CSE 355 Homework Two Solutions

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1 CSE 355 Homework Two Solution Due 2 Octoer 23, tart o cla Pleae note that there i more than one way to anwer mot o thee quetion. The ollowing only repreent a ample olution. () Let M e the DFA with tranition unction: State Tranition on Tranition on q tart, inal q q q inal q 2 q q 2 q q 2 Uing the GNFA method rom cla, produce a regular expreion that decrie the language recognized y M. Show the tep involved. Anwer: The irt tep i to add a new tart and inal tate, producing the ollowing GNFA (Tranition on are omitted): q q q 2 Then q 2 i removed reulting in the ollowing GNFA (Note: removing the tate in a dierent order will produce a dierent, ut equivalent, regular expreion): q q *

2 Then q i removed reulting in the ollowing GNFA: * q * * * * Finally q i removed reulting in the inal GNFA: * * ( * * ) * ( * ) Thereore, a regular expreion that decrie the language o M i ( ) (ɛ ). (Note: I you removed the tate in a dierent order, you would have otained a dierent, ut equivalent, inal regular expreion). 2

3 (2) Let M e the NFA with tranition unction: State Tranition on Tranition on Tranition on ε q tart {q, q { q inal {q 2 q 2 {q {q 4 q 4 { {q 6 { q 6 inal Uing the power et method rom cla, produce a DFA that i equivalent to M. Anwer: The DFA that i equivalent to M generated y uing the power et method i given elow (Only tate reachale rom the tart tate are hown): {q, {q 4 {, {q 4, q 6, {q, q, {q 2, q 4 {q,, {q 2, q 4, q 6 (3) Let M e the NFA in Quetion 2. Uing the GNFA method rom cla, produce a regular expreion that decrie the language recognized y M. Show the tep involved. Anwer: The irt tep i to add a new tart and inal tate, producing the ollowing GNFA (Tranition on are omitted): q q q 2 q 4 q 6 3

4 Then q 2, q 4 and q 6 are removed reulting in the ollowing GNFA (Note: removing the tate in a dierent order will produce a dierent, ut equivalent, regular expreion): q q Then q and are removed reulting in the ollowing GNFA: q ()* Next i removed reulting in the ollowing GNFA: q () * () * Finally q i removed reulting in the inal GNFA: * (() * () * ) Thereore, a regular expreion that decrie the language o M i (() () ). (Note: I you removed the tate in a dierent order, you would have otained a dierent, ut equivalent, inal regular expreion). 4

5 (4) Let w Σ. Write w = w w n with w i Σ or i n. The revere o w i the tring w n w. For language L Σ, the revere o L i {revere(w) : w L. Show that L i regular i and only i revere(l) i regular. Anwer: Firt, we ll aume L i regular and how revere(l) i regular. Since L i regular, there exit a DFA, D = (Q, Σ, δ, q, F ) uch that L(D) = L. We will contruct an NFA N to recognize revere(l) a ollow: Add a new tate to the tate o D and make thi the tart tate o N. Add ɛ-tranition rom thi new tart tate to all the inal tate o D. Thi allow a computation o N to nondeterminitically tart in all the poile inal tate o D. Revere all the tranition o D. That i or every tranition rom q to q on input a in D, add the tranition rom q to q on input a to N tranition unction. Thi allow the computation o N on input revere(w) to revere the computation o D on input w y nondeterminitically ollowing the tranition o D rom a tate ackward to all poile way the tate could have een reached on the given input. Finally, make the initial tate o D into the only inal tate o N. Thi enure a computation o N on input revere(w) only accept i it could end in the the tate that the computation o D on input w tarted, eectively having revered the computation ack to the eginning along one rach o N computation. Thu, we have contructed an NFA, N, that accept revere(w) i D accept w (y the argument mentioned in the point aove. A more ormal proo i given later or thoe intereted, ut i not required). Thereore, there i an NFA that recognize revere(l) and we conclude that revere(l) i regular. Next, we will aume that L i a language and revere(l) i regular and how that L i then regular. We have jut hown or any regular language, the revere o that language i regular. Thereore, ince revere(l) i regular, revere(revere(l)) i regular. However, revere(revere(l)) = L ince w = w w 2... w n L i revere(w) = w n... w 2 w revere(l) i revere(revere(w)) = w w 2... w n = w revere(revere(l)). Thereore, L i regular. Thu, we have hown that L i regular i and only i revere(l) i regular. [A more ormal proo or the orward direction ollow or thoe intereted, ut i not required. We will deine the NFA N = (Q {q, Σ, δ N, q, F N = {q ) with {q Q : δ(q, a) = q i q Q, a Σ δ N (q, a) = F i q = q, a = ɛ otherwie Next we will how L(N) = revere(l). revere(w) = w n... w 2 w revere(l) i w = w w 2... w n L (y deinition o revere) i w L(D) (ince L(D) = L y aumption) i there exit an accepting computation r r r 2... r n with r = q, 5

6 δ(r i, w i+ ) = r i+ or i n, and r n F on input w (y deinition o DFA acceptance) i r n δ N (r, ɛ) where r = q, r i δ N (r i, w i ) or i n, and r = q F N (y deinition o δ N aove) i r r n... r 2 r r i an accepting computation or N on input w n... w 2 w = revere(w) (y deinition o NFA acceptance) i revere(w) L(N). Thu, L(N) = revere(l) and thereore, revere(l) i regular.] (5) (not to e graded) Sometime tudent make the ollowing argument. (a) A DFA can only have a inite et o tate. () A DFA with n tate cannot rememer an integer value i it can take on any value etween and n. Inormally, a DFA could only count up to a inite numer le than it numer o tate. (c) Thereore any language whoe recognition require counting the occurrence o a utring, where that numer can e aritrarily large, cannot e a regular language. (d) Recognition o the language L = {w {a, : the numer o occurrence o a i the ame a the numer o occurrence o a involve counting occurrence o a and a, and thee count can e aritrary large. (e) So L cannot e regular. I thi a valid argument? Explain a preciely a you can why, or why not. Anwer: Thi argument i not valid. In act L i regular, which we ll how hortly. While point (a)-(c) aove are valid, the aumption in point (d) that recognizing L involve counting occurrence o a and a i not correct and, thereore, the concluion in (e) i invalid. To ee why the aumption in (d) i not correct, we will aume w L and, without lo o generality, w tart with an a (the ymmetrical argument hold i w tarted with a ). I w eventually contain a (that i an occurrence o a), then w mut contain another a (giving the countering occurrence o a). Additionally, it i impoile to have two occurrence o a without an intermediary occurrence o a in the input tring (ince the alphaet only contain a and ). Thereore, to recognize L we can jut match the next occurrence o an a in w with the ollowing occurrence o a and don t have to try and count all occurrence a tipulated in (d). Since, whatever ymol w tart with, i it ever contain a change o ymol, then the tarting ymol mut eventually ollow, we ee that L end up eing the ame a the language {w {a, : w tart and end with the ame ymol (an NFA or the related language over {, i given in the olution to Homework quetion 2e). A DFA i given or L elow. Thu, L i regular and we ee (e) i invalid. a a q q a q 2 a a 6

Solutions Problem Set 2. Problem (a) Let M denote the DFA constructed by swapping the accept and non-accepting state in M.

Solutions Problem Set 2. Problem (a) Let M denote the DFA constructed by swapping the accept and non-accepting state in M. Solution Prolem Set 2 Prolem.4 () Let M denote the DFA contructed y wpping the ccept nd non-ccepting tte in M. For ny tring w B, w will e ccepted y M, tht i, fter conuming the tring w, M will e in n ccepting