Practice Final Exam

Practice Final Exam

Prof. Carolina Ruiz
CS4123 Theory of Computation. B97

Department of Computer Science
Worcester Polytechnic Institute

1. (20 Points) Turing Machines

Construct the state diagram with transitions of a deterministic Turing machine that given an input number n in binary representation halts with just the binary representation of n+1 on its tape.

2. (20 Points) Recursive Functions

Show that the following integer approximation of the log function is PRIMITIVE recursive.

log(m,n) := "logarithm in base m of n" = "the least k for which m^k >= n"

m^k denotes m to the k-th power. You can assume that m >= 2 and n > 0.

Hint:s Use bounded minimization.

3. (20 Points) Decidability

Let Sigma = {0,1}. Show that the problem of testing whether a CFG generates some string in 1* is decidable. In other words,

show that ONESTARcfg = {< G > | G is a CFG and (1* INTERSECTION L(G)) <> empty} is decidable.

Hint:s Do not make the mistake of assuming that the intersection of CFLs is context free.

4. (20 Points) Undecidability

Solve either one of the following problems:

  1. Show that CONTEXTFREEtm = {< M > | M is a TM and L(A) is context free} is undecidable.

  2. Show that DECIDERtm = {< M > | M is a TM and M stops on all inputs} is undecidable.

5. (20 Points) P, NP, and NP-completeness

Let G = (V,E) be an undirected graph. A vertex cover for G is a is a subset V' of V such that for each edge (a,b) in E either a belongs to V', or b belongs to V' (or both).

VERTEX-COVER = {< G,c > | G is undirected graph and G has a vertex cover of size c}.

Show that VERTEX-COVER is an NP-complete problem

Hint: Show that:

  1. VERTEX-COVER is in NP.
  2. Show that CLIQUE <=_P VERTEX-COVER using the following transformation:

    < G,k> belongs to CLIQUE iff f(< G,k >) = < GC,c > belongs to VERTEX-COVER


    • G = (V,E).
    • GC = (V,EC),
      EC = {(a,b) | a and b belong to V but edge (a,b) does NOT belong to E}.
    • c = |V| - k.

    (Don't forget to show that the function f can be computed by a deterministic TM in polynomial time.)