S = {x | g(r(r(x))) is defined where g(x) is a PC function of one argument}

The problem of deciding whether or not the set S is Recursive is the same as deciding whether or not a. ~S is Recursive b. S is RE c. both. d. neither.

The problem of deciding whether or not g(x) is total is the same as deciding whether or not

a. S is Recursive. b. it is Computable. c. both. d. neither.

The problem of deciding whether or not the set K is many-one reducible to this set S is

the same as deciding whether or not a. S is m-Complete. b. S appears infinitely many times in the “Enumeration of All RE sets.” c. Both. d. neither.

The problem of deciding whether or not this set S is many-one reducible to the set K0 is

the same as deciding whether or not a. S is RE b. S is Recursive c. Both. d. neither.

The problem of deciding whether or not this set S is m-Complete is the same as deciding whether or not a. the set K is many-one reducible to this set S. b. the set K0 is many-one reducible to this set S. c. this set S is many-one reducible to the set K0 . d. all of above e. two of above. f. none of the above. Which is solvable?

a. whether or not Bounded Minimalizations preserve Computable functions (in all cases)

b. whether or not Bounded Minimalizations preserve PR functions c. both. d. neither.

Which preserves Computable functions?

a. Bounded Quantifications b. Proper Minimalizations c. both. d. neither.

Which preserves PC functions?

a. Unbounded Quantifications b. Bounded Minimalizations c. Proper Minimalizations. d. All. e. two.

Which is RE?

a. the set of all programs (with exactly one input) that compute a total function each b. the set K0 c. both. d. neither.

Which is Recursive? a. the set of all PR functions of exactly one argument. b. an RE set whose complement is RE. c. Both. d. Neither. Which is Computable? a. determining whether or not the complement of an RE set is Recursive. b. NSQ(X) = 1 if X is a square of an integer or else 0. (e.g., it is 1 if X=1,4,9,16,25,36, ...) c. both. d. neither. Which is m-Complete?

a. {l(x) | f(x,x) is defined} b. {<x,y> | f(x,y) is undefined} c. both. d. neither.

Next three questions are about the following DTMD with seven quads and the alphabet = {a,b}: N1: Q1 B R Q2 N2: Q2 a R Q3 N3: Q2 b R Q2 N4: Q2 B B Q2 // infinite loop N5: Q3 a R Q3 N6: Q3 b R Q2 N7: Q3 B b Q4 // accepting combination Which language is accepted by this TM? a. {a+ b+ a} b. {a* b* a} c. {(a + b}* a} //a) and b) included.

d. {(a + b)+ a } e. none When defined, which function is computed by this TM? a. f(x) = 2*x+2 b. f(x) = 2*x c. f(x) = x+2 d. none of above. 15. Q3 B a Q4 Suppose that above quadruple replaced the quadruple# N7 above. In this case, the resulting new TM would a. compute the same function as before. b. accept the same language as before. c. Both. d. neither. The next three questions are about the following DFSA with five states: (1, a, 1) (1, b, 2) (2, a, 3) (2, b, 4) (3, a, 2) (3, b, 5) (4, a, 5) (4, b, 4) (5, a, 4) (5, b, 5) where each triple is (originating-state, input-symbol, terminating-state) and state-4 and state-5 are accepting. There is no “Distinguishing” string (which can be the empty-string) between a. state-1 and State-2 b state-2 and state-4

c state-4 and state-5 d all above. e. two above. f. none above. Which two states are equivalent? a. 2 and 3 b. 4 and 5 c. both. d. neither. This DFSA accepts a. a ba* b (b* + aa* b)* b. a* ba* b (b* + aa* b)* c. a* b a* b b* a. none above. Which is a PR function (or predicate)? a. l(s) and <x1, x2> b. a composition of some PR functions. c. STP(x1,x2,y,t) and an unbounded minimalization of a PR predicate d. all above. e. two of the above. Which is solvable? a. HALT(x,x) and STP(x,x,t) //STP(x,x,t) is solvable b. TM State problem c. Member(x,S) for RE sets. d. Deciding if a PC function is total e. two of the above. f. none of the above.

FIRST FOUR OF following questions are about: Y = PHI (Xl, X2, ?1(X3),r(X3)>)

X3 is the number of a program that takes [B] a. one input

b. two inputs c. Neither. d. cannot tell Y is the value of a PC function(Y = PHI (Xl, X2, ?1(X3),r(X3)>)) [C] a. computed by X3 b. computed by ?1(X3),r(X3)> c. Both d. Neither

Y is the value of)(Y = PHI (Xl, X2, ?1(X3),r(X3)>)) [A] a. a PC function by a Universal program. b. a necessarily computable function c. Both. d. Neither

Y can be expressed as(Y = PHI (Xl, X2, ?1(X3),r(X3)>)) [A] a. PHI(Xl, S11(X2,X3)) b. PHI(X1,X2,S11(X3)) c. Both. d. Neither

A program Y in L that takes one input X stops [D] a. if and only if the first component of the current snapshot is n+1 where n isthe number of all instructions of the program. b. if the next instruction to be executed is supposed to have a label L and the program has no instruction with that particular label L. c. STP(X,Y,T) is true for some T d. All above e. Two above f. None

Which is a PR function? (D] a. SNAP(X,Y,T) where T is the number of steps already taken by program Y. b. SUCC(S,Y) where S is the current snapshot of the program Y. c. ?r(X),1(Y)> d. All. e. Two. f. None.

Which is a PR predicate? [B] a. STP(X,Y,T) where T is the number of steps already taken and Y is the current snapshot. b. TERM(S,Y) where S in the current snapshot. c. SKIP(S,Y) where S is the current snapshot and Y the current instruction. d. All. e. Two. f. None.

Which is PC? [D] a. HALT(X,X) b. A function that corresponds to the complement of the set K (defined on page 82). c. Both.

d. Neither.

Bounded minimization preserves [D] a. PR

b. Computability c. Partial Computability. d. All e. Two above f. None

Minimalization preserves [B] a. Computability

b. PC c. Both. d. Neither. Which is Recursive? [D] a. The complement of a Recursive set.

b. A finite RE set. c. The intersection of two Recursive sets. d. All. e. Two. f. None.

Which is NOT Recursive? (B] a. Some finite RE sets. b. An RE set whose complement is not RE. c. Both. d. Neither. For every non-empty RE set S, there exists (C] a. a PR function whose range is S b. a computable function whose range is S. c. Both. d. Neither. For every RE set S, not necessarily non-empty, there exists (D] a. a PC function of one argument whose range is S. b. a PC function f(X) such that f(X) is defined if and only if X is in S. c. An NTMD that accepts S. d. All e. Two. f. None above. Which is RE? [C] a. {?Y,X> I PHI(X,Y) is defined) b. {?X,Y> 1 PHI(X,Y) is defined) c. Both. d. Neither. 16. Which is unsolvable? [A] a. whether or not a program computes a total function. b. whether or not Ul computes a computable function. c. whether or not a program computes a PC function. d. All above. e. Two above. f. None

17. Which is unsolvable? (E] a. whether or not a DTMD accepts an input string. b. whether or not an NTMD reaches an arbitrary state of its own. c. whether or not an NTMD accepts an RE set of strings.

d. All above. e. Two above. f. None Which is PR? [E] a. determining whether an arbitrary quadruple of a DTMD is never applied or not. b. SQ(X) s 1 if X is a square of an integer or else O. (e.g., it is 1 if X.4,9,16,25,36, etc.) c. Determining whether or not a DTMD has an equivalent NTMN. d. All above. e. Two above. f. None All one-way infinite DTMD's All one-way infinite NTMD's All two-way infinite NTMD's All two-way infinite DTMD's a. Above are all equivalent. b. Three above are equivalent. c. Two above are equivalent. d. None above. 20. Which is m-Complete? (A] a. {x I PHI(x,x) is defined)

b. {x I PHI(x,y) is defined for a given y) c. {x I PHI(x,x) is undefined) d. {?x,y> ( PHI(x,y) is undefined) e. All. f. Three.

First three questions are about the following set: Set S is RE. S = {x <f(x),g(x)> is defined where g(x) and f(x) each is a PC function.} The problem of deciding whether or not the set S is Recursive is the same as deciding whether or not a.

both functions, f(x) and g(x), are Computable b the Complement of this set S is Recursive c. both. d. neither. S = {x <f(x),g(x)> is defined where g(x) and f(x) each is a PC function.} The problem of deciding whether or not f(x) is Computable is the same as deciding whether or not a. S is RE. b.it is total. c. both. d. neither.

,

S = {x <f(x),g(x)> is defined where g(x) and f(x) each is a PC function.} The problem of deciding whether or not the set K is many-one reducible to this set S is the same as deciding whether or not a. S is RE. b. S is m-Complete c. Both. d. neither. Which is solvable? a. whether or not Unbounded Minimalizations preserve Computable functions (in all cases) b. whether or not Unbounded Minimalizations preserve PR functions c. both. d. neither.

Which is solvable? a.deciding whether or not a PR function is total. b. deciding whether or not a PC function is total. c. both. d. neither. Which preserves Computable functions? a. Bounded Quantifications b. Bounded Minimalizations c.Both d. neither.

Which preserves PC functions? a.Unbounded Quantifications b. Bounded Minimalizations c.Unbounded Minimalizations. d.All e. two. 8. Which is Recursive? a. the set of all programs (with exactly one input) that compute a total function each b. the set Ko c. the set of all programs(with exatly one input) that compute a PC function each. d. All

e.two f.None Which is Solvable? a. Determining whether or not the complement of an arbitrary Recursive set is Recursive. b. determining whether or not an arbitrary RE set is Recursive. c. membership question for RE sets.(It is unsolvable and membership questions for recursive set is solvable) d. all above. e. two above. f. none above. Which is m-Complete? a. { l(x) Phi(x,x) is defined} b. { x|phi(x,y) is defined} c.both. d. neither.

Next three questions are about the following DTMD with 13 quads and the alphabet = {a,b}:

N1: Q1 B R Q2

N2: Q2 a R Q3

N3: Q3 a a Q3 II infinite loop

N4: Q3 b b Q4

N5: Q4 b b Q4 // infinite loop

N6: Q4 a R Q5

N7: Q5 B b Q6 // accepting state

N8: Q5 a a Q5 // infinite loop

N9: Q5 b a Q5

N10:

Q4 B B Q4 II infinite loop

N11:

Q3 B B Q3 II infinite loop

N12:

Q2 b b Q2

N13:

Q2 B B Q2

11. Which language is accepted by this TM? a {a+ b+ a} b {a* b÷ a} c {a+ b+ a b} d. none When defined, which function is computed by this TM?

a. f(x) = 2*x+2 b. f(x) = 2*x c. f(x) = x+2 d. none.

Q5 b b Q5 Suppose that above quadruple replaced the quadruple # N9 above. In this case, the resulting new TM would a. compute the same function as before. b. accept the same language as before. c. Both. d. neither.

14 Which preserves Computable functions? a. Bounded Quantifications b. Composition and Primitive Recursion c. Proper Minimalizations. d All. e. two.

15. Which pair can be unifiable? a Pl(V1, F1(V2)) Pl(V3, F1(C1)) b. P1(F1(V1), V1) P1(F1(F2(V3)),V3) c. both. d. neither

First five questions are about the following set:

S = {x| g (<r(x), 1(x)>) is defined where g(x) is a PC function of one argument}

1. The problem of deciding whether or not the set S is RE is the same as deciding whether or not a. —S is RE b. ~S is Recursive c. g(x) is Computable d. all above e. two above. f none above.

S = {x| g (<r(x), 1(x)>) is defined where g(x) is a PC function of one argument} The problem of deciding whether or not the set S is many-one reducible to the set K is the same as deciding whether or not a. the set S is m-Complete b.the set S is many-one reducible to an arbitrary m-Complete set. c. Both. d. neither.

S = {x| g (<r(x), 1(x)>) is defined where g(x) is a PC function of one argument} 3.The problem of deciding whether or not the set S is Recursive is the same as deciding whether or not a. ~S is Recursive b. ~S is RE c.Both d. neither.

S = {x| g (<r(x), 1(x)>) is defined where g(x) is a PC function of one argument} 4. The problem of deciding whether or not g(x) is total is the same as deciding whether or not a. S is Recursive. b. It is Computable. c. both. d. neither.

S = {x| g (<r(x), 1(x)>) is defined where g(x) is a PC function of one argument} 5. The problem of deciding whether or not the set K is many-one reducible to this set S is the same as deciding whether or not

a. S is m-Complete. b. S appears infinitely many times in the "Enumeration of All RE sets." c. Both. d. neither. 6. Which is solvable? a. deciding whether or not Bounded Minimalizations preserve Computable functions

b. deciding whether or not Unbounded Minimalizations preserve PR functions c. both. d. neither. 7. Which is solvable? a. the membership question of the set K. b. the membership question of the set of all PR functions of one argument. c. the membership question of the set of all Computable functions of one argument. d. all above. e. two above. f. none above.

8. Which is solvable? a. Turing Machine Halting Problem. b. TM State Problem. c. Deciding whether or not there exists a quintuple one-way infinite TM equivalent to an arbitrary quadruple two-way infinite tape TM. d.all above. e. two above. f. none above. 9.Which is Computable? a. any function that can be obtained by Compositions and Proper minimalizations on the three basic PR functions (Zero-functions, Successor-functions and Projection-functions) b. deciding whether or not an integer is a multiple of another. c. STP (x,y,t) d. all above. e. two above. f. none above.

10.

Which is minimally needed to compute the next snapshot of a program? a. the current snapshot b.the current snapshot and the program (under execution) c. the current snapshot, the program (under execution) and the statement currently being executed. d. none above.

11. Which are closed under Union and also under Complementation? a. the family of all RE sets of single integers. b. the family of all Recursive sets of single integers. c. any family of sets that is closed under Intersection and also under Complementation.

d. all above. e Two above. f. none above. 12. Which preserves Computable functions? a. Iterated Operations and Bounded Quantifications b. Proper Minimalizations c. Both. d. neither.

13. Which preserves PC functions? a. Iterated Operations and Unbounded Quantifications b. Unbounded Minimalizations c. Proper Minimalizations. d All e. two. 14. Which is RE? a. the set of all programs (with exactly one input) that compute a total function each b the set K and the set Ko c. both. d. neither. 15. Which is Recursive? a. the set of all Computable functions of exactly one argument. b RE set whose complement is RE. c. Both. d. Neither.

16.Which is Computable? a. determining whether or not an arbitrary RE set is Recursive. b. SQ(X) = 0 if X is a square of an integer or else 1. (e.g., it is 0 if X=1, 4, 9, 16, 25, 36, ...) c. both. d. neither. 17. Which is m-Complete? a. (1(x) I (1)(x,x) is defined} b. {<y, x> I (I)(x,y) is defined} d. neither. c. Both d.Neither

Next four questions are about the following DTMD with seven quads and the alphabet = {a,b}: N1: Q1 B R Q2 N2: Q2 a R Q3

N3: Q2 b R Q2 N4: Q2 B 13 Q2 // infinite loop N5: Q3 a R Q3

N6: Q3 b R Q2 N7: Q3 B b Q4 // accepting combination 18. Which language is accepted by this TM?(following DTMD with seven quads and the alphabet = {a,b}) a. {a+ b+ a} b. {a* b* a} ((a + b)* a) c. {(a + b)* a } d. {(a + b)+ a } e. none 19. The function computed by this TM(following DTMD with seven quads and the alphabet = {a,b}) a. is Computable. b. turns the input "abb" into the output "abbb" c. both. d. neither. 20. Q3 B a Q4 (following DTMD with seven quads and the alphabet = {a,b}) Suppose that above quadruple replaced the quadruple# N7 above. In this case, the resulting new TM would a. compute the same function as before. b. accept the same language as before. c Both. d. neither. 21. This TM computes(following DTMD with seven quads and the alphabet = {a,b}) a. a PC function strictly. b. a Computable function non-strictly. c. a PC function non-strictly. d. All above. e. two above. f. none above.

22 Which is Solvable? a. determining whether or not an arbitrary TM accepts an arbitrary input string. b. determining whether or not an arbitrary TM is truly deterministic. c. determining whether or not the family of all TM's and the family of all quintuple TM's with designated final states are equivalent. d. all above. above. e.two of above f. none above.

First three questions are about the following instruction: Y = phi(Xl, X2, X3 ,<l(X4),r(X4)>) 1. Y is the value of a PC function

a. computed by X4 b. computed by <r(X3),1(X3)> c. Botlh d. Neither Y = phi(Xl, X2, X3 ,<l(X4),r(X4)>) 2. Y is

a. is the value of a PC function computed by a Universal program. b. is a Partially Computable function c. Both d Neither Y = phi(Xl, X2, X3 ,<l(X4),r(X4)>) 3. Y can be expretssed as

a. phi (X 1, S 11(X2,X3,X4))

b. phi (X 1, X2,S22(X3,X4)) c. Both. d. Neither

Next four questions are about the following RE set:

S = {x I g(<r(x),l(x)>) is defined where g(x) is a PC function of one argument} //doubt The problem of deciding whether or not S is Recursive is the same as deciding whether or not

a. ~S is Recursive b. ~S is RE c.Both d. neither.

S = {x I g(<r(x),l(x)>) is defined where g(x) is a PC function of one argument} The problem of deciding whether or not g(x) is total is the same as deciding whether or not

a. —S is Recursive.

b. —S is the Universe set. c. both. d . neither

S = {x I g(<r(x),l(x)>) is defined where g(x) is a PC function of one argument} The problem of deciding whether or not the set K (defined on Page-82) is many-one reducible to this set S is the same as deciding whether or not a. ~S is many-one reducible to K. b. S appears exactly once in the "Enumeration of All RE sets." c. Both. d.neither.

S = {x I g(<r(x),l(x)>) is defined where g(x) is a PC function of one argument} The problem of deciding whether or not S is m-Complete is the same as deciding whether or not

a. the set K is many-one reducible to S1

b. S is many-one reducible to the set K

c. Both. d. neither.

8. DECR(x,y) is to check a. the type of the instruction specified by the given snapshot x of the program, y, only. c. both the type of the instruction specified by the given snapshot x , regardless of the program,

y, and whether or not the current value of the involved variable is at least one. c. both. d.Neither

9. BRANCH(x,y) is to check a. if the instruction specified by the snapshot x of the program, y, is an "if-goto". b. whether or not the conditional branch specified by the snapshot x of the program y is actually to take place.

c Both. d. Neither.

10. Which is solvable? a. whether or not Unbounded Minimalizations preserve Computable functions (in all cases) b. whether or not Bounded Minimalizations preserve PR functions c.Both d. neither. Which is Solvable?

a. deciding whether or not a PR function is Computable. b. whether or not a PC function is Computable. c. both. d. neither.

12. Which preserves Computable functions? a. Bounded Quantifications b. roper Minimalizations

c. both. d.neither.

3. Which preserves PC functions? a. Bounded Quantifications b. Bounded Minimalizations c. Proper Minimalizations. d.All e. two.

STP(xl,x2,y,t) is the same as a. TERM(SNAP(x1,x2,y,t),y)

b. l(SNAP(xl,x2,y,t)) > Lt(y) c. both. d. neither. 15. Which is RE? a. the set of all programs (with one input) that compute a total function each b. the set —K0 (page-91) c. the complement of the set K d. All e. two. f.none.

16. Which is Recursive? a. complement of a Recursive set.

b. The Union of two RE sets. c. both. d. neither. 17. Which is NOT Recursive? a. the set of all Computable functions of exactly one argument. b. an RE set whose complement is c. Both. either.

18. Which is RE? a. {<Y,X> I 4)(X,Y) is undefined) b. <X,Y> 4(X,Y) is defined) c. Both. d. Neither. 19. Which is PR? a. determining whether or not the complement of an arbitrary RE set is Recursive. b.NSQ(X) = 1 if X is a square of an integer or else 2. (e.g., it is 1 if X=1,4916,25,36 ...)

c.Both d. neither.

20. Which is m-Complete? a. {x I j(x,x,x) is undefined} b (<x,y> |phi(x,x,y) is defined} c. both. d. neither. //doubt 21. The Rice's Theorem states, in essence, a. a sufficient condition for Recursive sets of PC functions. b. a sufficient condition for Non-Recursive sets of PC functions. c. a sufficient condition for Non-recursive sets of Computable functions. d. All. e. two. f. None 22. The Parameter TH M essentially states, given a program P, that there is a systematic way of constructing a new program Q that computes the same function as this program P does except that a Q takes fewer inputs than P b. Q takes more inputs than P. c. both. d. neither. 19. Which WFF’s are equivalent? a. %X%Y(P(X)^Q(Y)) b. %X(P(X)) ^ %Y(Q(Y)) c. %Y(P(Y)^Q(Y)) d. all above. e. two above. f. none above. S = {x | g(<l(x), l(x)>) is defined where g(x) is a Computable function of one argument}

1. The problem of deciding whether or not the set S is Recursive is the same as deciding whether or not a. ~S is Recursive b. S is accepted by a TM. c. both. d. neither.

2. The problem of deciding whether or not g(x) is PR is the same as deciding whether or not

a. S is Recursive. b. It is PC. c. both. d. neither.

3. The problem of deciding whether or not the set ~S is RE is the same as deciding whether or not a. S is RE b. S is many-one reducible to the set K. c. ~S is accepted by a TM. d. all above. e. two above. e. none above.

4. Suppose that g(x) is PC instead. In this case the problem of deciding whether or not this set S is many-one reducible to the set K is the same as deciding whether or not

a. S is m-Complete. b. S is Recursive. c. Both. d. neither.

5. Which is solvable?

a. whether or not Bounded Minimalizations preserve Computable functions (in all cases) b. whether or not Bounded Minimalizations preserve PR functions c. both. d. neither.

6. Which preserves Computable functions?

a. Bounded Quantifications b. UnboundedMinimalizations c. both. d. neither.

7. Which preserves PC functions?

a. Unbounded Quantifications b. Unbounded Minimalizations c. Proper Minimalizations. d. All. e. two.

8. Which is RE?

a. the set of all programs (with exactly one input) that compute a total function each. b. the set K0 c. both. d. neither.

9. Which is Recursive? a. the set of all Computable functions of exactly one argument. b. an RE set whose complement is RE. c. Both. d. Neither. 10. Which is Computable? a. determining whether or not an RE set is Recursive. b. NSQ(X) = 1 if X is a square of an integer or else 0. (e.g., it is 1 if X=1,4,9,16,25,36, ...) c. both. d. neither.

11. Which is m-Complete? a. {l(x) | f(x,x) is defined} b. {<x, y> | f(y,x) is defined} c. both. d. neither.

12. Which is the Complement of the set K0 ?

a. {<x, y> | f(x,y) is undefined} b. {<x, y> | f(y,x) is defined} c. Both. d. neither.

Next three questions are about the following DTMD with seven quads and the alphabet = {a,b}: N1: Q1 B R Q2 N2: Q2 a R Q3 N3: Q2 b R Q2 N4: Q2 B B Q2 // infinite loop N5: Q3 a R Q3 N6: Q3 b R Q2 N7: Q3 B b Q4 // accepting combination 13. Which language is accepted by this TM? a. {a+ b+ a} b. {a* b* a} c. {(a + b}* a} d. {(a+ b)+ a } e. none 14. When defined, which function is computed by this TM? a. f(x) = 2*x+2 b. f(x) = 2*x c. f(x) = x+2 d. none. 15. Q3 B a Q4 Suppose that above quadruple replaced the quadruple# N7 above. In this case, the resulting new TM would a. compute the same function as before. b. accept the same language as before. c. Both. d. neither. 16. Which is Solvable? a. HALT (x,y) b. STP (x,y,t) c. both. d. neither. 17. Which is Solvable? a. determining whether or not a program computes a PR function. b. determining whether or not a PC function is Computable. c. determining whether or not a program computes a total function d. all above. e. two above. f. none above. 18. Which is Solvable?

a. Word Problem and PCP Problem. b. determining whether or not a string is in the language defined by a grammar, possibly type-0. c. determining whether or not a Recursive set is accepted by a TM. d. all above. e. two above. f. none above 19. Which pair can be unifiable? a. P1 (V1, V2, F1(V2)) P1 (F2(V4), V4, F1(V4))

b. P1 (V1, V1) P1 (V2, F1(V3,V2))

c. Both. d. neither 20. Which is Solvable? a. deciding whether or not an RE set is the range of a PC function. b. deciding whether or not all Quadruple TM’s and all Quintuple TM’s are equivalent. c. deciding whether or not a PC function can be computed by a Universal program. d. all above. e. two above. f. none above. PART-B. Other problems (40 points) Do four problems in this PART-B. A. Prove or disprove that there is a set that is not even RE when its complement is Recursive. B. Prove or disprove that each of the following is Computable: a. Factorial(Y,X) = Y*(Y+1)*(Y+2)*(Y+3)*…*X if Y <= X = 1 otherwise. b. GCD (X,Y) = gcd(X,Y) if X>0 and Y>0 = 0 otherwise. C. Consider the following TM with three states, Q1, Q2, and Q3 and six quadrupleson the alphabet {1,2}:

Q1 B R Q1 Q1 1 R Q2 Q2 1 1 Q2 Q2 B B Q3 Q3 B 1 Q3 Q3 2 R Q1 Devise and clearly show another TM whose TM State problem is equivalent to the Halting Problem of this TM. D. Give a TM that accepts the following language: L = {c, ababc, ababababc, ababababababc, ababababababababc, … }

E. Show that the set K0 is many-one reducible to the set K. F. Show that bounded minimalizations preserve Computable functions. G. Consider the following quadruple TM with five instructions on the alphabet A={a,b}: Q1 B a Q2 Q1 b L Q1 Q2 a R Q2 Q2 B b Q3 Q3 B B Q3 Clearly show an equivalent quintuple TM. 1. Which is useless? a. T b. P and Q c. both. d. neither. 2. Which rule is to be eliminated as a result of eliminating useless symbol or symbols? a. S -> a T b. P -> P T c. T -> T Q d. all above. e. two above. f. none above. 3. Which non-terminal is Nullable? a. Q and P b. S c. both. d. neither. 4. Which is an implicit (implied) Unit-Production Rule? a. P -> Q b. S -> Q c. S -> P d. all. e. two. f. none. 5. Which production rule is to be added to this grammar as a result of eliminating empty production-rule? a. S -> b Q b. S -> P a c. Q -> b d. all above. e. two. f. none above. 6. Which production rule is to be added to this grammar as a result of eliminating Unit production rule(s)?

a. S -> Q a Q b S -> P a P c. S -> Q

d. All above. e. two above. f. none above. 7. Which production rule satisfies the CNF but not the GNF? a. T -> T Q b. Q -> b Q c. Both. d. neither. *******************END-OF-CFG-REFERENCE*********************** 8. Which pair is equivalent? a. All DFSA’s and all Regular grammars. b. All PDA’s and all CFL’s. c. All Deterministic LBA’s and all CSL’s d. all. e. two. f. none. 9. Which is solvable? a. deciding whether or not the index of RL defined on a CFL is finite. b. deciding whether or not an arbitrary grammar defines a type-0 language. c. deciding whether or not an arbitrary CFG defines an empty language. d. All. e. Two. f. None. 10. Which is solvable? a. deciding whether or not the union of two DCFL’s is DCF. b. deciding whether or not the right-invariant equivalence relation, R L , defined on an arbitrary regular language is of finite index. c. both. D. neither. 11. Which WFF’s are equivalent? a. $X$Y(P(X)^Q(Y)) b. $X(P(X))^$Y(Q(Y)) c. $Y$X(P(X)^Q(Y)) d. all above. e. two above. f. none above. 12. Which is solvable? a. deciding whether or not the complement of an arbitrary CFL is CS. b. deciding whether or not the complement of an arbitrary DCFL is DCF c. deciding whether not the intersection of an arbitrary CFL and an arbitrary Regular language is CF. d. all above. e. two above. f. none above 13. Which WFF’s are equivalent? a. %X%Y(P(X)+Q(Y)) b. %X(P(X) + Q(Y)) c. %Y%X(Q(X)+P(Y)) d. all above. e. two above. f. none above. 14. Which WFF is valid?

a. $X$Y$Z ((P(X) -> Q(Y))^(Q(Y)->R(Z)) -> (P(X)->R(Z))) b. $X%Y (P(X,Y)) -> %X$Y (P(X,Y)) c. %X$Y (P(X,Y)) -> $Y%X (P(X,Y)) d. all above. e. two above. f. none above. $X$Z%Y ((P(X)+Q(Z)+~S(Y)) -> R(Y)) 15. Which clause is included in the above WFF ? a. ~P(X)+R(f(X,Z)) b. ~Q(Z)+R(f(X,Z)) c. ~P(X)^~Q(Z)^S(Y) + R(Y) d. all. e. two. f. none.

16. The Resolution Principle is to establish a. the validity of negated WFF’s b. the contradiction of original WFF’s without being negated. c. Both. d. neither. 17. Which strategy is complete? a. Input Strategy and Unit Preference Strategy b. AF Form Strategy c. Set of Support Strategy as the Set of Support should contain clauses without which other remaining input clauses together will be satisfiable. d. all above. e. two above. f. none above The next three questions are about the following language: { am bm cm | m>=1 } 18. This language is a. CF b. DCF c. Regular d. two above. e. none above 19. This language can be accepted by a. a DPDA by the final state(s). b. a DPDA by the empty-stack. c. an NPDA by the empty-stack. d. All above e. two above. f. none 20. The index of RL defined on this language is a. finite but greater than the number of non-terminals of a grammar that defines the language. b. infinite as the language is not regular. c. neither. The next two questions are about the following DFSA with five states: (1, a, 1) (1, b, 2) (2, a, 3) (2, b, 4) (3, a, 2)

(3, b, 4) (4, a, 5) (4, b, 4) (5, a, 5) (5, b, 4) where each triple is (originating-state, input-symbol, terminating-state) and only state-4 and is accepting. (state-1 is the initial state, of course) 21. Which pair of two strings is equivalent with respect to RM ? a. “aab” and “ab” b. “aa” and “abaa” c. both. d. neither. 22. This DFSA accepts a. a b a* b (b* + aa* b)* b. a* b a* b (b* + aa* b)* c. a* b a* b b* d. none above. 23. Which is closed under both Intersection and Complementation? a. Type-3 (regular) languages. b. DCFL’s c. CSL’s d. All bove. e. two above. f. none above. Some students are honest and study-hard. 24. Which most adequately describes the above assertion? a. $X (student(X) -> honest(X)^hard(X))

b. %X (student(X) -> honest(X)^hard(X)) c. $X (student(X) ^ honest(X) ^ hard(X)) d. %X (student(X) ^ honest(X) ^ hard(X)) Which is RE ? a the range of PC function b Empty c complement(Empty) d all of above e two or above PART-B. Other problems (28 points) Do three problems. A. { am bn cn | m,n >= 1 } Give a sufficiently long string of the above language and show a possible decomposition of this string that satisfies the CFL Pumping Lemma or the Regular Language Pumping Lemma.

B. Give all clauses for: ~ (%X$Y%Z ((P(X)+R(Z)) -> (S(Y)+Q(Z))) C. Prove each is a CFL. a. { am bm cn | m,n >=1 } b. { am bn cm | m,n >=1 } c. { am bn cm dn | m,n >=1 } D. Give two DCFL’s whose union is not a DCFL. And, prove that it is, in fact, not DCF. E. Give a CFL that is not a DCFL. And, prove that it is, indeed, not DCF. F. Reduce the following PCP system to CFG ambiguity problem. ab ba aba bab aa bbb bb aaa G. Prove the following argument by the Resolution Principle: P1: CFL’s are closed under Union. P2: CFL’s are closed under Complementation C: Therefore they are also closed under Intersection. ______________________________________________________________________ Ashish’s Question set Question 1 Which preserves Computable functions? Bounded Quantifications Question 2 Which is Complement(K)? Selected Answer: { <x,x> | Theta(x,x) is defined } Incorrect Correct neither. Question 3

Consider the following DTM with seven quadruples and an alphabet={a,b}:

N1: Q1 B R Q2

N2: Q2 a R Q3

N3: Q2 b R Q2

N4: Q2 B B Q2

N5: Q3 a R Q3

N6: Q3 b R Q2

N7: Q3 B b Q4 Which language is accepted by this TM? Selected Answer: none of above. Incorrect Correct Answer: b*a(a+bb*a)* Question 4 Consider the following set: S={g(l(x),r(x)) where g(x1,x2) is a PR function of two arguments}. The problem of deciding whether or not this function g(x1,x2) is PC is the same as deciding whether or not Correct Answer: g(x1,x2) is total. Question 5 Consider the following set: S={g(l(x),r(x)) where g(x1,x2) is a PR function of two arguments}. The problem of deciding whether or not this set is Recursive is the same as deciding whether or not Correct Answer: Complement(S) is RE. Question 6 Which is Computable? Correct Answer: r(x)

Question 7 Which is solvable? Correct Answer: neither. Question 8 0 out of 3 points Which is Recursive? Selected Answer: the set of all Computable functions of one argument. Incorrect Correct Answer: the set of all PC functions of one argument. Question 9 Suppose that we have an enumeration of all PR functions of one argument as follows: W0, W1, W2, W3,...,Winfinity. (note that no PR function of one argument is left out here.) And suppose that two functions are defined as follows:

f(x) = Wx(x) + 1 for x=0,1,2,3,4,...,infinity.

g(x) = Wx(x)*Wx(x) + 1 for x=0,1,2,3,4,...,infinity. In this case, which function is PR? Correct Answer: neither. Question 10 Consider the TM of program-3. Which I.D. would lead to a terminal I.D.? Selected Answer: b. BBBB11Qa2BBB Incorrect

Correct Answer: BBBQ1B1221BBB Response Feedback: choice (c) is incorrect. Question 11 Which is the simplest way to show that a set S is m-Complete? Selected Answer: many-one reducing S to the set K. Incorrect Correct Answer: many-one reducing K to the set S. Question 12 Which pair is equivalent? Correct Answer: two of above. Question 13 An I.D. of a TM should contain Correct Answer: all of above. Question 14 Diagonalization technique is the most useful in showing that Correct Answer: a certain object does not belong to a certain set or group in question. Question 15 Consider the following set: S={g(l(x),r(x)) where g(x1,x2)=x1 if x1>x2 or else it is x2}. In this case, Correct Answer:

both. Question 16 Consider the following DTM with seven quadruples and an alphabet={a,b}:

N1: Q1 B R Q2

N2: Q2 a R Q3

N3: Q2 b R Q2

N4: Q2 B B Q2

N5: Q3 a R Q3

N6: Q3 b R Q2

N7: Q3 B b Q4 Suppose that the N4 quadruple above has been deleted altogether. In this case, the resulting new TM would Selected Answer: compute the same function as before. Incorrect Correct Answer: neither. Question 17 Which is RE? Selected Answer: Incorrect the complement of an RE set. Correct Answer: two above. Response Feedback: choice-1 incorrect. Question 18 Which is simpler?

Correct Answer: reducing K to K0 Question 19 Which is RE? Selected Answer: Correct Answer: two of above. Question 20 Which is m-Complete? Correct Answer: both. Question 21 Consider the TM of program-3. Which input causes this TM to select and execute the quad N22 ( Qa 1 1 Qa)? Selected Answer: a. 2212 Incorrect Correct Answer: d. two of above. Response Feedback: choice (d) is incorrect Question 22 Which is PR? Selected Answer: STP(x,y) Incorrect Correct Answer: two above.

Question 23 Which is Computable? Selected Answer: Incorrect two of above. Correct Answer: SNAP(x1, x2, y, t) Question 24 Consider the TM of program-3. Which input is accepted by this TM? Selected Answer: Incorrect a. 1221 Correct Answer: Correct e. (a) and (b) above. Response Feedback: choice (c) is incorrect. Question 25 Consider the following set: S={g(l(x),r(x)) where g(x1,x2) is a Computable function of two arguments}. The problem of deciding whether or not this set S is many-one reducible to the set K is the same as deciding whether or not Correct Answer: S is RE Question 26 Consider the enumeration of all RE sets of single numbers. Which is possible? Correct Answer: two above.

Question 27 Which is RE? Correct Answer: neither. Question 28 Consider the following DTM with seven quadruples and an alphabet={a,b}:

N1: Q1 B R Q2

N2: Q2 a R Q3

N3: Q2 b R Q2

N4: Q2 B B Q2

N5: Q3 a R Q3

N6: Q3 b R Q2

N7: Q3 B b Q4 When defined, which function is computed by this TM? Correct Answer: fx)=2*x+2 Question 29 Which preserves PC functions? Selected Answer: Incorrect Proper Minimalizations. Correct Answer: all above.

Question 30 Consider the TM of program-3. Suppose this TM computes a function instead of accepting a language. In this case, the computed function is Correct Answer: a zero function, f(x)=0. Question 31 Suppose that the alphabet A={a, b, c,d} and that the four symbols are ordered as shown, namely, a<b<c<d, as usual. In this case, which string is equivalent to 500? Correct Answer: acbdd Question 32 Consider the following set: S={g(l(x),r(x)) where g(x1,x2) is a PR function of two arguments}. The problem of deciding whether or not this set is RE is the same as deciding whether or not Correct Answer: g(x1,x2) is Computable. Question 33 Which is solvable? Correct Answer: both. Question 34 An RE set is m-Complete if: Selected Answer: Incorrect K is many-one reducible to S. Correct Answer: two above.

Which preserves PC functions? Selected Answer: Proper Minimalizations. Correct Answer: all above.