Practice Final Exam

## Practice Final Exam

### 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

where

• 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.)