L2 ={w wis the representation of a prime number in base 3}

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Universität Koblenz-Landau FB 4 Informatik

Prof. Dr. Viorica Sofronie-Stokkermans^∗¹ 24.01.2018

M.Ed. Dennis Peuter^∗²

Exercises for Advances in Theoretical Computer Science Exercise Sheet 12

Due at 29.01.2018, 10:00 s.t.

Exercise 12.1

Give a function f : Σ^∗ →Σ^∗ which polynomially reducesL₁ to L₂, or explain why this is not possible:

I) Σ ={0,1,2};

L1 ={w∈ {0,1}^∗|w is the representation of a prime number in base 2}; L2 ={w∈ {0,1,2}^∗ |wis the representation of a prime number in base 3}. II) Σ ={0,1,2,3,4};

L₁ ={w∈ {0,1}^∗|w is the representation of a prime number in base 2}; L₂ ={w∈ {0,1,2,3,4}^∗ |wis the representation of a prime number in base 5}. III) Σ ={0,1};

L₁ ={w∈ {0,1}^∗|w is the representation of a prime number in base 2}; L2 ={w∈ {1}^∗ |wis the representation of a prime number in base 1}. IV) Σ ={a, b,0,1};

L1 ={aⁿbⁿ|n≥0}; L₂ ={1}.

Here, for IV), we require that for all w∈Σ^∗\L₁,f(w) = 0.

Exercise 12.2

We know that SAT is NP-complete. In the previous exercise we saw that satisabi- lity of formulae in DNF can be checked in polynomial time, so DNF-SAT = {F | F is a satisable formula of propositional logic in disjunctive normal form} is in P.

If we could construct a polynomial reduction of SAT to DNF-SAT (i.e. if we could prove that SAT ≺_pol DNF-SAT) then we could show that P = NP.

Formulae in propositional logic can be transformed to DNF using distributivity:

A∧(B₁∨ · · · ∨B_k)≡(A∧B₁)∨ · · · ∨(A∧B_k).

Why does this not lead to a polynomial reduction?

∗¹ B 225 sofronie@uni-koblenz.de https://userpages.uni-koblenz.de/~sofronie/

∗² B 223 dpeuter@uni-koblenz.de https://userpages.uni-koblenz.de/~dpeuter/

If you want to submit solutions, please do so until 29.01.2018, 10:00 s.t. via the cardboard box in the shelf in room B 222 or via e-mail (with Homework ACTCS in the subject) to dpeuter@uni-koblenz.de.