Computability Complexity Today s Agenda P SP ACE completeness

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Computability & Complexity 15

Today’s Agenda

• P SP ACE-completeness.

• An example of a P SP ACE-complete prob-
lem.

• The exam questions!

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Computability & Complexity 15

Deﬁning P SP ACE-completeness

A language B is P SP ACE-complete iﬀ

1. B ∈ P SP ACE and

2. A ≤P B, for every A ∈ P SP ACE. (P SP ACE-
hardness)

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Computability & Complexity 15

Fully Quantiﬁed Boolean Formulae

Quantiﬁed Boolean formulae are those gener-
ated by the grammar:

φ ::= ψ | ∃x φ | ∀x φ ,
where ψ is a boolean formula, and x is a boolean
variable.

A Boolean formula is fully quantiﬁed (closed,
a sentence) if every variable x that occurs in it
is within the scope of ∀x or ∃x.

Examples: The formulae
∀x∃y[(x ∨ y) ∧ y]
∃x∀y[(x ∨ y) ∧ y]
∃x∃y[(x ∨ y) ∧ y]
are fully quantiﬁed. Are they true?

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Computability & Complexity 15

An Example of a P SP ACE-complete Problem

The problem True Quantiﬁed Boolean Formula
(aka T QBF ) is deﬁned thus:

T QBF = { φ | φ is a true fully
quantiﬁed Boolean formula}

Theorem: T QBF is P SP ACE-complete.

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Computability & Complexity 15

Exam Questions

1. Decidable and Recognizable Languages: Def-
initions and Closure Properties. (Read at
least pages 127–135, and 140–141 in Sipser’s
book. Recall your solutions to exercises
3.14 and 3.15 on page 149 of Sipser’s book.)

2. The Halting Problem and Other Unsolv-
able Decision Problems for Turing Machines.
(Read at least pages 165–168 and 172–176
of Sipser’s book.)

3. Reductions via Computation Histories and
least pages 176–177 and 183–189 of Sipser’s
book.)

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Computability & Complexity 15

4. Mapping Reducibility and its Applications.
(Read at least pages 189–194 of Sipser’s
book.)

5. Rice’s Theorem and its Applications. (Read
u
at least Hans H¨ttel’s note)

6. The Complexity Class P: Deﬁnition, the
example of CFL recognition, and closure
properties. (Read at least pages 229–241,
including Thm. 7.14, of Sipser’s book, and
recall your solution to exercise 7.6 on page
271 of Sipser’s book.)

7. The Complexity Class N P , N P -Completeness
and the Cook-Levin Theorem. (Read at
least pages 241–259 of Sipser’s book.)

8. Space Complexity, Savitch’s Theorem and
the Classes PSPACE and NPSPACE. (Read
at least pages 277–282 of Sipser’s book.)

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Computability & Complexity 15

As guidelines for the exam preparation, you
should consider the following aspects in your
presentation (if at all possible):

• Overview: For instance the motivation and
the impact of the theory to be presented.

• Understanding: The ability to analyze, de-
compose and give the details of the proofs
involved at the right abstraction level.

• Applications: Give some examples of ap-
plications.

• Questions: Remember to leave some time
for questions.

I will make some of the slides available at the
exam. The available slides will be posted on
the “Exam” part of the course web-page.
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