CMPSCI 401: Theory of Computation

Final Exam, Spring 2013

David Mix Barrington

9 May 2013

Directions:

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 10 points
  Q6: 25 points
  Q7: 30+10 points
  Q8: 20 points
 Total: 125+10 points

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

A is Turing decidable if A = L(M) for some Turing machine M that always halts.

A function f 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 function f: Σ* → Σ* such that for any string w, w ∈ A ↔ f(w) ∈ B.

Although we did not prove this in lecture, you may assume both:

Define a weighted graph to be an undirected graph where each vertex has a positive integer weight -- you may assume that each weight is at most n, the number of vertices. The language WC (weighted clique) is the set {(G, S, k): S is a clique in the weighted graph G and the sum of the weights of the vertices in S is at least k}. The language MWC (maximum weighted clique) is the set {(G, k): there exists a clique in the weighted graph G such that the sum of the vertex weights in the clique is at least k}.

A directed graph is strongly connected if for every pair of vertices u and v, there is a (directed) path from u to v. The language STRCONN is the set of all strongly connected directed graphs.

Last modified 29 May 2013