NP-completeness
Theory of Computation
In the last lecture we showed the following theorem.
Lecture 30-31
Theorem
3SAT is NP-complete.
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This is our second problem that we have shown to be NP-complete.
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December 17, 2007
Sar
Lecture 40: Overview
Theory of Computation
Lecture #39
TQBF.
FORMULA-GAME.
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Generalized Geography.
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Space Complexity
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Theory o
Lecture 39: Overview
Theory of Computation
Lecture #39
TQBF
Prove that TQBF is PSPACE-complete.
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Other PSPACE-complete problems
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Space Complexity
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NP-completness
Theory of Computation
Recall that A is NP hard if for all B NP
Lecture 26
B p A.
Let us recall that a language A is called NP-complete if
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1
2
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Theory of Computation
A is NP-hard.
A is in NP.
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Sarmad Abbas
Lecture 35: Overview
Theory of Computation
Lecture #34
An approximation algorithm for the TSP problem.
Space Complexity.
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Th
Lecture 33: Overview
Theory of Computation
Lecture #34
Three versions of the TSP problem.
An approximation algorithm for the TSP problem.
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Space Complexity.
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TS
Lecture 32: Overview
Theory of Computation
Lecture #32
Review of NP-completeness results
Hamiltonian Path Problem
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NP-completeness results so far
1
IS is NP-co
Lecture 33: Overview
Theory of Computation
Lecture #33
TSP problem.
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TSP
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TSP
The traveling salesman p
The Cook-Levin Theorem
Theory of Computation
Lecture 28
Last time we were discussing the proof of the Cook-Levin Theorem.
Here is the basic outline of the proof.
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December 7, 2007
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Theory o
Satisability
Let us dene
Theory of Computation
SAT = cfw_< >: is satisable.
Lecture 27
It is easy to see that
Theorem
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SAT is in NP
We gave an easy verication algorithm. The algorithm takes a formula
and assignment and checks if satises .
De
Lecture 23: Overview
Theory of Computation
The Class NP
Polynomial Time veriers
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Examples of Problems in NP
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Theory of Computation
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The Class NP
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Theory of Computation
2 / 28
The Class NP
Sinc
Lecture 22: Overview
Theory of Computation
Non-deterministic Time
The Class P
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The Class NP
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Theory of Computation
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Time Complexity Classes
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Theory of Computation
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Time complexity
The pr
NP-completeness
Last time we nished the proof of the celebrated Cook-Levin theorem
which states that:
Theory of Computation
Lecture 29-30
Theorem
SAT is NP-complete.
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This means
1
December 17, 2007
SAT is in NP.
2
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Every lan
Space Complexity
Recall that our goal was to prove that
Theory of Computation
Lecture #43
Theorem
NL = co-NL.
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We simplied this problem by observing that
Theorem
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PATH is NL-complete (with respect to logspace reducibility).
Space Complexity
Theory of Computation
Lecture #45
Today we will discuss what we have learnt in this course and go over
the major results and accomplishments of computability and
complexity theory.
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Space Complexity
Theory of Computation
Lecture #43
Our goal today is to prove the following theorem:
Theorem
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NL = co-NL.
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Space Complexity
Theory of Computatio
Lecture 33: Overview
Theory of Computation
SPACE COMPLEXITY
Lecture #36
Deterministic Space Complexity
Non-deterministic space complexity
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Examples of Space efcient TMs.
Reachability Problem
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Savitchs Theorem
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Lecture 33: Overview
Theory of Computation
SPACE COMPLEXITY
Lecture #36
Deterministic Space Complexity
Non-deterministic space complexity
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Examples of Space efcient TMs.
Reachability Problem
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Savitchs Theorem
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Space Complexity
Let us look recall the game called generalized geography.
Two players play the game.
Theory of Computation
1
Player I starts the game and says the name of a city/town or
geographical region.
2
Player II does the same thing. The rule is pl
Space Complexity
Theory of Computation
Let us now consider some new complexity classes. Let us dene
Lecture #42
L = SPACE(log n)
and
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NL = NSPACE(log n).
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These turn our to be very interesting space complexity classes.
Sarma
Lecture 38: Overview
Theory of Computation
Lecture #38
Relationship between space and time complexity.
PSPACE
PSPACE-completeness
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TQBF
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Verication Algorithms
Let us recall that V is called a verication algorithm for A if
Theory of Computation
A = cfw_w : V accepts w, c for some c.
Lecture 25
In the last lecture studied verication algorithms for the following
problems.
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CLIQ
Lecture 24: Overview
The Class NP
Veriers and alternate characterization of NP
Theory of Computation
HAMPATH
CLIQUE
SUBSET SUM PROBLEM
Dr. Sarmad Abbasi
Department of Computer Science
National University of Computer and Emerging Sciences
Block B, Faisal T
Lecture 13: Overview
Theory of Computation
Decidability of Logical Theories
Presburger arithmetic
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Decidability of Presburger Arithmetic
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Theory of Computation
1/1
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Theory of Computation
2/1
Logical
Lecture 10: Overview
Theory of Computation
Linear Bounded Automata
Acceptance Problem for LBAs
Lecture #10
Emptiness Problem for LBAs
Computation Histories
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Emptiness Problem for CFGs
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An undecidable Problem for CFGs
Post Co
Lecture 14: Overview
Theory of Computation
Lecture #14
un-Turing-recognizability
A nice puzzle
Programs that print themselves
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Recursion Theorem
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Reducibility
S
Lecture 5: Overview
Theory of Computation
Variants of Turing Machines.
Lecture #5
k -tape TMs.
Show that k -tape TMs have the same power as 1-tape TMs.
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Study enumerators.
Prove equivalence of enumerators and TMs.
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Dovetaili
Lecture 10: Overview
Theory of Computation
Undecidability of the Halting Problem
Lecture #10
Russells Paradox
Reducibility
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Undecidable Problems
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Theory of Computation
Linear bounded automa
Lecture 7: Overview
Theory of Computation
Decidable Languages
Lecture #7
Decidable Problems concerning Regular Languages
Decidable Problems concerning Context Free Languages
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The Halting Problem
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Lecture 9: Overview
Theory of Computation
Halting Problem.
Lecture #9
Undecidability of the Halting Problem
Diagonalization
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Co-Turing Recognizability
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Theory of Computation
the Halting Pro