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
   Post Correspondence Problem. (Read at
   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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