CMPSCI 501: Theory of Computation

Final Exam, Spring 2016

David Mix Barrington

5 May 2016

Directions:

• Answer the problems on the exam pages.
• There are eight problems (some with multiple parts) for 120 total points plus 10 extra credit. Actual scale was A = 100, C = 50.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first five questions are statements -- in each case say whether the statement is true or false and give a convincing justification of your answer -- a proof, counterexample, quotation from the book or from lecture, etc. You get five points for the correct boolean answer (so there is no reason not to guess if you don't know) and up to five for the justification.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 10 points
  Q6: 25 points
  Q7: 35 points
  Q8: 10+10 points
 Total: 120+10 points

If C is any class of computers, such as DFA's, CFG's, TM's, strange variant TM's, etc.:

• A_C = {(M, w): M is a computer in C and w ∈ L(M)}

• E_C = {(M): M is a computer in C and L(M) = ∅}

• ALL_C = {(M): M is a computer in C and L(M) = Σ^*}

In particular, if C = P, then M must have a clock restricting it to some polynomial time bound, and if C = L, then M must have a marker restricting it to c log n worktape space for some constant c.

A language is Turing recognizable (TR) if it is equal to L(M) for some Turing machine M.

A language is Turing decidable (TD) if it is equal to L(M) for some Turing machine M that halts on every input.

A language is co-TR if and only if its complement is TR.

A function f from strings to strings is Turing computable if there exists a Turing machine M such that for any string w, M when started on w halts with f(w) on its tape.

Recall that if A and B are two languages, A is mapping reducible to B, written A ≤_m B, if there exists a Turing computable function f: Σ^* → Σ^* such that for any string w, w ∈ A ↔ f(w) ∈ B. If such an f exists that is computable in polynomial time, we say that A is poly-time reducible to B, written A ≤_p B. If f is computable in log space, we say that A is log-space reducible to B, written A ≤_L B.

The following languages are proved to be NP-complete in the text or in Exercises, and you may assume without proof that each of them is NP-complete.

• 3-SAT = {φ: φ is a satisfiable formula in 3-CNF}

• 3-COLOR = {G: G is an undirected graph and the vertices of G may each be assigned one of three colors so that no edge connects two vertices of the same color}

• DHAMPATH: = {(G, s, t): G is a directed graph in which there is a directed path from vertex s to vertex t such that the path visits each vertex exactly once}

• UHAMPATH: = {(G, s, t): G is an undirected graph in which there is an undirected path from vertex s to vertex t such that the path visits each vertex exactly once}

• SUBSET-SUM = {(a₁,..., a_k, t): the a_i's and t are positive integers written in binary and there is a subset of the a_i's that adds to exactly t}

• CLIQUE = {(G, k): G is an undirected graph and G has a set S of k vertices such that every two distinct vertices in S have an edge between them}

• VERTEX-COVER = {(G, k): G is an undirected graph and G has a set S of k vertices such that every edge in G has at least one endpoint in S}

Recall that a quantified boolean formula is a statement of the form ∃x₁: ∀x₂: ∃x₃;:... ∀x_n: φ(x₁,..., x_n) where each x_i is a boolean variable and φ is a boolean formula in conjunctive normal form. It was proved in the text and in lecture that the language TQBF of true quantified boolean formulas is PSPACE-complete, and you may assume this fact without proof.

A DogShow object consists of a set D of dogs, a set E of events, and a relation C ⊆ D × E such that C(d, e) means "dog d competes in event e".

A Schedule object for a DogShow (D, E, C) is a positive integer t and a function S from E to {1,..., t}. (Assume that t ≤ |E|.) A Schedule is valid if there do not exist a dog d and two distinct events e and e' such that C(d, e), C(d, e'), and S(e) = S(e') are all true. (Thus a schedule is valid if no dog competes in two different events that are scheduled at the same time.

The language DS-SCHED-OK is the set {(D, E, C, S): S is a valid schedule for (D, E, C)}.

The language DS-POSSIBLE is the set {(D, E, C, t): ∃S: (D, E, C, S) ∈ SCHED-OK and S has parameter t}.

Let M be any machine that takes an input string in {0, 1}^*. We define a number of games for M. The solitaire word game for M has White write down a string w and win the game if and only if w ∈ L(M). The bounded solitaire word game for M and n further requires that |w| = n. The bounded alternating word game for M and n is similar, except that now w is formed alternately by White and Black naming letters in {0, 1} until n letters have been named. White still wins if and only if w is in L(M).

We define a number of languages based on these games. If C is any class of computers, SWG_C is the set of computers M in C such that White wins the solitaire word game on M. Similarly BSWG_C is the set {(M, 1^t): M is a computer in C and White wins the bounded solitaire word game for M and n}, and BAWG_G is the similar language for the bounded alternating word game. (We make the second input component 1ⁿ rather than the binary for n so that the input size will be O(n).)

We define two particular context-free languages called the Dyck languages, which are essentially strings of balanced parentheses. The language D₁ has the context-free grammar with rules S → aSb, S → SS, and S → ε. The language D₂ has these three rules plus the additional rule S → cSd.

• Question 1 (10): True or false with justification: The language D₁ is regular.

• Question 2 (10): True or false with justification: The language D₁ is in the class L = DSPACE(log n), but the language D₂ is not.

• Question 3 (10): True or false with justification: The language SWG_TM is Turing decidable.

• Question 4 (10): True or false with justification: The language SWG_L is turing recognizable but not Turing decidable.

• Question 5 (10): True or false with justification: The language SWG_DFA is in the class P.

• Question 6 (25): These questions use the definitions of DogShow and Schedule objects above, and their associated languages.
  • (a, 5) Suppose that a DogShow object (D, E, C) has at most n dogs and at most n events. Explain why the size of the input strings to DS-SCHED-OK or DS-POSSIBLE for this object have size polynomial in n.

  • (b, 10) Prove that the language DS-SCHED-OK is in the class P.

  • (c, 10) Prove that the language DS-POSSIBLE is NP-complete.

• Question 7 (35): These questions all involve the solitaire and alternating word games defined above, and their associated languages.

  • (a, 10) Prove that BSWG_TM ≤_m BAWG_TM.

  • (b, 10) Prove that the language BSWG_L is NP-complete.

  • (c, 15) Prove that the language BAWG_L is PSPACE-complete.

• Question 8 (10+10): These questions involve the Dyck language D₂ defined above. Given a string w of length n, define a predicate P(i, j), for all i and j with 1 ≤ i ≤ j ≤ n, to be true if and only if the string w_i...w_j is in the language D₂.
  • (a, 10) Give a recursive definition of the predicate P(i, j), with base cases for P(i, i) and an inductive case defining each value P(i, j) by boolean operations on other values P(k, l) and predicates of the form w_i = a, w_i = b, w_i = c, and w_i = d. Explain why you are guaranteed to reach a base case.

  • (b, 10XC) Describe a circuit family, based on your definition in part (a), such that the circuit C_n will input 4n boolean variables describing a string w ∈ {a, b, c, d}ⁿ and output a boolean that says whether w ∈ D₂. What are the size and depth of your circuits, in big-O terms as a function of n?

Last modified 25 July 2016