1 Exercises about introductory part

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1 1 Exercises out introductory prt 1. Pigeonhole Principle sys: If you hve more pigeons thn pigeonholes, nd ech pigeon ies into some pigeonhole, then there must e t lest one hole tht hs more thn one pigeon. Wht hppens, if you hve s mny pigeonholes s there re nturl numers, nd s mny pigeons, s there re integers? Wht out, if you hve s mny pigeons s there re nturl numers, ut ech pigeon tries to mke nest with every other pigeon into dierent hole? (Only one nest cn e mde into one hole.) 2. There is coming n innite numer of usses, with innite numer of people in ech of them. How would you llocte the guests into rooms of Hilert's Hotel? 3. In the quest ook of Hilert's Hotel there is only nite numer of nmes in ech pge nd new quests must lwys write their nmes into the next empty line. How mny pges there must e in the ook so tht there is room for the new nmes (without reorgnizing the nmes) s long s there is room in the hotel (mye fter reorgnizing)? 4. In the Cntor's plnet everything is mesured with innite precision. Ech inhitnt hs unique nme, which consists of innite string in lphet {, }. Inhitnts... nd... hve decided to rrnge group tour to Hilert's Hotel, nd they hve invited ll inhitnts whose nmes re in lphetic order etween their own nmes. For exmple...'s little sister... nd...'s cousin's son... re invited. Cn you ccommodte everyody into Hilert's Hotel? Justify your nswer crefully! (Hint: Cntor's digonliztion method.) 5. Find the error in the following proof tht 2 = 1. Consider the eqution =. Multiply oth sides y to otin 2 =. Sutrct 2 from oth sides to get 2 2 = 2. Now fctor ech side, ( )( + ) = ( ), nd divide ech side y ( ), to get + =. Finlly, let nd equl 1, which shows tht 2 = Let X e set nd X the size of n = X. Prove y induction tht the size of the powerset of X is P(X) = 2 n. 1

2 7. Wht is wrong in the following induction proof tht ll cts re of the sme colour? Let n e the numer of cts. If n = 1 the clim holds clerly (one ct is lwys of the sme colour). Let's now suppose tht for ny group of n cts the clim holds. Then let's consider group of n + 1 cts. By selecting ny n cts from this group (which cn e done in n + 1 dierent wys) we get y the induction ssumption group in which ll the cts hve the sme colour. So ll n + 1 cts must e of the sme colour. 8. Prove the following clim. If there re n(n 2) people in the prty, then t lest two people hve equl numer of friends in the prty. 9. Prove y contrposition: If c is n odd integer numer, then the eqution n 2 + n c = 0 doesn't hve ny integer solution for n. 10. Construct decision prolem, which cn e used for evluting the diculty of the following computtionl prolem! () Clculte fctoril of n, n!, when n is nturl numer. () Order given word list into lphetic order. (c) Serch from given grph ll cliques i.e. such vertex sets tht there is n edge etween ll vertices in the set. Consider lso, how to code the input of decision prolem s string. Wht is the corresponding forml lnguge like? 11. Invent exmples out non-computtionl prolems! Could you still use computers somehow to help in solving the prolem? 12. Red the story out decision prolems nd complete it! 13. Let's consider the logic school of Kissstn gin. This time the topic is more complicted MIAU-system, which consists of the following rules: xuax xauy xuux xiuy 2

3 x MxM x xui xx x xi xua The tsk is to show tht even n empty string cn crete proper miow (MIAU) y the rules of the system! 14. Let's consider lphet Σ = {m, i, u}. We dene the the powers of lphet in the following wy: Σ 0 = {ɛ} (empty string) Σ k+1 = Σ Σ k = {x Σ nd x Σ k }. E.g. Σ 1 = {m, i, u}, Σ 2 = {mm, mi, mu, im, ii, iu, um, ui, uu}. How mny items (words) does Σ n contin? Wht out the whole lnguge Σ = i=0? 2 Exercises out regulr lnguges 1. By the UNIX commn egrep (extended grep) you cn serch for ptterns in the text, which cn e dened s regulr expressions. The sic syntx of egrep is following: egrep <expression> <file>, in which the expression cn e list of chrcters in the rckets, e.g. [cd]: ny of the chrcters,, c, d (expression1)(expression2): conctention of two expressions (expression1) (expression2): either expression1 or expression2 (expression) : 0 or more times the expression (closure of the expression) \: empty string in the edge of word \B: empty string in the middle of word Notice! The expression might need single quottion mrks ('expression') More informtion y commnd mn egrep. Test the commnd egrep with input le 3

4 Wht kind of ptterns do you nd with the following commnd? egrep '(uu) (st) (iss) (oit) (ll)' esim.c Do you invent simpler commnd, which nds exctly the poem lines? 2. Construct n egrep-commnd to nd the following lines from esim.c: ) Rows, which consist numers. ) Rows, which consist either word while or for c) Rows, which consist numer 10 d) Rows, which consist integers. (Notice! Your commnd should not ccept desiml numers.) 3. Construct n egrep commnd for nding emil nd street ddresses nd telephone numers from html le. Use the homepges of Kissstn Logic School s your input dt ( 4. Let's consider the following lnguges of the lphet Σ = {, }. For ech of the lnguges give two strings tht re memeers nd two strings tht re not memeers! ) ) () c) d) () e) (ɛ ) f) Σ Σ Σ Σ 5. Which strings elong to the lnguge descried y the following expression? (c h m r)t((c t) (s t)o)ught(m l tw r)ice 6. Which strings elong to the lnguge L( )? Wht out L(ɛ )? 7. Find shortest string which elongs to the lnguge descried y the following expression! 4

5 ) ( ) ) ( () ) c) ( )( ) 8. Construct the regulr expressions corresponding the following lnguges: ) {w {, } the third lst chrcter of w is } ) {w {, } w contins either sustring or } c) {w {, } w does not cointin string } 9. Construct the regulr expressions corresponding the following lnguges: ) {w {, } w contins n even numer of chrcters } ) {w {, } the length of w is odd} c) {w {, } numer of chrcters is multiple of 3} 10. Let's hve lphet Σ = {, }. Construct nite utomton, which recognizes the following lnguge: ) L(M) = L( ) ) L(M) = L( ) c) L(M) = L(ɛ) d) L(M) = L(ɛ ) e) L(M) = L(Σ ) 11. Simulte the given nite utomton s gme! Ech student corresponds to one stte, which reds the next input chrcter nd trnsfers the control to the successor stte. Accepting or rejecting stte reports out result. (One point for ech prticipnt!) 5

6 W E Sim Suomenlhti 12. Construct lock utomton for chnnel, which opens nd closes the locks nd increses nd decreses the wter level utomticlly. The Western lock gte cn e opened only when the wter is up nd the Estern gte only when the wter is down. You cn control the wter level only when oth gtes re closed. The utomton gets the following input dt from the control system: WD = wter down WU = wter up SW = ship from West SE = ship from Est GC = gtes closed SL = ship in lock You my suppose tht etween the ships the lock utomton wits one gte open, until the next ship rrives. (Notice! The utomton doesn't hve ny specil initil or nl sttes.) 13. Crete nite utomton tht descries the function of lift moving etween two oors. The lift my e either up or down. On oth oors, there is simple "come here" utton, nd in the lift there is n "up" nd "down" utton. In ddition, the lift hs door tht cn e opened nd closed; the lift only moves if the door is closed. (The utomton does not hve to hve specic fvorle sttes.) Input: pushing the uttons, opening nd closing the door. Sttes: up/down; door open/closed. Nottion: U = up, D = down, o = door open, c = door 6

7 closed. 14. Let Σ = {0, 1}. Construct deterministic nite utomton M, which recognizes the following lnguge ) L(M) = {w the length of w is odd} ) L(M) = {w numer of 1s in w is multiple of three} c) L(M) = {w w hs even numer of 1s nd 0s} 15. Crete minimum utomton tht is equivlent to the following deterministic nite utomton. A B D C 16. Crete minimum utomton tht is equivlent to the following deterministic nite utomton. 7

8 Generting regulr lnguge: "The Poem Automt" Implement progrm tht cretes rows ccording to the regulr expression (the)(attr) SUBJ PRED the (ATTR) OBJ( ATTRIB ɛ) Lnguges ATTR, SUBJ, OBJ, PRED nd ATTRIB consist of the following words, for exmple: ATTR = {ft, lck} SUBJ = {ct, moon, sh} OBJ = {ct, moon, sh} PRED = {shines, wtches, swims} ATTRIB = {in the sky, in the lke} Your progrm cn now produce 'verses' like the following: The lck ct wtches the ft sh in the lke The moon swims in the sky NB! Specify tht not ll predictes my e followed y n oject (e.g. swims). Expnd the exmple s you wish with words tht t into the poem! Add the exclmtions "Wht (ATTR) (SUBJ)!" 8

9 nd the questions "Are you (ATTR) (SUBJ)?" Return the code of your progrm nd exmples of your outputs (poems) to the instructor! Be redy to represent your progrm/poems in the rt exhiition! 18. Let's consider the comic in the lecture mteril out nondeterministic utomton consulting the doctor. Which middle phses does the opertion consist of? Does everything go s it should? 19. Invent t lest one phenomenon in your everydy life, which cn e descried s nondeterministic utomton. Drw the trnsition digrm of your utomton nd try to determinize it! 20. Crete non-deterministic nite utomt tht tests whether lphet {, } contins the chrcter string "" nd mke it deterministic. 21. Trnform the following utomton to deterministic! Give nondeterministic nite utomt to ccept the following lnguges. Try to tke dvntge of nondeterminism s much s possile. ) The set of strings over lphet {0, 1,..., 9} such tht the nl digit hs ppered efore. ) The set of strings over lphet {0, 1,..., 9} such tht the nl digit hs not ppered efore. c) The set of strings over lphet {0, 1} such tht there re two 0's seprted 9

10 y numer of 1's tht is multiple of 4. Note tht 0 is n llowle multiple of Remove ll ɛ-trnsitions from the Golins' Gingerred utomton nd trnsform it to deterministic! (See Wht kind of lnguge does the Golins' Gingerred utomton recognize? Descrie the lnguge s regulr expression! 25. Give the regulr expressions corresponding the following utomt! () () 26. Construct the nite utomt corresponding the following regulr expressions! )() () )(( ) ) ) 27. Show tht the set of regulr lnguges is closed under cut nd ctention! I.e. if L 1 nd L 2 re regulr lnguges, then lso L 1 L2 nd L 1 L 2 re regulr. (Hint: utomt.) 28. Are the following lnguges regulr? Justify your nswer crefully! 10

11 () {w w is chrcter string of length 3 consisting of 's nd 's} () {ww w {, } } (c) {w there re s mny 1's s 0's in w (d) {w there re twice s mny 0's s 1's in w} 29. Wht hppens if you try to prove n ctully regulr lnguge s non-regulr y the Pumping Lemm? Consider for exmple lnguges L 1 =, L 2 = {, }, L 3 = { }. 30. Wht is wrong in the following Pumping lemm proofs? ) Let L = {() i () j i, j 0}. Clim: L is nonregulr. Proof: Let x = () n () k = 2n 2n, x = 4n. Now x cn e divided into prts uvw only in one wy: u = i, v = j, j 1, w = 2n i j 2n. Now uv 0 w = 2n j 2n / L, thus L is non-regulr. ) Let L = {c r k k r 1, k 0} { k l k, l 0}. Clim: L is regulr. Proof: L = L 1 L 2, in which L 1 = {c r k k r 1, k 0} nd L 2 = { k l k, l 0}. For ll x L either x L1 or x L2. Let's consider oth cses: 1) If x L 1, let's select x = c n k k. x = n + 2k > n. x cn e divided into prts uvw only in one wy: u = c i, v = c j, j 1, w = c n i j k k. Now uv k w L for ll k = 0, 1, 2,..., i.e. we cn lwys pump x. 2) If x L 2, let's select x = n l, x = n + l. x cn e divided into prts in only one wy u = i, v = j, j 1, w = n i j l. uv k w L for ll k = 0, 1, 2,... L is regulr. 31. We know tht in generl the prolem REG(L) (i.e. is the given lnguge L regulr or not) is unsolvle. Wht is the reson for tht? Could you still invent progrm, which would help people to prove y the Pumping Lemm tht the given lnguge is nonregulr? 32. The stronger version of the Pumping Lemm gives oth sucient nd necessry conditions for the regulrity of lnguge: 11

12 Pumping Lemm 2: The lnguge A Σ is regulr if nd only if there is constnt n 1 such tht for ll x Σ, if x n then there exists u, v, w such tht x = uvw nd v 1, nd for ll i 0 nd ll yinσ xy A if nd only if uv i wy A. Bsed on this, could you mke progrm, which decides for ny given lnguge A, if it is regulr or not? 33. Prove with the Pumping lemm tht the following lnguges re non-regulr: () { n n c k n, k = 0, 1,...} () { n k c k n, k = 0, 1,...} (c) { n n m m n, k = 0, 1,...} 34. Let's denote y w R the string w written ckwrds (i.e. if w = n, then w R = n ). The string is plindrome if w = w R (e.g. "mdmimdm"). Consider the plindrome lnguge L pl = {ww R w {, } }. Prove with the Pumping Lemm tht L pl is non-regulr! 35. Write progrm, which trnsforms simple HTML le into corresponding ltex le. HTML le <html><ody> text </ody></html> corresponds to ltex le \documentclss[4pper,12pt]{rticle} \egin{document} text \end{document}. Text contins only norml text nd title denitions. Title denitions <h1>title</h1>, <h2>title</h2>, <h3>title</h3> re generted in ltex y \section{title}, \susection{title} nd \sususection{title}. Is n ppliction sed on nite utomt powerful enough for checking, if the HTML le is lso correct? 36. How could you use regulr expressions nd nite utomt in prctice? Give t lest three pplictions! 37. Wht cn you sy out time requirements of prolems, which elong to the clss of regulr lnguges? Wht out their spce requirements? (Hint: Consider the progrm representtion of nite utomton.) Is there ny dierence if the corresponding utomton is deterministic or undeterministic? 12

13 3 Exercises out context-free lnguges 1. Let's hve context-free grmmr G = (V, Σ, P, S), V = {A, B, C, D, S} Σ, in which Σ = {Jim, ig, green, cheese, te}, nd rules P re: A B CA S ADA C ig green B cheese Jim D te Wht kind of strings elong to the lnguge descried y G? Give some exples! Do the following sentences elong to the lnguge L(G): ig ig green cheese, te Jim ig cheese? 2. Construct nite utomton, which recognizes the lnguge descried y the following grmmr. Wht is the corresponding regulr expression? S A B A S A B B ɛ 3. Give right-liner grmmr tht descries the lnguge L = {w {, } w does not contin sustring } 4. All words of the lnguge of outerspce liens follows the Blurs normlform. Blur is Whoozit, followed y one or more Whtzit. Whoozit is letter 'x', which cn e followed y ny numer of letters 'y' (lso zero). Whtzit is letter 'q', which is followed y either letter 'z' or 'd', followed y Whoozit. Give the context-free grmmr, which descries the Blurs lnguge. Cn you now give the corresponding regulr expression? (Hint: construct rst the utomton from the grmmr!) 5. For regulr expression holds the following rule:: If r = rs t, then r = ts, when ɛ / L(s). Show, why this rule holds with the help of context-free grmmrs! 13

14 6. Crete context-free grmmr tht produces progrmming lnguge expressions consisting of n innite numer of 'for' loops inside ech other nd elementry opertions, in the sme wy s in the exmple. for (i = N; i < N; i + +) { for (j=n; j<n; j++) { ; } } in which N is n integer. 7. Descrie the following type of C-progrms s context-free grmmr! the function tkes the form fcttype fctnme(prtype pr (,prtype pr) ) { fcttrunk } in which fcttype my e void or int nd prtype is int. fcttrunk is n ifexpression or ssignment the ssignment tkes the form vrile=vlue; pr nd the vrile re vrile nmes consisting of one letter the vlue is n integer the ifexpression tkes the form if (condition) ifexpression ssignment; or if (condition) ifexpression ssignment; else ifexpression ssignment; or n empty chrcter string the condition tkes the form (vrile=vlue) 8. Drw the following pushdown utomton M: M = ({q 0, q 1, q 2 }, {, }, {A}, δ, q 0, {q 1, q 2 }}, in which δ(q 0,, ɛ) = {(q 0, A)} δ(q 0, ɛ, ɛ) = {(q 1, ɛ)} 14

15 δ(q 0,, A) = {(q 2, ɛ)} δ(q 1, ɛ, A) = {(q 1, ɛ)} δ(q 2,, A) = {(q 2, ɛ)} δ(q 2, ɛ, A) = {(q 2, ɛ)} Wht kind of lnguge does M recognize? Trce ll computtion pths with input strings, nd! Does M ccept strings nd? 9. Write tht w R = the chrcter string w ckwrds (i.e. if w = n, then w R = n ). The chrcter string is plindrome if w = w R (e.g. "mdmimdm"). Consider the lnguge PAL = { w {, } w = w R }. tht is formed y plindromes in the lphet {, }. ) Give context-free grmmr, which produces the lnguge. ) Construct pushdown utomton, which recognizes the lnguge. 10. Crete context-free grmmr tht produces Romn numers tht re less thn 90. The numers re L=50, X=10, V=5, I=1. XII=12, for exmple, nd XIV=14, XL=40, XLIX=49. For thew ske of simplicity, you my ssume tht n empty chrcter string signies 0 nd the chrcter string LXL the numer Construct pushdown utomton, which recognizes the Romn numers tht re less thn 90, s descried in the previous tsk! 12. In the Logic school of Kissstn the current topic is context-free lnguges. In this clss the cts consider the following lnguge: One miow cn egin with one or more miu nd in the end there is equl numer of mu's. Between them cn e 0 or more u's. Or the miow is mow, which is followed y one or more u. Mow egins with miu or mu, which is followed y 0 or more u's. ) Give context-free grmmr, which produces the lnguge. ) Give some exmples words of this lnguge. c) Construct pushdown utomton, which recognizes this lnguge. 13. Wht kind of lnguge does the following pushdown utomton M recognize? M = ({q 0, q 1 }, {, }, {A, A 1, B, B 1, A, A 1, B, B 1 }, δ, q 0, {q 0 }) 15

16 δ(q 0,, ɛ) = {(q 1, A 1 )} δ(q 0,, ɛ) = {(q 1, B)} δ(q 1,, ɛ) = {(q 1, A 1 )} δ(q 1,, B) = {(q 1, B 1 )} δ(q 1,, B 1 ) = {(q 1, ɛ)} δ(q 1,, B) = {(q 1, B 1 )} δ(q 1,, A 1 ) = {(q 1, A)} δ(q 1,, A 1 ) = {(q 1, A)} δ(q 1,, ɛ) = {(q 1, B)} δ(q 1,, A) = {(q 1, ɛ)} δ(q 1,, A 1 ) = {(q 1, B 1 )} δ(q 1,, A 1 ) = {(q 1, B 1 )} 14. Show tht the following grmmrs re miguous! Cn you give n unmiguous grmmr which descries the sme lnguge? () S S SS ɛ () S S S ɛ 15. Construct grmmr which produces lnguge { m n c m+n m, n 0}. Is your grmmr miguous or unmiguous? 16. Wht does it mtter if lnguge is inherently miguous? 17. Show tht the following lnguges re context-free!: ){ m n m n} ){ m n c p d q m + n = p + q} c){uw u, w {, }, u = w } d){ m n n m 2n} 18. Let Σ = {m, i, u, }. Let's consider the grmmr: S miun mus SmiuA ɛ N miunmu U U uu ɛ A Au ɛ Give the leftmost nd the rightmost derivtion nd the prsing tree for the string 16

17 miumiuuumumumiumiuu! Is the grmmr unmiguous or miguous? 19. Does the following grmmr stisfy the LL(1) condition? S (L) p q L LndS LorS S Wht kind of lnguge does it decrie? 20. Give (s pseudocode) recursive prser for the lnguge descried y the previous grmmr. (N.B. Trnsform it rst to LL(1) if needed!) 21. Let L e lnguge, the words of which re constructed from ny text nd legl prntheses structures. I.e. s in the prolem 8, ut now there cn e text in the middle of prntheses. E.g. sentence (ig) ct niml (either lion or tiger {which re rre } or ser-toothed tiger {which extincts [red further Kurten ] shme } or leoprd) is n ttrctive (ut dngerous!) friend elongs to the lnguge. You cn suppose for simplisity tht the text prts consist of only lowercse letters..z nd spces. Give LL(1) grmmr, which descries the lnguge, nd design for it recursive prser! 22. Let's consider simple progrmming lnguge, in which ll progrms re of form egin SENTENCE; end. SENTENCE cn e simple ssignment VARI- ABLE:=FACTOR;, VARIABLE:=FACTOR+FACTOR; or VARIABLE:=FACTOR- FACTOR;, in which FACTOR is either vrile,..., z or unsigned integer. Or SENTENCE is while structure: while (CONDITION) do egin SENTENCE end. CONDITION compres two fctors with opertor <,<=,>,>=,==,<>. Construct grmmr for the lnguge nd descrie, how it is prsered recursively! 23. Crete context-free grmmr, which descries the polynoms of vrile x. For the ske of simplicity, you my ssume tht the coecients nd the exponents of the terms re integers of one digit nd tht the rst term is without prex. The terms do not hve to e in specic order, nd there my e temrs of the sme denomintion in the polynom. Give the derivtion trees of the following strings in your grmmr: 2 x 2 2 x + 1, x x x 2 3x, x, 5! Cn you construct recursive prser for the lnguge? 17

18 24. Write progrm which trnsforms HTML list structures into corresponding ltex lists. HTML lists re of form <ul>(<li>text)+</ul> nd <ol>(<li>text) + </ol> nd in ltex the corresponding lists re \egin{itemize}(\item text) + \end{itemize} j \egin{enumerte} (\item text) + \end{enumerte}. Notice tht there cn e severl nested lists! 25. Show tht the following lnguges re deterministic: ) { m n m n} ) {wcw R w {, } } c) { m c m } { m d 2m } 26. Trnsfor the grmmr S (S) A A SS ɛ into Chomsky norml form. Give lso the middle steps (removing ɛ-productions nd unit productions)! 27. Trnsfor the grmmr S ABC A A ɛ B B ɛ C cc c into Chomsky norml form. Give lso the middle steps! 28. Simulte the CYK-lgorithm, when it decides, if the strings, nd elong to the lnguge descried y the grmmr S AS A SA If the nswer is yes, give lso the corresponding prse tree. 29. Progrm the CYK lgorithm for grmmr S AB BC A BA B CC C AB Test your progrm with dierent strings! 18

19 30. How would you prse the following lnguge? It is enough to give the sic strtegy. ) HTML ) Cluses of propositionl logic, which consist of tomic cluses A, B,..., Z, opertions,,,,, nd prntheses. c) C- or Pscl-source code d) SQL-query 31. Let L 1 = { n 2n c m n, m o} nd L 2 = { n m c 2m n, m o}. Is L 1 L2 context-free lnguge? Justify your nswer! 32. Design n ecient lgorithm, which recognizes, if the given nonterminl symol is nullle (i.e. cn it produce ɛ in some derivtion)! 33. Prove tht context-free lnguges re closed under union, conctention nd closure. I.e. if L 1 nd L 2 re context-free, then ) L 1 L2 ) L 1 L 2 c) L 1 re context-free. (Hint: Suppose tht there exists pushdown utomt M(L 1 ) nd M(L 2 ) nd construct pushdown utomt for comined lnguges.) 34. Lnguge { n n c n n 0} cnnot e recognized y common pushdown utomton. Could it e recognized, if you hd two stcks ville? If it could, drw the trnsition digrm of the utomton nd simulte its ehviour! If not, then justify, why not! 35. Prove y the Pumping Lemm for context-free lnguges tht {ww w {, } } is not context-free! 36. Prove y the Pumpinh Lemm for context-free lnguges tht { p p is prime} is not context-free! 4 Exercises out Turing mchines 1. Simulte the ehviour of given Turing mchine s gme. (Your exercise techer will hve some trnsition digrms of dierent size of Turing mchines.) If there re enough students in the group, you cn mke competition 19

20 etween the mchines in recognizing lnguge. Ech student corresponds stte, performs required writing opertion on the tpe, moves the reding hed nd gives the control to the correct successor stte. The ccepting or rejecting nl stte reports the result. (One point for ech prticipnt of the gme!) 2. Wht does the following Turing mchine do? Simulte its ehviour with dierent input strings of s nd s! /A,R /, R /, R A/A,R B/B,R /,L /,L </A,L A/A,L B/B,L A/,R B/,R A/A,R B/B,R /,L /,L >/>,R </<,L /B,R </B,L /,R /,R A/A,R B/B,R </<,L 3. Construct Turing mchine, which reds the input string, until it nds two consecutice s. The input lphet is {, }. 4. Construct stndrd Turing mchine, which sutrcts one from the input inry string. I.e. n integer n is given s inry string x, in which the most signicnt its re left nd lest signicnt right (e.g. 8=1000). If n > 0, the 20

21 mchine replces x y the inry representtion of integer n 1. If n = 0, the tpe remins the sme, nd the mchine moves to the rejecting nl stte. 5. Construct Turing mchine, which divides the input numer represented s inry numer y two, if it is even. With odd numers the mchine trnsfers to the rejecting nl stte. The inry numers re represented ) the most signicnt it on right ) the most signicnt it on left 6. Construct stndrd Turing mchine, which recognizes the lnguge {ww R w {, } }. 7. Construct stndrd Turing mchine, which trnsform the string w into string ww R (w {, } ). 8. Construct stndrd Turing mchine, which recognizes the lnguge {w {, } w contins equl numer of 's nd 's}. 9. Construct stndrd Turing mchine, which lists ll words in lnguge 1 n, n = 0, 1,... The mchine egins with emty tpe, nd genertes unry numers 1, 11, 111, 1111,... (Notice! Your mchine will never hlt.) 10. Construct Turing mchine, which genertes inry representtions of ll nturl numers 0,1,00,01,10,11,000,001,... You cn represent the inry numers s the lest signicnt it on left. 11. Construct 2-tpe Turing mchine, which gets input string w {, } on its 1. tpe, nd writes string w R (=w in reversed order) onto 2. tpe. You cn use specil direction S=sty. 12. Construct 3-tpe Turing mchine, which clcultes it-nd opertion. Binry strings p nd q re given on the rst two tpes nd the mchine clcultes the result p&q onto thrid tpe. You cn ssume tht p nd q re of equl length. 13. Construct 3-tpe Turing mchine for multiplying unry numers. Unry presenttions of numers p nd q re given on the rst two tpes (e.g. p = 5 is coded s 11111) nd the mchine clcultes the unry presenttion of the 21

22 product n = p q onto third tpe. You cn use specil direction S=sty. How would you implement the sme with 1-tpe stndrd mchine? 14. Let's consider the following nondeterministic Turing mchine: M = ({q 0, q 1, q 2, q f }, {0, 1}, {0, 1}, δ, q 0, q f, q no ), whose trnsition digrm is dened s δ(q 0, 0) = {q 0, 1, R), (q 1, 1, R)} δ(q 1, 1) = {q 2, 0, L)} δ(q 2, 1) = {q 0, 1, R)} δ(q 1, <) = {q f, <, R)} Wht does the mchine do? Hint: simulte its ehviour with dierent inry strings. (You cn use JFLAP, if you wnt.) 15. Construct nondeterministic Turing mchine, which recognizes the lnguge {ww w {, } }. 16. Consider the nondeterministic Turing mchine TEST_COMPOSITE (look t the English mteril given to you), which recognizes composite numers. Could you mke prime tester mchine y chnging the ccepting nd rejecting sttes of the mchine? Justify your nswer! 17. Descrie (unformlly) nondeterministic Turing mchine, which recognizes the following lnguge: The words of the lnguge re of form w 1 #w 2 #...#w n for ny n such tht for ll i w i {, } nd for some j w j is the inry representtion of integer j. N.B.! Utilize the nondeterminism s much s possile, i.e. prefer much rnched ut short pths. (The mchine guesses the correct pth nondeterministiclly.) 18. Construct nondeterministic 2-tpe Turing mchine, which solves suset sum prolem. Fctors x 1,..., x n re given on the rst tpe, s unry numers, seprted y chrcter #, nd numer K is given on the second tpe s unry numer. The mchine should check if the rst tpe contins suset sum, which is K. E.g. in 1#1111#11 we cn nd suset sum K = 111. (I.e. we should select non-deterministiclly the fctors, which sum up to K, if such exist.) 22

23 19. Construct stndrd Turing mchine, which egins with empty tpe nd genertes s mny 1s s possile on its tpe efore hlting.(the mchine must hlt nlly!) The mchine cn e composed of t most 3 sttes nd the ccepting nl stte. 20. Give n unrestricted (or context-sensitive) grmmr, which produces lnguge L = { i 2i i i > 0}. Give the derivtion for string in the grmmr! 21. Chomsky thought tht nturl lnguge cn e descried y context-sensitive grmmrs, in which ll productions cn e represented in form: S ɛ or αaβ αωβ, in which A is non-terminl symol nd ω ɛ. Give some exmple of (syntctic) structure in nturl lnguge, which requires such context α_β! (You cn consider exmples in your own ntive lnguge, ut explin the feture lso in English.) Do you elieve tht ll structures of nturl lnguge cn e descried with context-sensitive grmmrs? 5 Exercises out solvility nd unsolvility 1. The Universl Turing mchine hs invited ll Turing mchines of the universum into Universl Truring Symposium, which is hold in Hilert's Hotel. The room reservtions hve een done with the codes c M of the Turing mchines. Is there enough rooms for ll Turing mchines, if there re no other guests in the hotel? How would you llocte the rooms? 2. Drw the Turing mchine descried y the following code! Wht re the codes of the following Turing mchines? () δ(q 0, 0) = (q 0, 0, R) δ(q 0, 1) = (q 0, 1, R) δ(q 0, <) = (q yes, <, L) () δ(q 0, 0) = (q 1, 0, R) δ(q 0, 1) = (q no, 1, R) 23

24 δ(q 0, <) = (q no, <, L) δ(q 1, 0) = (q yes, 0, R) δ(q 1, 1) = (q 0, 1, R) δ(q 1, <) = (q yes, <, L) 4. Write progrm, which gets trnsition function of Turing mchine s its input, nd outputs its code. You cn give the trnsition function in text le in the following form: on the 1. row the numer of sttes n (thus q no = q n ) nd on the following rows the trnsitions δ(q, ) = (q,, ) written s qq. 5. Simulte the ehviour of the universl Turing mchine, when it is given s its input c M w the code of the mchine in -prt of the previous tsk nd string 0101! 6. Construct nite utomton, which recognizes the legl codes of Turing mchines. Wht is the corresponding regulr expression? 7. Construct stndrd Turing mchine, which recognizes the legl codes of Turing mchines. 8. Let the lnguges A nd B e recursive enumerle. (i.e. there exists Turing mchines M A nd M B, which recognize the given lnguges nd hlt in Yescse, ut not necessrily in No-cse.) Prove tht lnguges A B nd A B re lso recursive enumerle i.e. you cn construct for them Turing mchines, which hlt t lest in Yes-cse for ll input strings! 9. Let A e recursive enumerle lnguge. Cn you construct Turing mchine for its complement lnguge A from the mchine M A y chnging the ccepting nd rejecting nl sttes? 10. Prove tht lnguge {c M M hlts on input ɛ} is recursive enumerle ut not recursive! I.e. invent for it Turing mchine, which hlts in Yes-cse, ut not necessrily in No-cse! 11. Give n exmple of Turing mchine, which ) ccepts its own code. ) does not ccept its own code. 24

25 12. Are the following properties of Turing mchines syntctic or semntic? If the property is syntctic, how it cn e solved? ) Fron M's initil stte there is t lest one trnsition with, which does not led to error stte. ) If M is strted with input, it trnsfers to rejecting stte. c) The computtion of M tkes with ny input t most 10 steps. d) M hs t most 10 trnsitions. e) From M's initil stte you cn get into the nl stte without writing nything. 13. Show tht the following semntic properties of recursive enumerle lnguges re nontrivil (trivil property holds for none or for ll lnguges). ) L contins string w. ) L is nite. c) L is regulr. d) L is {0, 1}. 14. Give two exmples of trivil properties: one, which doesn't hold for ny recursive enumerle lnguge nd the other, which holds for ll! 15. Which of the following properties of Turing mchines re solvle? (Hint: either invent n lgorithm ide or consider, if the property is semntic.) ) M hlts on ll even inry numers. ) If M is strted with empty input, it reches stte q on t most 10 steps, or if it is strted with input it reches stte p on t most 20 steps. c) M doesn't contin ny trnsition into stte q, in which it would write end chrcter <. d) Stte q cn e reched from stte p on t most 3 steps. e) M hs less thn 100 sttes nd M hlts on input We know tht for recursive reduction of prolems m holds: i) If A m B nd B is recursive enumerle, then A is recursive enumerle. i) If A m B nd B is recursive, then A is recursive. Is lnguge X recursive, recursive enumerle or totlly unsolvle in the following cses? 25

26 ) For universl lnguge U holds U m X. ) For lnguge H nd for n unknown recursive enumerle lnguge Y holds H m Y nd Y m X. (H = {c M w M hlts on input w}) c) For H's complement H holds H m X. d) For Hmiltonin Cycle -lnguge HC = {x x codes grph G, which contins Hmiltonin cy holds X m HC. e) HC m X N.B.! Rememer the contrposition rule: A B B A. 17. Prove tht the following property is unsolvle: Given code of Turing mchine c M, stte q nd input w. Does M enter y input w into stte q? 18. Are the following lnguges recursive, recursive enumerle or totlly unsolvle? ) L 1 = {c M The code of mchine M c M is plindrom}. ) L 2 = {c M c M is code of mchine M nd M recognizes ll plindroms in lphet {0, 1}}. (Denition of plindroms: Denote w = n, if n = n n , then w is plindrom.) 19. Are the following prolems solvle, prtilly solvle or totlly unsolvle? ) Does the given Turing mchine hlt on ll input? ) Does the given Turing mchine hlt on no input? c) Does the given Turing mchine hlt on t lest one input? c) Does the given Turing mchine fil to hlt on t lest one input? 20. The following prolems re known to e unsolvle, ut re they prtilly solvle or totlly unsolvle? ) Does the lnguge L(M) contin t lest two strings? ) Is L(M) nite? c) Is L(M) context-free lnguge? 1 Hmiltonin cycle=cycle, which goes through ll vertices exctly once. 26

27 21. Invent some concrete exmples of unsolvle prolems! (Other thn hlting prolem.) N.B.! Justify the unsolvility. Is the prolem prtilly solvle (i.e. recursive enumerle lnguge) or totlly unsolvle prolem? 22. Are the following prolems solvle or unsolvle? If they re unsolvle, tell if they re prtilly solvle or totlly unsolvle. If they re solvle, give the wekest type of mchine (nite utomton, pushdown utomton, Turing mchine), which solves the prolem! ) Give the integer nollkoht? for ritrry polynom 1 x n + 2 x n x + 0! ) Progrm for compiler n error checker, which gets s its input code of progrm nd checks tht the progrm doesn't perform division y 0 during it execution. c) Sech if the given text le contins either words ct nd lck or words ird nd white, ut not oth. d) Check tht in the given text le word niml is followed y word wise, only if word ct ppers in the sme sentence. e) Construct generl virus tester, which recognizes ll dngerous progrms i.e. such progrms, which cn chnge system les during their execution. 27