Theory of Computation CSE 105

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1 $ $ $ Theory of Computtion CSE 105 Regulr Lnguges Study Guide nd Homework I Homework I: Solutions to the following problems should be turned in clss on July 1, Instructions: Write your nswers clerly nd completely. Plese use inches pper. Use stpler or clip to ttch the individul pges. Write your nme. When presenting ny construction, for exmple, n lgorithm or n utomton, plese give n overview of the min ides nd then present the construction. Alwys support the correctness of your construction with short informl proof. 1. For ech of the lnguges given below, design finite stte utomt nd regulr expressions to recognize them. In ll cses the lphbet is. () does not contin the substring 110 (b) contins n even number of 0 s, or exctly two 1 s. Problem 1., Pge 88.. Problem 1.4, Pge 90. Provide short proof of the correctness of your construction. 4. Prove the following lnguges non-regulr:!"# () is prime ; (b) Let $ %'& )( & *( & +( &,(.-'/ Here, contins ll columns of 0 s nd 1 s of height two. A string of symbols in gives two rows of 0 s nd 1 s. Consider ech row to be binry number nd let 1 0 the bottom row of is the reverse of the top row of / 1

2 Study Guide: In the following, the mteril on regulr lnguges (chpter 1) is broken down into number of short topics. For ech topic, list of specific items nd problems re provided. If you understnd these items nd solve the problems, you would do very well in the course. 1 Deterministic Finite Automt (DFA) Topics: The notion nd definition of DFA, presenttion of DFA by trnsition digrms, the notion of cceptnce by DFA, the clss of regulr lnguges, techniques for designing DFAs, nd closure opertions. Designing DFAs , 1., nd 1., pges 8 nd 84.. Exercise 1.4, Pge 84.. For ech of the following regulr expressions, drw DFA recognizing the corresponding lnguge : ;7 () 9 <+ : 7 (b) <5 9 : 7 (c) 4. Drw DFA tht recognizes the lnguge of ll strings of 0 s nd 1 s of length = tht, if they were interpreted s binry representtions of integers, would represent integers evenly divisible by. Leding 0 s re permissible. 5. Show tht if is regulr lnguge nd > is finite lnguge, then nd BAB> re regulr.. Show tht if is non regulr lnguge nd > is finite lnguge, then > nd BAB> re non regulr. 7. Problems 1.5 nd 1.7, Pge , 1.0, 1.41 Closure Properties of Regulr Lnguges 1. For ech sttement below, decide whether it is true or flse, If it is true, prove it; if not, give counter exmple. All prts refer to lnguges over "DC. () If EEFG nd E is not regulr, then H is not regulr. (b) If EEFG is not regulr, then is regulr. (c) If nd re nonregulr, then is nonregulr.

3 (d) If nd re nonregulr, then?i is nonregulr. (e) If is not regulr, then J, the complement of, is not regulr. (f) If E is regulr nd H is nonregulr, then H is nonregulr. (g) If E is regulr, H is nonregulr, nd E?5H is nonregulr, then E nonregulr. :K:K:K (h) If E re regulr, then.l MON M is regulr.. Problem 1.4, 1.4 is Nondeterministic Finite Automt (NFA) Topics: The notion of nondeterminism, definition of cceptnce for NFAs, economy of sttes by using NFA, equivlence of DFAs nd NFAs, nd exmples tht illustrte the conversion of n NFA to n equivlent DFA. Notion of Nondeterminism 1.9, 1.10, pge 85 Prctice in Designing NFAs 1.5, 1., 1.7, 1.8, pges 84 nd 85. Prctice in converting n NFA to n equivlent DFA 1.1, pge 85. Regulr Expressions (RE) Topics: The definition of regulr expressions, writing regulr expressions, equivlence with finite utomt: every regulr expression hs n equivlent finite utomt nd every finite utomt hs n equivlent regulr expression. Bsics of Regulr Expressions 1. Wht is the shortest string of s nd C s not in the lnguge corresponding to the regulr expression CP7 QCPCP7 87P 7? 7TS5CP7 CP7TS5CW 7XS. Consider C 7S 7 the C: 7 following two regulr expressions R nd UV. () Find string corresponding to R but not to U. (b) Find string corresponding to but not to. (c) Find string corresponding to both R nd U. (d) Find string corresponding to neither R nor U.. Simplify the following regulr expressions:

4 1 1 Y <+ : Z, Y 9 : U+ : 9 Y 87 () [ (b) (c) Y Y 7 Y Y 7 ] Y 7 \ Y <,^ 7 (d) : 4, Y 7 : 9 Y : _, Y 7 (e) Y 7 4. Wht is true of the lnguge corresponding to regulr expression tht does not involve the opertors ` or S? Why? Designing Regulr Expressions , pge 8. Find regulr expressions corresponding to ech of the lnguges defined recursively below. () ^1 ; if, then b C CPC nd re elements of ; nothing is in unless it cn be obtined from these two sttements. (b) +1 ; if, then QQC QQC CPC,, nd re elements of ; nothing is in unless it cn be obtined from these two sttements.. Find regulr expression corresponding to ech of the following subsets of 7. : 7 () The lnguge of strings contining exctly two 0 s. (b) The lnguge of strings contining t lest two 0 s. (c) The lnguge of strings tht do not end with 01. (d) The lnguge of strings tht begin or end with 00 or 11. (e) The lnguge of strings contining no more thn one occurrence of the string 00. (The string 000 should be viewed s contining two occurrences of 00.) (f) The lnguge of strings in which the number of 0 s is even. (g) The lnguge of strings in which every 0 is immeditely followed by 11. (h) The lnguge of strings tht do not contin the substring 110. (i) The lnguge of strings tht do contin both the substring 11 nd the substring 010. Interpreting Regulr Expressions Describe s simply s possible the lnguge corresponding to ech of the following regulr expressions. 7 7 : dc 7 ^ei 4+. \

5 = <I Y 87 Z, Y [ [ [ 5 [ [ [ Prctice the Trnsltion Algorithm from REs to NFAs 1.14, pge 8 Prctice the Trnsltion Algorithm from DFAs to REs 1.1, pge 8 4 Non regulr Lnguges Topics: Pumping lemm, exmples of nonregulr lnguges nd pplictions of pumping lemm. Appliction of Pumping Lemm 1. Using the Pumping Lemm show tht ech of these lnguges is not regulr. M CW () f M g Q bhic8j kply m,npoqsr (b) f 1t XDC u7; vw y{zov D (c) s x x where vw x ( v x ) is the number of occurrences of the letter ( C ) in. 1 "DC 7 (d) f no initil substring of hs more C s thn s 1 "DC 7 (e) f is plindrome V} s1) "DC u7 (f) f. Here is proof using the pumping lemm, tht the lnguge of ll strings of s nd C s of length 100 is not regulr. Since the result being proved is flse (ll finite lnguges re regulr), the proof cnnot be correct. Wht is the flw in the proof? Assume tht is regulr. By the pumping lemm, if we choose n element of, sy 9 \~, there re string, nd, with ~] n ~Ql, so tht every string of the form (where m = ) is in. Since there re infinitely mny different strings of this form, this contrdicts the fct tht is finite. Therefore, is not regulr , pge , 1.8, 1., 1., pges 88 nd , pge , 1.4, pge 90 Regulr or Nonregulr Below re number of lnguges over "DC. In ech cse, decide whether the lnguge is regulr or not, nd prove tht your nswer is correct.

6 1. is the set of strings beginning with non-null string of the form Z.. is the set of ll strings hving some non null string of the form V.. is the set of strings hving some non null substring of the form ZV. 1 "DC 7 4. f is not plindrome 1 "DC 7 5. f begins with plindrome of length = 1 "DC u7uv w. f x is perfect squre 1 "DC u7 7. f in every initil string of the number of s nd the number of C s differ by no more thn 1 "DC u7 8. f in every substring of, the number of s nd the number of C s differ by no more thn 1 "DC 7 vw 9. f x nd v x re both divisible by 5 1 "DC u7 10. f there is some integer m,nƒ so tht vw x nd v x re both divisible by m 5 Decision Algorithms Describe decision lgorithms to nswer ech of the following questions. 1. Given two DFAs nd, re there ny strings tht re ccepted by neither?. Given n NFA nd string, does ccept?. Given two NFAs, do they ccept the sme lnguge? 4. Given n NFA nd string, is there more thn one sequence of trnsitions corresponding to tht cuses to ccept? Miscellneous Problems Myhill-Nerode Theorem: 1.4 nd 1.5. Number of sttes: 1.9 nd Trnsducers: 1.19, 10, 1.1 nd 1.