CMPSCI 601: Theory of Computation

David Mix Barrington

Summer, 2003

Final Exam

Posted 25 August 2003

Types of variables: Unless otherwise indicated, a variable A or B denotes a language, D denotes a DFA, G denotes a context-free grammar, M denotes a Turing machine, n denotes a (non-negative)number, and w denotes a string. Remember that N is the set of all numbers, which we may also think of as the set Σ^* of all strings.

Directions: Question 1 consists of five statements, each of which is either provably true, provably false, or unresolved given the results presented in the course. Indicate TRUE, FALSE, or UNKNOWN (choose one only) for each question. No justification is needed or wanted, and there is no penalty for a wrong guess.

Questions 2 and 3 consist of true statements that you are to prove. These facts were proved in lecture, but of course you may not simply quote them as facts -- give an explanation in your own words of why they are true that demonstrates your understanding.

Questions 4 through 8 are similar to those on the midterm. There is a statement which is either true or false. You will get five points for a correct boolean answer, and there is no penalty for guessing. Then JUSTIFY YOUR ANSWER -- the five remaining points will be awarded based on the degree to which your justification is correct and convincing.

A list of useful facts and definitions is provided after the questions. You may assume facts proved in lecture if you state them carefully and correctly. Make sure that if a fact from lecture requires hypotheses, you show that those hypotheses hold in this case.

• Question 1 (15):

  • (a,3) The class NC¹ is strictly contained in the class PSPACE.

  • (b,3) The class PSPACE is strictly contained in the class NC¹

  • (c,3) If A is the language of a poly-time alternating Turing machine, A is also the language of some poly-time deterministic Turing machine.

  • (d,3) There exists a PTAS for MAX-3-SAT.

  • (e,3) If B is the language of some nondeterministic Turing machine (with no resource bounds) then B is also the language of some deterministic Turing machine.

• Question 2 (15): Let M be a one-tape Turing machine that always halts within some polynomial time bound p(n) on input of size n. Prove that for any n, there exists a circuit C_n with n inputs such that C_n(w) = 1 iff w is in L(M). Prove that the size of your C_n is polynomial in n. Explain briefly why this circuit family is log-space uniform.

• Question 3 (20): The following are two definitions of what it means for a language $E$ to be in the class NP:

  • Definition 1: There exists a poly-time nondeterministic Turing machine N such that E = L(N).

  • Definition 2: There exists a poly-time computable predicate F(x,y), where x and y are strings, such that for any string x, x is in E iff ∃y: F(x,y).

  Prove that these two definitions are equivalent, that is, that for any language E, E satisfies Definition 1 iff it satisfies Definition 2.

• Question 4 (10): (True/false with justification) There exists a context-free language B such that B is log-space reducible to the language REACH.

• Question 5 (10): (True/false with justification) Every regular language is in the class NC¹.

• Question 6 (10): (True/false with justification) Every language in the class NC¹ is regular.

• Question 7 (10): (True/false with justification) If a language G is in the class BPP, then there exists a poly-time alternating Turing machine M such that G = L(M).

• Question 8 (10): (True/false with justification) If the language MAX-3-SAT-DECISION = {(φ,k): φ is a 3-CNF formula and there exists a setting of φ's variables that satisfies at least k clauses} is in the class P, then the optimization problem MAX-3-SAT can be solved exactly in polynomial time.

Crib Sheet:

• L(M) is the set of numbers n such that M halts on input n with output 1. Note that the set {M_n: n is in N} contains all possible Turing machines in some standard encoding and that each M_n is a valid Turing machine.
• L(D) is the set of strings w such that D running on w finishes in an accepting state.
• The following classes are defined as the languages of the following kinds of Turing machines:

  • P: poly-time deterministic TM's

  • NP: poly-time nondeterministic TM's

  • L: space O(log n) deterministic TM's

  • NL: space O(log n) nondeterministic TM's

  • PSPACE: poly-space deterministic TM's

• If A and B are languages, then A≤ B (A is log-space reducible to B) means that there is a log-space computable function f such that for any x, x is in A iff f(x) is in B.

• The following classes are defined as the the languages decideable by the following kinds of circuit families:

  • NCⁱ: circuits with AND, OR, and NOT gates, fan-in two, polynomial size, and O(logⁱ n) depth,

  • ACⁱ: circuits with AND, OR, and NOT gates, unbounded fan-in, polynomial size, and O(logⁱ n) depth,

  • ThCⁱ: circuits with threshold gates, unbounded fan-in, polynomial size, and O(logⁱ n) depth.

• L(G) is the set of strings w such that w can be generated from G's start symbol according to G's rules.

• A language is r.e. if it is equal to L(M) for some Turing machine M.

• A language is recursive if it is equal to L(M) for some Turing machine M that always halts.

• A function f is total recursive if there is a Turing machine that always halts and gives output f(n) on any input n.

• The language A is reducible to B if there exists a total recursive function f such that for any n, n is in A iff f(n) is in B.

• The language A is r.e.-complete iff A is r.e. and for any language B, if B is r.e. then B is reducible to A.

• An optimization problem has a poly-time approximation scheme or PTAS if for any positive real number ε, there exists a poly-time algorithm that approximates an optimal solution to the problem within a ratio of ε.

• We refer to some or all of the following specific languages and problems:

  • REACH is the set {(G,s,t): G is a directed graph and there is a path from s to t in G}.

  • MAX-3-SAT is the optimization problem where the input is a 3-CNF formula, the output is a setting of the formula's variables, and the goal is to maximize the number of the formula's clauses satisfied by the setting.

  • K = {n: n is in L(M_n). This language is r.e.-complete.

Last modified 25 August 2003