COT4210 Final Review

Consider the following language L = {a^n b^n c^n d^n | n >= 1} L is: a) CFL but not regular b) CSL but not CFL c) Regular d) Type 0 language but not type 1 e) None of the above

CSL but not CFL

if L = Σ* is undecidable if a) L is context free but not regular b) L is regular c) It does not matter the type of L d) All of the above

L is context free but not regular

Hilbert's Tenth asking for an algorithm to find the integral roots of polynomials with integral coefficients, is a) Decidable b) Undecidable c) Semi-decidable d) Not a computation problem e) Does not exist

Semi-decidable

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) a leftmost derivation b) a rightmost derivation c) both leftmost and rightmost derivation d) neither leftmost nor rightmost derivation e) None of the above

d, neither leftmost nor rightmost derivation

Which is NOT the correct statement(s)? (i) Every context sensitive language is recursive. (ii) There is a recursive language that is not context sensitive a) (i) is true, (ii) is flase b) (i) is true and (ii) is true c) (i) is false, (ii) is false d) (i) is false and (ii) is true e) None of the above

(i) is true and (ii) is true

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, then is ~L also regular? 4) If L is recursive language, then, is ~:=L also recursive? a) 1,2,3,4 b) 1,2 c) 2,3,4 d) 3,4 e) None of the above

3,4

Function x * y defined as x * 0 = 0 x** (y + 1 ) x ** y + x is an example of a) A base function b) A primitive recursive function c) A μ-recursive function d) An undefined funtion e) Not a function

A primitive recursive function

Recursive languages are a) A proper superset of CFL b) Always recognized by PDA c) Are also called type 0 languages d) Always recognized by FSA e) None of the above

A proper subset of CFL

Which of the following statement(s) is/are correct? A. L= {a^n b^n a^n | n = 1, 2, 3...} is recursively enumerable B. Recursive languages are closed under union C. Every recursive language is closed under union D. None of these E. A,B, and C

A,B, and C

Which of the following is not primitive recursive but partially recursive? a) Carnot function b) Rierman function c) Bounded function d) Ackermann's function e) None of the above

Ackermann's function

Which of the following statements is wrong? a) Any regular language can be generated by a context-free grammar b) Some non-regular languages cannot be generated by any CFG c) The intersection of a CFL and 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

The following languages are re excepta) a)For Type 0, emptiness and even membership problems b) Membership in L1/L2, L1 and L2 CFLs c) L regular, for CFL (CSL), L d) ˜L CFL, for CFL, L? e) All of the above

All of the above

Assume S1 and S2 are defined as: S1: L2-L1 is recursive enumerable where L1 and L2 are recursive and recursive enumerable respectively S2: The set of all turing machines is countable. Which of the following is true? a) S1 is correct and s2 is incorrect b) Both S1 and S2 are correct c) Both S1 and S2 are not correct d) S1 is not correct and S2 is correct e) None of the above

Both S1 and S2 are correct

Recursively enumerable languages are not closed under a) Union b) Homomorphism c) Complementaton d) Concatenation e) None of the above

Complementaton

Which of the following is the most general phase structured grammar? a) Regular b) Context-sensitive c) Context free d) None of the above e) None of the above

Context-sensitive

Sets N (natural numbers) and R (real numbers) are respectively: a) Both countably finite b) Both countable infinite c) Countably infinite and uncountably infinite d) Both uncountably infinite e) Have the same cardinality

Countably infinite and uncountably infinite

Which of the following problems is solvable? A. Writing a universal Turing machine B. Determining if an arbitrary Turing machine is a universal Turing machine C. Determining if a universal Turing machine can be written for fewer than k instructions for some k D. Determining if a universal Turing machine and some input will halt E. None of the above

Determining if a universal Turing machine can be written for fewer than k instructions for some k

Bounded minimization is a technique for A. Proving whether a primitive recursive function is Turing computable or not B. Proving whether a primitive recursive function is a total function or not C. Generating primitive recursive functions D. Generating partial recursive functions

Generating primitive recursive functions

Which of the following statements is false? A. Halting problem of Turing machines is undecidable B. Determining whether a context-free grammar is ambiguous is undecidable C. Given two arbitrary context-free grammars G1 G2 and it is undecidable whether L(G1) = L(G2) D. Given two regular grammars G1 G2 and it is undecidable whether L(G1) = L(G2) E. All of the above

Given two regular grammars G1 G2 and it is undecidable whether L(G1) = L(G2)

The following CFG is in S -> aBB B -> bAA | b A -> a a) Chomsky normal form but not strong Chomsky normal form b) Weak Chomsky normal form but not chomsky normal form c) Strong Chomsky normal form d) Greibach normal form e) None of the above

Greibach normal form

The Halting PRoblem -- Given an arbitrary program P, in some Language L, and an input x to P, will P eventually stop when run with input x? - can be defined as a) Halt (P,x) = 1 if φp(x) is defined 0 if φp(x) is not defined b) Halt (P,x) = 1 if φp(x) is not defined = 0 if φp(x) is defined c) Halt (P,x) = terminates if φp(x) is defined = runs forever if φp(x) is not defined d) None of the correct form e) Cannot be formulated using μ-recursive function

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 following statements is true? a) I only b) I and II c) I and III d) II and III e) None of the above

I and II

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 above statements are true? a) I only b) I and II c) II and III d) I and III e) None of the above

I and II

Which of the following problems are undecidable? a) 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

Hilbert's 10th problem was to devise an algorithm to determine the roots of a polynomial. Is this problem decidable? Why, or Why not?

No, while an algorithm can probably determine some of the roots for some problems, it would be impossible to test for all roots for all problems

A total recursive function is A) partial recursive function B) primitive recursive function C) both a and b D) none of the above

None of the above

Which of the following denotes Chomskian hierarchy? a) REG -> CFL -> CSL -> type0 b) CFL -> REG -> Type0 -> CSL c) CSL -> type0 -> REG -> CSL d) CSL -> CFL -> REG -> type0 e) None of the above

REG -> CFL -> CSL -> type0

Which of the following is complement of a A. Recursive language is recursive B. Recursively enumerable language is recursively enumerable C. Recursive language is either recursive of recursively enumerable D. a. and b. E. None of the above

Recursive language is recursive

If there exists a language L, for which there exists a TM, T, that accepts every word in L and either rejects or loops for every work that is not in L, L is called A. Recursive B. Recursively enumerable C. NP-HARD D. None of these

Recursively enumerable

Suppose S != {} then the following are equivalent except: a) S is re b) S is the range of primitive rec. function c) S is the same class of language as TOTAL d) S is the domain of a partial rec. function e) S is the range/domain of a partial rec. function whose domain is the same as its range and which acts as an identity when it converges

S is the same class of language as TOTAL

The following CFG is in S -> AB B -> CD | AD | b D -> AD | d A -> a C -> a a) Chomsky normal form but not strong Chomsky normal form b) Weak Chomsky normal form but not Chomsky normal form c) Strong Chomsky normal form d) Greibach normal form e) None of the above

Strong Chomsky normal form

The statement, "A TM can't solve the halting problem" is A. True B. False C. Still an open question D. All of these

True

There exists a TM which when applied to any problem in the class, teminates, if correct answer is yes and may or may not terminate otherwise is called A. Stable B. Unsolvable C. Partially solvable D. Unstable

Unsolvable

Hilbert's 10th problem was to devise an algorithm to determine the roots of a polynomial. Is this problem turing recognizable?

Yes, there exists some TM that will be able to reach an accepting state based on input for this problem

If every string of a language can be determined whether it is legal or illegal in finite time the language is called a) decidable b) Undecided c) Interpretive d) Non deterministic e) None of the above

decidable

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 a) type 3 b) is type 2 but not type 3 c) is type 1 but not type 2 d) is type 0 but not type 1 e) None of the above

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 a) is type 3 b) is type 2 but not type 3 c) is type 1 but not type 2 d) is type 0 but not type 1 e) None of the above

is type 2 but not type 3

The running time T(n), where n is input size of a recursive function T(n) = {c + T(n - 1) if n > 1, d if n <= 1} The order of the algorithm is A. n^2 B. n C. n^3 D. n^n E. log n

n

CSGs are not closed under: a) Init b) Final c) Mid d) Quotient with regular sets e) none of the above

none of the above

Consider a language L for which there exists a Turing machine, T, that accepts and either rejects or loop for every word that is not in L. The language L is a) NP hard b) NP complete c) Recursive d) recursively enumerable e) None of the above

recursively enumerable

Which of the following CF languages is inherently ambiguous? a) {a^n b^n c^m d^m | n,m>=1} b) {a^n b^m c^p d^q | n = p or m = q, n, m, p, q >= 1} c) {a^n b^m c^p d^q | n >= m >= p>= q} d) a^n b^m c^p d^q | n <= m <= p <= q} e) All of the above

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

The next move function δ of a Turing machine M = (Q, Σ, Γ...) A. δ : Q X Σ → Q X Γ B. δ : Q X Γ → Q X Σ X {L, R} C. δ : Q X Σ → Q X Γ X {L, R} D. δ : Q X Γ → Q X Γ X {L, R}

δ : Q X Γ → Q X Γ X {L, R}