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Question 1. Which of the following statements correctly describes the Church‑Turing Thesis? A) Every problem solvable by a physical computer can be solved by a deterministic Turing machine in polynomial time. B) Any effectively calculable function can be computed by a Turing machine. C) Nondeterministic Turing machines are strictly more powerful than deterministic ones. D) The class P equals the class NP. Answer: B Explanation: The Church‑Turing Thesis posits that any function that can be computed by an algorithmic process can be computed by a Turing machine, regardless of the machine’s deterministic or nondeterministic nature. Question 2. The Time Hierarchy Theorem guarantees that: A) P = EXP. B) Giving a Turing machine more time can solve strictly more languages. C) Space and time hierarchies are equivalent. D) All problems in NP can be solved in linear time. Answer: B Explanation: The theorem shows that for any time-constructible function f, there exists a language decidable in O(f(n)·log f(n)) time that is not decidable in o(f(n)) time. Question 3. Which language is known to be complete for the class coNP? A) SAT B) TAUT (the set of tautologies) C) 3‑COLORING D) CLIQUE
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Answer: B Explanation: TAUT is the complement of SAT; since SAT is NP‑complete, TAUT is coNP‑complete. Question 4. In the definition of a nondeterministic Turing machine (NTM), the transition function can be: A) A partial function that maps a state and tape symbol to a unique next state. B) A relation that may map a state and tape symbol to multiple possible next moves. C) A deterministic function that also updates a random bit. D) A function that only moves the head left. Answer: B Explanation: An NTM’s transition relation can have multiple possible moves, representing nondeterministic branching. Question 5. Which of the following is a correct statement of the Space Hierarchy Theorem? A) For any space‑constructible s(n), there exists a language decidable in O(s(n)) space but not in o(s(n)) space. B) SPACE(s) = SPACE(s log s) for all s. C) All problems solvable in linear space are also solvable in constant space. D) The theorem only applies to nondeterministic space. Answer: A Explanation: The Space Hierarchy Theorem asserts that more space yields strictly more computational power for space‑constructible functions. Question 6. Which problem is the canonical NP‑complete problem proved by the Cook‑Levin theorem? A) 3‑SAT
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Question 9. Which of the following problems is known to be PSPACE‑complete? A) 3‑SAT B) TQBF (True Quantified Boolean Formula) C) VERTEX‑COVER D) LONGEST‑PATH in directed graphs Answer: B Explanation: TQBF, the problem of evaluating fully quantified Boolean formulas, is PSPACE‑complete. Question 10. Savitch’s Theorem states that for any function s(n) ≥ log n, NSPACE(s(n)) is contained in: A) DSPACE(s(n)²) B) DSPACE(2^{s(n)}) C) DSPACE(s(n)/log n) D) DSPACE(s(n) + log n) Answer: A Explanation: Savitch proved that nondeterministic space s(n) can be simulated deterministically using at most s(n)² space. Question 11. The Immerman‑Szelepcsényi theorem establishes that: A) NL = coNL B) L = NL C) P = NP D) PSPACE = NPSPACE Answer: A
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Explanation: The theorem shows nondeterministic logarithmic space is closed under complement, i.e., NL = coNL. Question 12. Which class captures problems solvable by a deterministic Turing machine using O(log n) space? A) P B) L C) NL D) PSPACE Answer: B Explanation: L (logarithmic space) consists of languages decidable by deterministic machines with O(log n) workspace. Question 13. In the polynomial hierarchy, Σ₂^P can be described as: A) P with an NP oracle. B) NP with a coNP oracle. C) NP with an NP oracle. D) coNP with a PSPACE oracle. Answer: C Explanation: Σ₂^P = NP^{NP}, i.e., an NP machine that can query an NP oracle. Question 14. Which of the following statements is equivalent to saying that the polynomial hierarchy collapses to its first level? A) P = NP B) NP ⊆ P/poly C) Σ₁^P = Π₁^P
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C) NL = coNL D) BPP ⊆ P/poly Answer: B Explanation: By counting the number of possible circuits versus the number of Boolean functions, Shannon showed many functions need exponential‑size circuits. Question 18. The class RP consists of languages for which there exists a polynomial‑time randomized algorithm that: A) Never accepts an incorrect input and accepts correct inputs with probability ≥ 1/2. B) May reject correct inputs with probability ≤ 1/2 but never accepts incorrect inputs. C) Accepts both correct and incorrect inputs with probability exactly 1/2. D) Always halts with a correct answer. Answer: B Explanation: RP has one‑sided error: for YES instances the algorithm accepts with probability ≥ 1/2, for NO instances it always rejects. Question 19. BPP is the class of languages decidable by a probabilistic polynomial‑time algorithm with: A) One‑sided error at most 1/3. B) Two‑sided error bounded by 1/3. C) Zero error but expected polynomial time. D) Error probability exactly 1/2. Answer: B Explanation: BPP allows both false positives and false negatives, each with probability at most 1/3 (any constant < 1/2 can be amplified).
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Question 20. ZPP can be characterized as the intersection of which two classes? A) RP ∩ coRP B) BPP ∩ P C) RP ∪ coRP D) P ∩ NP Answer: A Explanation: ZPP consists of problems solvable with expected polynomial time and zero error; equivalently, ZPP = RP ∩ coRP. Question 21. The Adleman’s Theorem (also called the BPP ⊆ P/poly result) shows that: A) Every BPP language has a polynomial‑size circuit family. B) BPP = P. C) BPP ⊆ NP. D) BPP ⊆ PSPACE. Answer: A Explanation: By fixing the random bits that make a BPP algorithm succeed for each input length, one obtains a polynomial‑size circuit family. Question 22. The Schwartz‑Zippel Lemma is primarily used in which context? A] Derandomizing BPP algorithms. B] Proving lower bounds for circuit depth. C] Polynomial identity testing. D] Constructing zero‑knowledge proofs. Answer: C Explanation: The lemma gives a bound on the probability that a non‑zero polynomial evaluates to zero on a randomly chosen point, which underlies randomized polynomial identity testing.
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Explanation: Arithmetization represents logical operations as polynomial operations, enabling the verifier to check polynomial identities probabilistically. Question 26. A zero‑knowledge proof guarantees that the verifier learns: A) Nothing beyond the validity of the statement. B) The entire witness. C) The prover’s private key. D) The exact runtime of the prover’s algorithm. Answer: A Explanation: Zero‑knowledge means the verifier gains no additional knowledge other than that the statement is true. Question 27. The PCP theorem states that every language in NP has a proof that can be verified using: A) O(log n) random bits and a constant number of queries. B) Polynomial time and polynomial queries. C) Exponential time and no randomness. D) Linear space and linear queries. Answer: A Explanation: The classic formulation is NP = PCP(log n, 1), meaning logarithmic randomness and a constant (often 1) query to the proof. Question 28. Which of the following is a direct corollary of the PCP theorem? A) MAX‑3SAT cannot be approximated within any constant factor unless P = NP. B) There exists a PTAS for the Traveling Salesperson Problem. C) Approximating MAX‑3SAT within some constant > 1 is NP‑hard.
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D) All NP‑complete problems have exact polynomial‑time algorithms. Answer: C Explanation: The PCP theorem implies hardness of approximation for many optimization problems, including that achieving any constant factor > 1 for MAX‑3SAT is NP‑hard. Question 29. The Unique Games Conjecture (UGC) concerns the hardness of approximating which type of problem? A) Unique constraint satisfaction problems with a permutation constraint. B) General linear programming. C) Graph coloring with a fixed number of colors. D) Subset sum with unique weights. Answer: A Explanation: UGC posits that certain “unique” constraint satisfaction problems are hard to approximate beyond specific thresholds. Question 30. The class #P captures problems that ask: A) Whether a solution exists. B) How many accepting paths a nondeterministic polynomial‑time TM has. C) Whether a language is in P. D) The space required to solve a problem. Answer: B Explanation: #P counts the number of accepting computation paths of an NP (or NPTM) machine. Question 31. Toda’s Theorem states that the entire polynomial hierarchy is contained in:** A) P^{#P}
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Question 34. Which of the following problems is known to be NL‑complete under log‑space reductions? A) ST‑CONNECTIVITY (reachability) in directed graphs. B) 3‑SAT. C) Hamiltonian Cycle. D) Graph Isomorphism. Answer: A Explanation: Directed s‑t reachability is NL‑complete. Question 35. The class coNP is closed under which of the following operations? A) Union B) Intersection C) Complement D) Concatenation Answer: C Explanation: By definition, coNP consists of complements of NP languages, so taking complements maps back into NP. Question 36. Which of the following statements is true regarding deterministic vs. nondeterministic space? A) NSPACE(s(n)) ⊆ DSPACE(s(n) / log n) for s(n) ≥ log n. B) DSPACE(s(n)) ⊆ NSPACE(s(n)²). C) NSPACE(s(n)) = DSPACE(s(n)) for all s(n). D) NSPACE(s(n)) ⊆ DSPACE(s(n)²) (Savitch’s theorem). Answer: D Explanation: Savitch’s theorem gives the inclusion NSPACE(s) ⊆ DSPACE(s²).
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Question 37. In the definition of a polynomial‑time reduction f : Σ* → Σ, which of the following must hold? A) f must be injective. B) f must be computable in logarithmic space. C) For all x, x ∈ L₁ iff f(x) ∈ L₂. D) f must be a bijection. Answer: C Explanation: The reduction preserves membership: x is in the source language iff f(x) is in the target language. Question 38. Which of the following problems is known to be PSPACE‑complete even when restricted to planar graphs? A) 3‑SAT B) Generalized Geography C) Vertex Cover D) Hamiltonian Path Answer: B Explanation: Planar Generalized Geography remains PSPACE‑complete. Question 39. The class P/poly contains which of the following languages? A) The unary language {1^n | n is prime}. B) The language of all strings (Σ). C) The language of Kolmogorov‑random strings. D) The language of all strings that encode a Turing machine that halts. Answer: B
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D) RP = coRP = BPP. Answer: C Explanation: The intersection of RP and coRP yields exactly ZPP (zero‑error probabilistic polynomial time). Question 43. The concept of “hardness of approximation” is most directly derived from which theorem? A) Cook‑Levin theorem. B) PCP theorem. C) Savitch’s theorem. D) Immerman‑Szelepcsényi theorem. Answer: B Explanation: The PCP theorem establishes that checking a proof with few queries yields hardness of approximation results for many optimization problems. Question 44. Which of the following is a complete problem for the class Σ₂^P? A) QSAT₂ (Quantified Boolean formula with one alternation ∃∀). B) 3‑SAT. C) TAUT. D) GRAPH‑ISO. Answer: A Explanation: QSAT₂, where the formula has an ∃∀ quantifier prefix, is Σ₂^P‑complete. Question 45. In a public‑coin interactive proof (AM), the verifier’s randomness is: A) Secret and unknown to the prover. B) Sent to the prover before the prover’s message.
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C) Generated after the prover’s first message. D) Not used at all. Answer: B Explanation: In AM protocols, the verifier first sends random bits (public coin) to the prover, who then responds. Question 46. Which of the following problems is known to be complete for the class coNP under polynomial‑time many‑one reductions? A) SAT B) TAUT C) CLIQUE D) VERTEX‑COVER Answer: B Explanation: TAUT (the set of tautologies) is coNP‑complete. Question 47. The class EXPTIME consists of languages decidable in time: A) 2^{poly(n)} B) n^{log n} C) 2^{O(n)} D) n! Answer: C Explanation: EXPTIME = DTIME(2^{O(n)}), i.e., deterministic exponential time. Question 48. Which of the following statements about the relationship between P and BPP is currently known? A) P = BPP.
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Question 51. The class NEXP consists of languages decidable by a nondeterministic Turing machine in: A) Exponential time, i.e., 2^{poly(n)}. B) Polynomial time. C) Double‑exponential time. D) Sub‑exponential time. Answer: A Explanation: NEXP = NTIME(2^{poly(n)}), nondeterministic exponential time. Question 52. Which of the following statements about the relationship between PSPACE and EXP is correct? A) PSPACE = EXP. B) PSPACE ⊆ EXP. C) EXP ⊆ PSPACE. D) PSPACE and EXP are incomparable. Answer: B Explanation: By Savitch’s theorem and the fact that polynomial space can be simulated in exponential time, PSPACE ⊆ EXP. Question 53. The class coNL is equal to: A) NL B) P C) L D) PSPACE Answer: A Explanation: Immerman‑Szelepcsényi theorem shows NL = coNL.
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Question 54. Which of the following is a known complete problem for the class #P? A) COUNT‑SAT (count the number of satisfying assignments of a Boolean formula). B) SAT. C) TAUT. D) CLIQUE. Answer: A Explanation: Counting the satisfying assignments of a CNF formula is #P‑complete. Question 55. In the context of interactive proofs, a “public‑coin” protocol means: A) The verifier’s random bits are hidden from the prover. B) The verifier’s random bits are sent to the prover. C) No randomness is used. D) The prover chooses the random bits. Answer: B Explanation: Public‑coin protocols (e.g., AM) expose the verifier’s random bits to the prover. Question 56. Which of the following reductions is stronger (i.e., more restrictive) than a polynomial‑time many‑one reduction? A) Turing (oracle) reduction. B) Logspace many‑one reduction. C) Randomized reduction. D) Truth‑table reduction. Answer: B
