states of N along with q0 = Q ∪ {qr} Σ = alphabet of NR = same as N qr = start state of NR {qn} = set of final states of NR = start state of N Transition function: δr(q, a) = {q1 : δ(q1, a) = q} δr(qr, λ) = F 3
- 4. δr(qr, a) = ∅, if a = λ Now we will show that LR = L(NR). w ∈ LR iff wR ∈ L iff there is a walk on the transition graph of N with label wR from qn to some qi ∈ F iff there is a walk on the transition graph of NR from qr to qi with label λ and a walk from qi to qn with label w (Following the reverse of every transition in the original graph) iff w ∈ L(NR). Since LR can be represented by a nfa, it is regular (by equivalence of nfa to dfa, and dfa to regular language). Problem 3: Linz 2.1.8 2.1.8: A run in a string is a substring of length at least two, as long as possible and consisting entirely of the same symbol. For instance, the string abbbaab contains a run of b s of length three and a run of a s of length two. Find dfa’s for the following languages on {a, b}. (a): L = {w : w contains no runs of length less than four}. a b a aaaa a a a a a a b b b b b b b b b b b 4
- 5. (b): L = {w : every run of a s has length either two or three}. b a b b bb aaaa (c): L = {w : there are at most two runs of a s of length three}. a a a a a a a a a a aaaa a a,b b b b b b b b b b b b b b b b b 5
- 6. (d): L = {w : there are exactly two runs of a s of length 3}. b a a a a a aaa a a a a a a a a,b b b b b b b b b b b b b b b Problem 4: Linz 2.2.22 2.2.22: Let L be a regular language on some alphabet Σ, and let Σ1 ⊂ Σ be a smaller alphabet. Consider L1, the subset of L whose elements are made up only of symbols from Σ1, that is, L1 = L ∩ Σ∗ 1. Show that L1 is also regular. Since L is a regular language, there should be a dfa, N, representing L such that L(N) = L, where N = (Q, Σ, δ, q0, F). Since L1 is made up of strings with alphabets from Σ1, Σ1 ⊂ Σ, and L1 is a subset of L, L1 contains only strings that are accepted by L as well. We can construct a dfa, M, for L1 as follows: 1. From the transition graph of N, remove every transition that is labeled with some a /∈ Σ1. Now we will show that L(M) = L1. w = a1a2 . . . an ∈ L1 iff there is a walk on the transition graph of N with label w from q0 to some qi ∈ F and every ai ∈ Σ1 iff there is a walk on the transition graph of M from q0 to qi with label w (it will be the exact same path as it was in N) iff w ∈ L(M). Since L1 can be represented by a dfa, it is regular. 6
- 7. Problem 5: Linz 2.3.3 and 2.3.8 2.3.3: Convert the following nfa into an equivalent dfa (see textbook for the diagram). 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 {0,1,2} {1,2}{0,1} {1} {2} Φ {0,2}{0} 2.3.8: Find an nfa without λ-transitions and with a single final state that accepts L = {a} ∪ {bn : n ≥ 1}. Noting that λ /∈ L, we can use the technique given in 2.3.9 (Problem 2) and we get the nfa given by the following transition graph: b b a b a,b 7