Klausur zur Vorlesung Vertiefung Theoretische Informatik Sommersemester 2016

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1 Prof. Dr. Viorica Sofronie-Stokkermans Dipl.-Inform. Markus Bender AG Formale Methoden und Theoretische Informatik Fachbereich Informatik Universität Koblenz-Landau Klausur zur Vorlesung Vertiefung Theoretische Informatik Sommersemester Name: Vorname: Matrikelnummer: Studiengang: Hinweise Legen Sie Ihren Studentenausweis und ein Ausweisdokument mit Lichtbild (Personalausweis, Pass, Führerschein) auf den Tisch. Schalten Sie Ihr Handy aus. Prüfen Sie Ihr Exemplar der Klausur auf Vollständigkeit (9 Aufgaben). Schreiben Sie mit einem dokumentenechten, schwarzen oder blauen Stift. Es sind keine Hilfsmittel erlaubt. Weitere leere Blätter sind bei der Aufsicht erhältlich. Diese sind mit Namen und Matrikelnummer zu beschriften und müssen abgegeben werden. Heften Sie bei der Abgabe sämtliche Blätter zusammen. Blätter, die versehentlich nicht abgegeben wurden, können nicht nachträglich abgegeben werden. Viel Erfolg! Aufgabe Σ Punkte erreicht von möglichen Note: Unterschrift:

2 Exercise 1 (7P + 6P + 5P + 20P + 3P = 41 Punkte) Remark: In Exercises wrong crosses lead to loss of points. The minimum number of points for each of the exercises is 0 (one cannot carry negative points into the next exercise). /7 P 1.1 Register Machines, Recursive Functions Please indicate (by crossing the right box) whether the following statements are true or false. In the LOOP program loop x 1 do P end the number of times P is executed is not influenced by the fact that P changes x 1. The while instruction can be simulated with a LOOP program. There exist GOTO computable functions which are not µ-recursive. There exist µ-recursive functions which are not primitive recursive. Every µ-recursive function is LOOP computable. There exist total TM-computable functions which are not µ-recursive. Every primitive recursive function is total. 1.2 Complexity Theory Please indicate (by crossing the right box) whether the following statements are true, false, or open (not known in the scientific community). true false true false open Assume that f is a computable function. Then every language from NSPACE(f(n)) is decidable. There are NP-hard problems which are in P. NP = co-np. P = NP. Every NP-complete problem is in NP. /6 P /5 P QBF (the problem of checking whether a quantified Boolean formula is true) is PSPACE-hard. 1.3 Decidability Please indicate (by crossing the right box) whether the following statements are true or false. The Post correspondence problem over an alphabet Σ ( Σ 2) is decidable. The set of all valid formulae in propositional logic is decidable. The set of all valid formulae in first-order logic is not recursively enumerable. It is undecidable whether for a context-free language L over an alphabet Σ with Σ 2 we have L = Σ. It is decidable whether for a context-free language L and a regular language R over an alphabet Σ with Σ 2 we have R L. true false 1

3 1.4 Register Machines, Recursive Functions, Complexity Please indicate (by crossing the right box) whether the following statements are true or false and give a brief justification of your answer (this can for instance be an example or a counterexample). For each correct cross you get 1 point; for the correct justification you get 1 additional point. For every LOOP program there exists an equivalent WHILE-IF program with one while instruction only. true false There exist total functions which are not µ-recursive. There exists a total WHILE computable function which is not primitive recursive. The language {n M n does not halt on the empty input } (where M n is the Turing machine with Gödel number n) is undecidable. There exists at least one WHILE computable function f for which there do not exist primitive recursive functions g, h, such that f = g µh. The function{ f : N N N N defined by: n + m if n 3 = m 2 + k 2 f(n, m, k) = k otherwise is primitive recursive. 2

4 Let g 1, g 2 : N N N be a primitive { recursive function. Then the function ( i<n g1 (i, m) g 2 (m, i) ) if n > 0 f : N N N defined by: f(n, m) = 0 if n = 0 is primitive recursive. true false Let f : N N be defined by f(n) = n 5. Then every language in NTIME(f(n)) is decidable. Let L 1, L 2 be languages with L 1 pol L 2 and L 1 NP. Then L 2 NP. Let L 1 be a language. If there exists a language L 2 which is NP-hard and a language L 3 NP and polynomial reductions L 2 pol L 1 pol L 3 we may conclude that L 1 is NP-complete. /20 P 1.5 Indicate the relationships between the classes LOOP, WHILE part, GOTO part, TM part, P, and F part µ. /3 P /41 P 3

5 LOOP/WHILE/GOTO code In the context of this exam, register machine instructions and register machine programs for the languages LOOP, WHILE and GOTO are defined as follows: LOOP prog ::= LOOP instr ; LOOP prog LOOP instr LOOP instr ::= ass loop reg do LOOP prog end if cond then LOOP prog else LOOP prog end if cond then LOOP prog end WHILE prog ::= WHILE instr ; WHILE prog WHILE instr WHILE instr ::= ass while reg = 0 do WHILE prog end if cond then WHILE prog else WHILE prog end if cond then WHILE prog end GOTO prog ::= nat num : GOTO instr ; GOTO prog nat num : GOTO instr GOTO instr ::= ass goto nat num if reg = 0 goto nat num ass ::= reg := nat num reg := reg reg := nat num op reg reg := reg op nat num reg := reg op reg reg ::= x 1 x 2 x 3... nat num ::= cond ::= reg rel reg reg rel nat num op ::= + rel ::= = You are allowed to define and use auxiliary LOOP/WHILE/GOTO programs. You are not allowed to use any other LOOP/WHILE/GOTO instruction or LOOP/WHILE/GOTO programs. Summary In LOOP programs you can use the following instructions: x i := c P 1 ; P 2 if x i = c then P 1 end if x i = c then P 1 else P 2 end x i := x j loop x i do P 1 end if x i c then P 1 end if x i c then P 1 else P 2 end x i := c op x j if x i = x j then P 1 end if x i = x j then P 1 else P 2 end x i := x j op c if x i x j then P 1 end if x i x j then P 1 else P 2 end x i := x j op x k (where x i, x j are registers, c N, op {+,, }, and P 1, P 2 are LOOP programs) In WHILE and WHILE-IF programs you can use the following instructions: x i := c P 1 ; P 2 if x i = c then P 1 end if x i = c then P 1 else P 2 end x i := x j while x i 0 do P 1 end if x i c then P 1 end if x i c then P 1 else P 2 end x i := c op x j if x i = x j then P 1 end if x i = x j then P 1 else P 2 end x i := x j op c if x i x j then P 1 end if x i x j then P 1 else P 2 end x i := x j op x k (where x i, x j are registers, c N, op {+,, }, and P 1, P 2 are WHILE programs) In GOTO programs you can use the following instructions: x i := c x i := c op x j goto l x i := x j x i := x j op c if x i = 0 goto l x i := x j op x k (where x i, x j are registers, c N, op {+,, }, and l is a label) A GOTO program has the form l 1 : B 1,..., l k : B k (k 1) where B 1,..., B k are GOTO instructions and l 1,..., l k are labels. 4

6 Exercise 2 (4P + 4P = 8 Punkte) Let f : N N N be defined as follows: { 1 if m 3 f(m, n) = m m+n + (m 3) n+m otherwise a) Write a WHILE program which computes f. If it is not possible to write a WHILE program for f say so and justify why it is not possible. /4 P 5

7 b) Write a LOOP program which computes f. If it is not possible to write a LOOP program for f say so and justify why it is not possible. /4 P /8 P 6

8 Exercise 3 (3P + 4P + 3P = 10 Punkte) Let P be the following WHILE program: x 3 := 1; x 4 := 1; x 5 := x 1 1; while x 5 0 do x 5 := x 5 1; x 6 := x 5 ; x 3 := x 3 + x 4 ; x 4 := x 3 ; while x 6 0 do x 6 := x 6 1; x 4 := x 4 + x 3 end end; x 4 := x 2 + x 3 ; while x 4 0 do x 4 := x 4 1 end a) Enter the value that is calculated by P for the given inputs into the table I) II) III) x x Output /3 P b) Determine the function f : N N N computed by P. f(n 1, n 2 ) = /4 P 7

9 c) Use the transformation presented in the lecture to construct a GOTO program which has the same semantics as P. /3 P /10 P 8

10 Recursive functions We use the following notation: is function composition. if j k, π k j is the projection function defined by π k j (n 1,..., n k ) = n j. (+1) : N N is defined by: (+1)(n) = n + 1. { 0 if n = 0 ( 1) : N N is defined by: ( 1)(n) = n 1 otherwise : N N N is defined for all n 1, n 2 N by: (n 1, n 2) = n 1 n 2. + : N N N is defined for all n 1, n 2 N by: +(n 1, n 2) = n 1 + n 2. : N N N is defined for all n 1, n 2 N by: (n 1, n 2) = n 1 n 2 = for all s, k N, c k s : N k N is defined for all n N k by: c k s(n) = s. { 0 if n 1 n 2 n 1 n 2 otherwise In what follows we will assume known that all the functions above are primitive recursive. We also assume known that definitions by case distinction and definitions using the bounded µ operator (as defined in the lecture, in which only primitive functions are used) define primitive recursive functions. Exercise 4 (2P + 2P + 2P = 6 Punkte) Determine the arity of the following recursive functions by drawing a circle around the corresponding number. Give a brief justification for your answer. I) f 1 = PR[c 1 1, + (π3 1, π3 1 )] arity of f 1 :

11 II) f 2 (n) = µ i<12 i (g 2 (n, i) = 0) with g 2 = ( (+ (π1 2, π2 2 )), ( (π2 1, π2 2 ))) arity of g 2 : arity of f 2 : III) f 3 = µg 3 with g 3 = ( (π 3 1, (π3 2, π3 3 ))) ( (π2 1, π2 2 ), (π2 2, π2 1 ), π2 2 ) arity of g 3 : arity of f 3 : /6 P 10

12 Exercise 5 (2P + 3P + 3P = 8 Punkte) Define which function is computed by the following recursive functions. Give a brief justification for your answer. I) f 1 = PR[ (π 2 1, π2 2 ), (+ (c4 1, π4 3 ), (π4 2, π4 4 ))] f 1 (n 1, n 2, m) = II) f 2 : N N N N defined by f 2 (n 1, n 2, m) = µ i<m i (n 1 (i + n 2 ) = 0) f 2 (n 1, n 2, m) = /3 P 11

13 III) f 3 : N N defined by f 3 (n) = µ i ( g(n, i) = 0 ) n 2 2 n if i = 0 where g(n, i) = µj(3 (i + 1) (j + n) = 0) if i = 1 n i if i 2 f 3 (n) = /3 P /8 P 12

14 Exercise 6 (5 Punkte) Let f : N N be defined as follows: 7 if n = 0 f(n) = 3 if n = 1 (2 f(n 1)) (2 + f(n 2)) if n 2 Is f primitive recursive? Justify your answer. /5 P /5 P 13

15 14

16 Exercise 7 (2P + 2P + 2P = 6 Punkte) a) State the theorem of Rice (the formal variant). b) In what follows, L(M n ) is the language accepted by the Turing machine M n. For each of the languages L below answer the following questions: (i) Can the theorem of Rice be used to prove that L is undecidable? Justify your answer. If the theorem of Rice can be used to prove undecidability give the proof. (ii) Is the language decidable? If yes, explain why. I) L = {n L(M n ) = } (i): (ii): 15

17 II) L = {n ɛ L(M n )}. (i): (ii): /6 P 16

18 Exercise 8 (2P + 1P + 1P + 2P = 6 Punkte) a) State the Post correspondence problem. b) Let G = (Σ, R) be a semi-thue system with Σ = {a, b, c} and R = {aa bc, bb ab, bc ab}. 1) Show that aab G abb by writing all the steps and underlining the occurrence of the left hand side of the rule in the current word at every step. /1 P 2) Construct the correspondence system P G,w,w as explained in the lecture, where w = aab and w = abb. Assume that rule 4 is (X, Xw X) and rule 5 is (w XX, X). /1 P 17

19 3) Construct a solution for P G,w,w with start 4 using the derivation aab G abb. /6 P 18

20 Exercise 9 (2P + 2P + 4P + 2P = 10 Punkte) A cook wants to create a new dish. There are n ingredients {v 1,..., v n }, and he would like to use as many as possible. However, some ingredients do not go well with others: there is a symmetric n n matrix D describing the discord between any two ingredients: D(v i, v j ) is a real value between 0 and 1; 0 means that the ingredients v i and v j go perfectly well together and there is no discord and 1 means that they go very badly together (complete discord). Any dish prepared with ingredients I = {v i1,..., v ik } {v 1,..., v n } incurs a penalty which is the sum of the discords between all pairs of ingredients in the dish: penalty D (I) = D(v i, v j ). v i,v j I,i<j We are interested in the following problem: Can we prepare a dish with at least k ingredients with total penalty at most p? Consider the following language: CUISINE = {(k, T, D, p) k is a natural number, T = {v 1,..., v n } is a finite set, D : T 2 [0, 1] with D(v i, v j ) = D(v j, v i ) for all v i, v j T, there exists I T, with I k, such that penalty D (I) p} a) Decide, by ticking the appropriate option, which of the tuples (k, T, D, p) below is an instance of CUISINE? YES NO (2, {v 1, v 2, v 3 }, D, 1) (2, {v 1, v 2, v 3 }, D, 0) (3, {v 1, v 2, v 3 }, D, 2) (3, {v 1, v 2, v 3 }, D, 0) b) Is CUISINE in NP? where in all the cases D is defined by: D(v 1, v 2 ) = D(v 2, v 1 ) = 0, D(v 1, v 3 ) = D(v 3, v 1 ) = 1, D(v 2, v 3 ) = D(v 3, v 2 ) = 0.5, D(v 1, v 1 ) = D(v 2, v 2 ) = D(v 3, v 3 ) = 0. Justify your answer briefly (you do not need to construct a Turing machine). 19

21 c) In the lecture we studied the clique problem: Clique = {(G, k) G = (V, E) is an undirected graph which contains a clique of size k} Let f be the map which associates with every pair (G, k) where G = (V, E) is an undirected graph with V = {v 1,..., v n } the tuple (k, T, D, p) where T = V, p = 0 and: D(v i, v j ) = { 0 if (vi, v j ) E or v i = v j 1 if (v i, v j ) E Prove that f defines a polynomial reduction of Clique to CUISINE. /4 P d) We know that the Clique is an NP-complete problem. Prove or refute the following: CUISINE is an NP-complete problem. /10 P 20

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