First Midterm Examination

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1 Çnky University Deprtment of Computer Engineering Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does the following DFA recognize? Descrie verlly q 0 q q 2 q 3 q 4 3) Let A e lnguge over Σ = {0, }. A = {w w ends with or 000}. ) Find n NFA tht recognizes A. ) Find n NFA tht recognizes A. 4) Find the regulr expression equivlent to the following NFA: ε q q 2 c d q 4 q 3 d 5) Show tht the lnguge A = {w w = w R } over Σ = {0, } is non-regulr. (Here, w R shows the reverse of the string w, for exmple 0 R = 0)

2 Answers ) q Strt c c c q c Reject,,c q c 2) Mny nswers re possile. Some of them, strting with the simplest re: Contins 0 nd ends with. Contins 0 nd 0 nd ends with. Contins t lest two symol chnges nd ends with. Contins t lest 3 symols,, 0 nd in this order (ut not necessrily consecutively) nd ends with. 3) 0, 0 0 0

3 0, ε 0 ε ε 0 0 4) GNFA: ε ε ε q q 3 q 2 d d c q 4 Eliminte q 4 nd q 2 : ε q q 3 c ε dd Eliminte q : q 3 ε Finlly: dd c (dd c ) So the nswer is: (dd c ) Other possile nswers re: [(dd) c ] (dd) or [(dd) (c ) ] 5) Suppose A is regulr. Let the pumping length e p. Choose w = 0 p 0 p. We know tht w = xyz nd xy p therefore y consists of zeros only. y = 0 k where k p. w = xyz A ut xyyz = 0 p+k 0 p / A for ny possile choice of y. So there is contrdiction. A is non-regulr y pumping lemm.

4 Çnky University Deprtment of Computer Engineering Fll Semester Second Midterm Exmintion ) Eliminte rules contining ε from the following grmmr: S AB 00A A BB 0 AB ε B B A 2) Give either PDA (pushdown utomton) or CFG (context free grmmr) for the lnguge { n m c m n n, m 0} over Σ = {,, c} 3) Let A = { n n+ c 4n n 0}. Show tht A is not context-free lnguge. 4) Descrie Turing mchine tht recognizes the lnguge A, given in question 3. 5) A Turing mchine with douly infinite tpe hs single tpe, ut its tpe is infinite to the left nd to the right. The tpe is initilly lnk except for the input. Show tht this is equivlent to n ordinry Turing mchine. Bonus) Give PDA (pushdown utomton) for the lnguge over Σ = {0, } mde of strings tht contin equl numers of 0 s nd s.

5 Answers ) S AB 00A B 00 A BB 0 AB B B B A 2) S S T T T c ε OR, ε, ε ε, ε $ ε, ε ε q q 2 q 3 ε, ε ε q 6 ε, $ ε q 5 ε, ε ε q 4, ε c, ε

6 3) Suppose A is context free. Let p e the pumping length. Choose w s w = p p+ c 4p. How cn we choose v nd y such tht p p+ c 4p = uvxyz? ) They contin more thn one symol. In this cse, the pumped string uv 2 xy 2 z will hve symols out of order. For exmple, it will contin s fter s. Therefore uv 2 xy 2 z / A. 2) They contin single symol. In tht cse, we cn pump t most two of the symols {,, c}. The remining one will hve the sme power in the pumped string. For exmple, if we choose v = nd y = we otin uv 2 xy 2 z = n+ n+2 c 4n / A Therefore we cnnot pump this string. By pumping lemm, A is not context free. 4). Sweep from left to right. IF ny symol is out of order (for exmple, s fter s) REJECT. 2. Go to strt. Serch for. IF found, cross it. (Replce y ) ELSE, Go to Serch for. IF found, cross it. ELSE, REJECT. 4. Repet 4 times: Serch for c. IF found, cross it. ELSE, REJECT. 5. Go to Sweep from left to right. IF there is one nd only one AND there is no c, ACCEPT. ELSE, REJECT.

7 5) We cn divide the doule tpe into two prts: positive nd negtive. Then, we cn mp it into norml Turing tpe y zig-zgging nd plcing positive squres to odd squres nd negtive squres to even squres s follows: { 2n n > 0 f(n) = 2n n < 0 OR Doule Tpe Norml Tpe We cn use the multi-tpe ide for doule tpe nd seprte the contents y #: Doule Tpe Norml Tpe n # 2 Note tht in this cse the # moves s the progrm proceeds. Bonus) 0, ε 0, 0 ε q 3 0, ε 0 ε, $ ε ε, ε 0, $ $ ε, ε $ q q 2 q 5 q 6 q ccept, ε 0, $ $ ε, ε ε, $ ε q 4, ε 0, ε Stte q 3 : Up to now, more 0 s thn s rrived. Stck contins 0 s. Stte q 4 : Up to now, more s thn 0 s rrived. Stck contins s.

8 Çnky University Deprtment of Computer Engineering Fll Semester Finl Exmintion ) Find n equivlent DFA for the following NFA:, q q 2 q 3 q 4 2) Give CFG tht genertes the following lnguge over Σ = {0, }: {w w is of odd length nd contins more 0 s thn s.} 3) Define the lnguge L s: A, k where A is n NFA, k is n integer nd A rejects ll strings of length k. Show tht L is decidle. (Hint: Give TM for L tht hlts) 4) Let A = the set of ll integer coefficient polynomils. B = the set of ll functions f : N N. C = the set of ll infinite strings on Σ = {0,, 2}. Choose one of these sets. Clim tht it is countle or uncountle. Prove your clim. 5) TS (Trveling Slesmn) PROBLEM: Given n cities, nd distnces d(i, j) etween them, wht is the minimum distnce of pth tht visits ech city once? HAMPATH (Hmiltonin Pth) PROBLEM: Given grph G, is there pth tht goes through ech node exctly once? ) Reformulte the TS prolem s Yes/No prolem. Cll it TS2. ) Given tht the HAMPATH prolem is NP-complete, show tht TS2 is NP-complete.

9 Answers ) q q 2 q 23 q 34 q 234, 2) S T 0T T 0T T 0 0T 0 T T ε 3). Simulte A on the TM. 2. Mrk strt stte. 3. For i = to k Mrk ll sttes tht cn e reched from mrked sttes in one step. If n ccept stte is mrked, Return REJECT. EndFor 4. Return ACCEPT. Another ide (tht is d style ut still works) is to mke list of ll strings of length k, give them one y one to our simultion of A nd reject if it ccepts ny one of them.

10 4) ) The set {...,, 0,, 2,...} is countle. Therefore there re countly mny 0 terms in the polynomil. Similrly, there re countly mny x terms. In tht wy, we hve to find countle union of countle sets which is countle. We cn count them in the sme wy we counted rtionl numers, y counting n infinite mtrix. ) We cn think of ech function s n infinite string. The set of such strings is uncountle. If we ssume there is list, we cn generte n element tht is not on the list using Cntor s digonl ide. This contrdicts the ssumption tht the list contins everything. This is the sme ide we used in proving uncountility of rel numers on [0, ]. c) The sme s ). 5) ) TS2 (Trveling Slesmn) PROBLEM: Given n cities, nd distnces d(i, j) etween them, nd positive numer k, is there pth tht visits ech city once nd hs length k? ) We hve to show tht HAMPATH P TS2 Suppose we cn solve ny TS2 prolem. Given HAMPATH prolem, trnsform it into TS2 s follows: If there is connection etween vertex i nd vertex j, set d(i, j) =. If there is no connection etween vertex i nd vertex j, set d(i, j) = 0. Set k = n (numer of vertices) If TS2 finds pth of length n (ctully, we hve length = n) it is the one we re looking for. Clerly, this reduction is of polynomil type.

11 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK Give the stte digrm of DFA or n NFA tht recognizes the lnguge A over lphet Σ = {0, } where A = {w w strts with 0 nd hs odd length, or strts with nd hs length t most 5} NFA: 0, q 0 q 2 q 3 0, 0, 0, 0, 0, q 4 q 5 q 6 q 7 q 8 DFA: 0, q 0 q 2 q 3 0, 0, 0, 0, 0, 0, 0, q 4 q 5 q 6 q 7 q 8 q 9

12 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK Give the stte digrm of DFA or n NFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w = } NFA: q q 2 DFA:, q q 2 q 3

13 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 2 Let A nd B e lnguges over Σ = {0, }. A = {w w contins } B = {w w does not contin 000} Give the stte digrm of n NFA tht recognizes A B. 0 0, strt ε q q 2 q 3 0 0, ε r r 2 r 3 r 4

14 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 2 Let A nd B e lnguges over Σ = {0, }. A = {w w contins 00} B = {w w contins even numer of zeros} Give the stte digrm of n NFA tht recognizes A B. 0 0, 0 0 ε 0 q q 2 q 3 q 4 q 5 q 6 0 A B

15 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 3 Let Σ = {, }. A lnguge is descried y the regulr expression: (Σ Σ ) ) Give string tht is in this lnguge. ) Give string tht is NOT in this lnguge. c) Give n NFA tht recognizes this lnguge. ) or or or ) or or c),, ε ε

16 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 3 Let Σ = {0, }. A lnguge is descried y the regulr expression: (ε 000 )(0) ) Give string tht is in this lnguge. ) Give string tht is NOT in this lnguge. c) Give n NFA tht recognizes this lnguge. ) 000 or or 0000 or 000 ) or 0000 or 0 or. c) 0 ε 0 0 0

17 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 4 Give CFG generting the following lnguge over Σ = {0, }: {w w is of even length nd strts nd ends with the sme symol.} S 0A0 A ε A 00A 0A 0A A ε

18 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 4 Give CFG generting the following lnguge over Σ = {0, }: {w w is of odd length nd contins t lest two 0 s.} S A0A0A A0B0B B0A0B B0B0A A 0B B B ε 00B 0B 0B B (Here, A is ny string of odd length nd B is ny string of even length.)

19 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 5 Give PDA (pushdown utomton) tht recognizes the lnguge { n 2n n 0} over Σ = {, }., ε ε, ε $ ε, ε ε ε, $ ε, ε, ε ε

20 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 5 Give PDA (pushdown utomton) tht recognizes the lnguge { 2n n n 0} over Σ = {, }., ε ε, ε $ ε, ε ε ε, $ ε, ε ε, ε

21 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 6 Let A = { n! n 0}. Use pumping lemm to show tht A is not context-free lnguge. Let the pumping length e p. Choose the string s s = p!. Now divide into five prts s s = uvxyz. Any choice for v nd y consist of s only. We know tht therefore Clerly, therefore vxy p uv 2 xy 2 z p! + p p! + p < (p + )! uv 2 xy 2 z (p+)! uv 2 xy 2 z / A So we cnnot pump this string.

22 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 6 Let B = { n2 n 0}. Use pumping lemm to show tht B is not context-free lnguge. Let the pumping length e p. Choose the string s s = p2. Now divide into five prts s s = uvxyz. Any choice for v nd y consist of s only. We know tht therefore Clerly, therefore vxy p uv 2 xy 2 z p 2 + p p 2 + p < (p + ) 2 uv 2 xy 2 z (p+)2 uv 2 xy 2 z / A So we cnnot pump this string.

23 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 7 Give description of Turing mchine M tht, given n input from {0,, #}, elimintes ll # s. For exmple, given input 0#0#0, the tpe should contin 000 when M hlts.. Move right until meeting the first #. If there is no #, ccept. 2. Move right, red symol, move left, write symol, move right. (This moves one symol one unit left, the symol could e 0,, # or lnk). 3. If symol is lnk, go to 4. Else, go to 2. (This moves ll symols to the right of # one unit left). 4. Move the hed to the strt nd go to.

24 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 7 Give description of Turing mchine M tht, given n input from {}, doules the numer of symols. For exmple, given input, the tpe should contin when M hlts.. If the first symol is lnk, ccept. If it is, put mrk on it nd move right. Put mrk on ll s until meeting lnk. 2. Move left until meeting the first (mrked ). Erse the mrk. If there s no mrked, ccept. 3. Move right until meeting lnk. Replce lnk with. 4. Go to 2., R L R, R, R, L q q 2 q 3 q 4, R, L R q ccept (Here, denotes the very first )

25 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 8 Let E DF A = { A A is DFA nd L(A) = }. In other words, A is DFA tht does not ccept ny string. Show tht E DF A is decidle lnguge. The following Turing mchine decides this lnguge. (Try to uild pth from ccept stte ckwrds to strt stte) On input A where A is DFA:. Mrk the ccept sttes of A. If there is no ccept stte, ACCEPT. 2. Repet until no new sttes get mrked: 3. If stte hs n rrow pointing to mrked stte, mrk it. 4. If strt stte is mrked, REJECT; otherwise, ACCEPT.

26 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 8 Let E CF G = { G G is CFG nd L(G) = }. In other words, G is CFG tht does not generte ny string. Show tht E CF G is decidle lnguge. The following Turing mchine decides this lnguge. (Try to uild pth from terminls ckwrds to S) On input G, where G is CFG:. Use Chomsky form. Mrk ll terminl symols. If there is no terminl, ACCEPT. 2. Repet until no new vriles get mrked: 3. Mrk ny vrile A if there is rule A, OR if there is rule A BC nd oth B nd C re mrked. 4. If S is not mrked, ACCEPT; otherwise, REJECT.

27 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 9 The VERTEX COVER prolem is defined s: Given grph nd n integer k, is there collection of k vertices such tht ech edge is connected to one of the vertices in the collection? Is VERTEX COVER in NP? Given solution, we cn verify it in polynomil time y the following method: INPUT A totl of n vertices, list of k vertices (cover), list of l edges. For i = to k For j = to l If vertex i is connected to edge j Mrk edge j EndIf EndFor EndFor For j = to l If edge j is unmrked Return REJECT EndIf EndFor Return ACCEPT We know tht k n nd l n(n )/2 so this lgorithm tkes O(n 3 ) + O(n 2 ) = O(n 3 ) time, i.e. polynomil time. Therefore VERTEX COVER is in NP. Note: We ssume connection etween vertices cn e checked in O() opertion.

28 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 9 The COLORING prolem is defined s: Given grph nd n integer k, is there wy to color the vertices with k colors such tht djcent vertices re colored differently? Is COLORING in NP? Given solution, we cn verify it in polynomil time y the following method: INPUT A list of n vertices with ech one hving one of k colors, list of l edges. For i = to n For j = i + to n If vertex i is connected to vertex j AND Color(i)==Color(j) Return REJECT EndIf EndFor EndFor Return ACCEPT This lgorithm tkes O(n 2 ) time, i.e. polynomil time. Therefore COLORING is in NP. Note: We ssume connection etween vertices cn e checked in O() opertion.

29 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 0 INDEPENDENT SET prolem: Given grph G nd n integer k, is there suset S of k vertices such tht no two vertices in S re connected? (All vertices re independent) VERTEX COVER prolem: Given grph G nd n integer k, is there suset S of k vertices such tht every edge hs t lest one endpoint in S? Show tht INDEPENDENT SET P VERTEX COVER Suppose we hve n lgorithm tht decides VERTEX COVER in polynomil time. INPUT A grph G with n vertices, n integer k. VERTEXCOVER(G, n k) // If reject, reject, if ccept, ccept. (Note tht, if there is vertex cover of n k elements, the remining k elements form n independent set. )

30 Çnky University Deprtment of Computer Engineering Nme-Surnme: CLASSWORK 0 PARTITION Prolem: Given finite set of integers, determine if it cn e prtitioned into two sets such tht the sum of ll integers in the first set equls the sum of ll integers in the second set. SUBSET SUM Prolem: Given set of integers nd numer k, determine if there is suset of these integers whose sum is k. Show tht PARTITION P SUBSET SUM Suppose we hve n lgorithm tht decides SUBSET SUM in polynomil time. INPUT A set S of integers, n integer k. Add ll elements of S. Cll the result TOTAL. If TOTAL is odd Return REJECT else SUBSETSUM(S, T OT AL/2) // If reject, reject, if ccept, ccept. EndIf Prepred y: Dr. Emre Sermutlu (204)

First Midterm Examination

First Midterm Examination 24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet

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