🅱 ig Discrete

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Which of the following problems are decidable? 1) Does a given program ever produce an output? 2) If L is context-free language, then, is ~L also context free? 3) If L is regular language, then, is ~L also regular? 4) If L is recursive language, then, is ~L also recursive?

3) If L is regular, is !L also regular? 4) If L is recursive, is !L also recursive?

Function x * y is defined as: x * 0 = 0 x * (y+1) = x*y + x is an example of:

A primitive recursive function

Recursive languages are

A proper superset of CFL

Which of the following is not primitively recursive but partially recursive? a. Carnot function b. Riemann function c. Bounded function d. Ackermann's function All of the above

Ackermann's function

Which of the following statements is wrong? Which of the following statements is wrong? a. Any regular language can be generated by context-free grammar b. Some non-regular languages cannot be generated by any CFG c. the intersection of a CFL and a regular set is a CFL d. All non-regular languages can be generated by CFGs. e. None of the above

All non-regular languages can be generated by CFGs.

CSGs are not closed under:

All of the above

Suppose A <= m B, where A and B denote 2 problems, one of which is known to be (un)decidable Which of the following is correct?

All of the above.

Which is NOT correct? Every context sensitive language is recursive. There is a recursive language that is not context sensitive

Both 1 and 2 are true.

Assume statements S1 and S2 defined as: S1: L2-L1 is recursive enumerable where L1 and L2 are recursive and recursively enumerable respectively. S2: The set of all Turing machines is countable. Which of the following is true? S1 is correct and S2 is not correct. Both S1 and S2 are correct. Both S1 and S2 are not correct. S1 is not correct and S2 is correct. None of the above.

Both S1 and S2 are correct.

Consider the following language: L = {anbncndn | n >= 1}. L is:

CSL but not CFL

Recursively enumerable languages are not closed under:

Complementation

Which of the following is the most general phrase structured grammar?

Context-sensitive

Set N (natural numbers) and R (real numbers) are respectively:

Countably infinite and uncountably infinite

If every language can be determined whether it is legal or illegal in finite time, the language is called:

Decidable

Bounded minimization is a technique for:

Generating primitive recursive functions.

Which of the following statements is false? a. Halting problem of Turing machine is undecidable b. Determining whether a context-free grammar is ambiguous is undecidable c. Given two arbitrary context-free grammar G1,G2, it is undecidable whether L(G1)= L(G2) d. Given two regular grammar G1,G2, it is undecidable whether L(G1)= L(G2) e. All of the above

Given two regular grammar g1, g2, it is undecidable whether L(G1) = L(G2)

The following CFG is in: S→ aBB B→ bAA | b A→ a

Greibach normal form

The Halting problem -- Given an arbitrary program P, in some language L, and an input x to P, Problem: "will P eventually stop when run with input x?" can be defined as:

Halt(P,x) = 1 if ϕp(x) is defined 0 if ϕp(x) is not defined

Consider the following statements: I. Recursive languages are closed under complementation. II. Recursively enumerable languages are closed under union. III. Recursively enumerable languages are closed under complementation. Which of the statements are true

I (Recursive languages are closed under complementation) II (Recursively enumerable languages are closed under union

Consider the following statements I. Recursive languages are closed under complementation II. Recursive enumerable languages are closed under union III. Recursive enumerable languages are closed under complementation. Which of the above are true?

I and II

L = ∑* is undecidable if

L is context free but not regular

Which of the following problems is undecidable? Membership problem for CFL b. Membership problem for regular sets c. Membership problem for CSL d. Membership problem for type 0 languages e. None of the above

Membership problem for type 0 languages.

A Unstable is

None of these

42. The running time T(n), where 'n' is input size of a recursive algorithm, is given as T(n) = { c + T( n - 1), if n > 1 d, if n <= 1 } The order of the algorithm is

O(n)

If there exists a TM which when applied to any problem in the class, terminates if correct answer is yes and may or may not terminate otherwise is called:

Partially solvable

Which of the following denotes Chomskian hierarchy?

REG -> CFL -> CSL -> type0

Which of the following is a complement of a? a. Recursive language is recursive b. Recursively enumerable language is recursively enumerable c. Recursive language is either recursive or recursively enumerable d. (a) and (b) e. None of these

Recursive language is either recursive or recursively enumerable

Consider a language L for which there exists a TUrning machine, T, that accepts every word in L and either rejects or loops for every word that is not in L. The language is:

Recursively enumerable

If there eixsts a language L, for which there is a TM T that accepts every word in L, and either rejects or loops for every word that is not in L, L is called:

Recursively enumerable

Suppose S != {} then the following are equivalent except:

S is the same class of languages as TOTAL

Hilbert's Tenth asking for him to find the integral roots of polynomials with integral coefficients is:

Semi-decidable

The following languages are undecidable except:

They are all undecidable

The statement "A TM can't solve halting problem" is

True

The following CFG is in: S→ AB B→ CD | AD | b D→ AD | d A→ a C→ a

Weak Chomsky normal form but not Chomsky normal form

Which of the following problems is solvable? a. Writing a universal Turing machine b. Determining of an arbitrary Turing machine is a universal Turing machine c. Determining of a universal Turing machine can be written for fewer than k instructions for some k d. Determining of a universal Turing machine and some input will halt e. None of the above

Writing a universal Turing machine

Consider the following CFG S→ aB | bA B→ aBB | bS | b A→ bAA | aS | a Consider the following derivation S→ aB → aaBB → aaBb → aabSb → aabbAb → aabbab This derivation is:

a rightmost derivation

Which of the following is/are correct? a. L = {anbnan | n = 1,2,3 ...} is recursive enumerable b. Recursive languages are closed under union c. Every recursive is closed under union d. None of these e. (a), (b), and (c)

a, b, and c

The Following grammar G = {N, T, P, S}, where N = {S, A, B, C, D, E}, T= {a, b, c} P: S → aAB AB → CD CD →CE C → aC C → b bE → bc is:

is type 1 but not type 2

The following grammar: G = (N, T, P, S), where N = {S, A, B}, T = {a, b, c} P: S → aSa S → aAa A → bB B → bB B → c is:

is type 2 but not type 3

Which of the following CF languages is inherently ambiguous?

{a^n b^m c^p d^q | n=p or m=q, n,m,p,q >= 1}

43. Next move function of a Turing machine M = {Q, Σ, Γ, δ, q0, B, F} is a mapping

δ : Q x Γ -> Q x Γ x {L,R}