Hint:Takeanarbitrarylanguagein Σ andreduceitto Σ .Notethatwe These alternationbounded problemswillhelpustounderstandthepolynomial ChristmasExercise

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WS 2015/2016 16.12.2015 Exercises to the lecture

Complexity Theory Sheet 8 Prof. Dr. Roland Meyer

M.Sc. Peter Chini Delivery until 06.01.2016 at 12h

Christmas Exercise

Exercise 8.1 (Alternation bounded QBF and collapsing of the polynomial hierarchy) Consider the following definition:

• Σ

i

QBF = {ψ | ψ = ∃ x

1

∀ x

2

. . . Q

i

x

i

ϕ(x

1

, . . . , x

i

) is true },

• Π

i

QBF = {ψ | ψ = ∀ x

1

∃ x

2

. . . Q

i

x

i

ϕ(x

1

, . . . , x

i

) is true },

where x

_j

denotes a finite sequence of variables and Q

_i

is a quantor. Note that there are at most i − 1 alternations of quantors.

These alternation bounded QBF problems will help us to understand the polynomial hierarchy in more detail:

a) Show that Σ

i

QBF is in Σ

^P_i

and that Π

i

QBF is in Π

^P_i

.

b) Prove that Σ

_i

QBF is Σ

^P_i

-hard with respect to polytime reductions and that Π

_i

QBF is Π

^P_i

-hard with respect to polytime reductions.

Hint: Take an arbitrary language in Σ

^P_i

and reduce it to Σ

i

QBF . Note that we showed that QBF is PSPACE-complete. Extract the idea from this proof.

Note that we now have the following situation:

• SAT = Σ

₁

QBF and this is NP = Σ

^P₁

-complete,

• co-SAT = Π

₁

QBF and this is co-NP = Π

^P₁

-complete.

• The alternation bounded QBF instances are complete for Σ

^P_i

and Π

^P_i

,

• and the general QBF which allows unbounded alternation is PSPACE-complete.

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We can make use of this to show that in some situations, the polynomial hierarchy collapses:

c) Assume we have a k so that Σ

^P_k

= Σ

^P_k+1

, then we also have Π

^P_k

= Π

^P_k+1

.

d) If we have a k so that Σ

^P_k

= Π

^P_k

, then we have for any k

⁰

≥ k that Σ

^P_k0

= Π

^P_k0

= Σ

^P_k

. So the polynomial hierarchy collapses to this level.

Hint: Prove this by induction on k

⁰

. Show that Σ

_k+1

QBF is already in Σ

^P_k

. e) If we have a k so that Σ

^P_k

= Σ

^P_k+1

Hint:Takeanarbitrarylanguagein Σ andreduceitto Σ .Notethatwe These alternationbounded problemswillhelpustounderstandthepolynomial ChristmasExercise

WS 2015/2016 16.12.2015 Exercises to the lecture

Complexity Theory Sheet 8 Prof. Dr. Roland Meyer

M.Sc. Peter Chini Delivery until 06.01.2016 at 12h

Christmas Exercise

Exercise 8.1 (Alternation bounded QBF and collapsing of the polynomial hierarchy) Consider the following definition:

• Σ

QBF = {ψ | ψ = ∃ x

∀ x

. . . Q

x

ϕ(x

, . . . , x

) is true },

• Π

QBF = {ψ | ψ = ∀ x

∃ x

. . . Q

x

ϕ(x

, . . . , x

) is true },

where x

denotes a finite sequence of variables and Q

is a quantor. Note that there are at most i − 1 alternations of quantors.

These alternation bounded QBF problems will help us to understand the polynomial hierarchy in more detail:

a) Show that Σ

QBF is in Σ

and that Π

QBF is in Π

.

b) Prove that Σ

QBF is Σ

-hard with respect to polytime reductions and that Π

QBF is Π

-hard with respect to polytime reductions.

Hint: Take an arbitrary language in Σ

and reduce it to Σ

QBF . Note that we showed that QBF is PSPACE-complete. Extract the idea from this proof.

Note that we now have the following situation:

• SAT = Σ

QBF and this is NP = Σ

-complete,

• co-SAT = Π

QBF and this is co-NP = Π

-complete.

• The alternation bounded QBF instances are complete for Σ

and Π

,

• and the general QBF which allows unbounded alternation is PSPACE-complete.

We can make use of this to show that in some situations, the polynomial hierarchy collapses:

c) Assume we have a k so that Σ

= Σ

, then we also have Π

= Π

.

d) If we have a k so that Σ

= Π

, then we have for any k

≥ k that Σ

= Π

= Σ

. So the polynomial hierarchy collapses to this level.

Hint: Prove this by induction on k

. Show that Σ

QBF is already in Σ

. e) If we have a k so that Σ

= Σ

then we already have that the polynomial

hierarchy collapses to this level.

Hint: Use parts d) and c).

We wish you a merry Christmas and a good start to the new year. Enjoy your vacation!

Delivery until 06.01.2016 at 12h into the box next to 34-401.4