Turing awards seminar
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The document discusses the evolution and principles of computational complexity, including key concepts such as deterministic time, NP-completeness, and the power index of problems. It also highlights the work of Richard Edwin Stearns in shaping the foundations of complexity theory and provides insights on various complexity classes and their relationships. Observations suggest that not all NP-complete problems have the same level of difficulty and some may require less time than previously thought.
![Richard Edwin Stearns
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Graduation:
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BA in Mathematics at Carlton
College in 1958
Ph.D(on game theory) from
Priceton University in 1961
Joined in General Electric's Research
Laboratory, Newyork in 1961
Received 1993 ACM Turing Award
along with Juris Hartmanis “in
recognition of their seminal paper
which established the foundations for
the field of computational complexity
theory”
Now Distinguished Professor Emeritus
of Computer Science at the University
at Albany[1]
10/27/2013
RAJ KUMAR RAMPELLI
Born 5 July 1936
3](https://image.slidesharecdn.com/turingawardsseminar-131027084849-phpapp01/85/Turing-awards-seminar-3-320.jpg)
![R.E.Stearns Interests
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Game Theory
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Computer Science
John von Neumann
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J. von Neumann and
Morgenstern, Theory of Games
and Economic Behaviour.
Competition
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Von Neumann stored program
computer model -- Father of
computer science[2]
Computation
How well do our models reflect the salient features of the object or situation we wish to describe?
10/27/2013
RAJ KUMAR RAMPELLI
4](https://image.slidesharecdn.com/turingawardsseminar-131027084849-phpapp01/85/Turing-awards-seminar-4-320.jpg)
![Complexity model
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Analysis of algorithm: Upper bound on running time of the algorithm
Complexity model: Place the problem into a complexity class using
algorithm
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Easiness classes
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Speed-up theorem
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O(T(n)) not T(n)
Result: n->
8
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DTIME(T(n)) = DTIME(c * T(n)) for all c>0
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10/27/2013
[T(n) * log(T(n))]/U(n) = 0
DTIME(U(n)) contains the language which is not in DTIME(T(n))
RAJ KUMAR RAMPELLI
6](https://image.slidesharecdn.com/turingawardsseminar-131027084849-phpapp01/85/Turing-awards-seminar-6-320.jpg)
![PSPACE
Set of all decision problems which can be solved by a Turing machine using a polynomial amount of space[3].
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PSPACE-hardness > NP-hardness
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Might require exponential time even if
it unexpectedly turns out that the NPcomplete problems can be solved in
polynomial time
PSPACE-Complete problems:
Hardest problems in PSPACE
1st problem in PSPACE completeness
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Figure: A representation of the relation among
complexity classes, which are subsets of each other.
10/27/2013
No reason in the sense that
PSPACE prob require more time
QSAT: Deciding if a quantified
CNF formula is true
RAJ KUMAR RAMPELLI
9](https://image.slidesharecdn.com/turingawardsseminar-131027084849-phpapp01/85/Turing-awards-seminar-9-320.jpg)
![Power Index
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R.E.Stearns and Harry Hunt III [4]
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Problem has power index k if it can be solved in time 2^nk
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k=0 : Problems with polynomial algorithms
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k>0 : Problems with exponential algorithms
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NP-complete problem has k>0 and P-complete problem has k=0. SO, P !=
NP
Theorem: If L1 has k value r and L1 redicuble to L2 by polynomial reduction
time reduction of size ns, then L2 has k value atleat n/s
Example: Standard reduction from SAT to CLIQUE [5] have n2 size and no
better reduction is known.
10/27/2013
RAJ KUMAR RAMPELLI
10](https://image.slidesharecdn.com/turingawardsseminar-131027084849-phpapp01/85/Turing-awards-seminar-10-320.jpg)
Turing awards seminar
- 1. IT'S TIME TO RECONSIDER TIME -Richard Edwin Stearns RAJ KUMAR RAMPELLI 10/27/2013 RAJ KUMAR RAMPELLI 1
- 2. Outline ● Introduction ● Interests ● Complexity ● Deterministic Time ● Complexity Model ● Hardness concepts ● NP-complete ● PSPACE ● Power Index ● Observations ● References 10/27/2013 RAJ KUMAR RAMPELLI 2
- 3. Richard Edwin Stearns ● Graduation: ● ● ● ● ● BA in Mathematics at Carlton College in 1958 Ph.D(on game theory) from Priceton University in 1961 Joined in General Electric's Research Laboratory, Newyork in 1961 Received 1993 ACM Turing Award along with Juris Hartmanis “in recognition of their seminal paper which established the foundations for the field of computational complexity theory” Now Distinguished Professor Emeritus of Computer Science at the University at Albany[1] 10/27/2013 RAJ KUMAR RAMPELLI Born 5 July 1936 3
- 4. R.E.Stearns Interests ● Game Theory ● Computer Science John von Neumann ● ● J. von Neumann and Morgenstern, Theory of Games and Economic Behaviour. Competition ● ● Von Neumann stored program computer model -- Father of computer science[2] Computation How well do our models reflect the salient features of the object or situation we wish to describe? 10/27/2013 RAJ KUMAR RAMPELLI 4
- 5. Complexity ● “Computational Complexity” ● ● Richard Edwin Stearns and Hartmanis On the computational complexity of algorithms paper at the fifth Annual Smposium1964 ● Abstract view of complexity classes ● Deterministic time: To define complexity classes ● Definition: DTIME(T(n)) is the set of all languages L for which there is a multitape Turing machine suth that the machine ● ● 10/27/2013 Answers the question “does input w belong to L” and Answers the question in at most T(|w|) moves where |w| is length of input w RAJ KUMAR RAMPELLI 5
- 6. Complexity model ● ● Analysis of algorithm: Upper bound on running time of the algorithm Complexity model: Place the problem into a complexity class using algorithm ● Easiness classes ● Speed-up theorem ● ● O(T(n)) not T(n) Result: n-> 8 ● DTIME(T(n)) = DTIME(c * T(n)) for all c>0 ● 10/27/2013 [T(n) * log(T(n))]/U(n) = 0 DTIME(U(n)) contains the language which is not in DTIME(T(n)) RAJ KUMAR RAMPELLI 6
- 7. Hardness Concepts ● ● Hard to show that there is no good method at all for solving a particular problem Stephen Cook introduced the concpets in 1971 ● ● ● ● ● NP-hardness NP-completeness Relate hardness of particular problem to the hardness of some set of difficulty problems Good Algorithm: takes only polynomial time Hardness concept is now “NP-completeness” ● NP: Non-deterministic Polynomial time ● Set of all decision problems 10/27/2013 RAJ KUMAR RAMPELLI 7
- 8. NP-complete ● ● Definition: If any of the NP-complete problems can be solved in polynomial time then any problem which has a nondeterministic polynomial algorithm can be solved in polynomial time. Prove: a problem X is NP-complete ● ● Take a problem Y already known to NPcomplete Polynomial reduction of any instance of Y into an instance of X having the same answer ● SAT: Satisfiability problem for Boolean formula ● PSPACE hardness ● 10/27/2013 Based on the concept of PSPACE RAJ KUMAR RAMPELLI completeness 8
- 9. PSPACE Set of all decision problems which can be solved by a Turing machine using a polynomial amount of space[3]. ● PSPACE-hardness > NP-hardness ● ● ● ● Might require exponential time even if it unexpectedly turns out that the NPcomplete problems can be solved in polynomial time PSPACE-Complete problems: Hardest problems in PSPACE 1st problem in PSPACE completeness ● Figure: A representation of the relation among complexity classes, which are subsets of each other. 10/27/2013 No reason in the sense that PSPACE prob require more time QSAT: Deciding if a quantified CNF formula is true RAJ KUMAR RAMPELLI 9
- 10. Power Index ● R.E.Stearns and Harry Hunt III [4] ● Problem has power index k if it can be solved in time 2^nk ● k=0 : Problems with polynomial algorithms ● k>0 : Problems with exponential algorithms ● ● ● NP-complete problem has k>0 and P-complete problem has k=0. SO, P != NP Theorem: If L1 has k value r and L1 redicuble to L2 by polynomial reduction time reduction of size ns, then L2 has k value atleat n/s Example: Standard reduction from SAT to CLIQUE [5] have n2 size and no better reduction is known. 10/27/2013 RAJ KUMAR RAMPELLI 10
- 11. EXPSPACE Subdivided by power Index 10/27/2013 RAJ KUMAR RAMPELLI 11
- 12. Observations ● All NP-complete problems are not equally hard ● All PSPACE-complete problems are not equally hard ● ● All PSPACE-complete problems can be easier than an NP-complete problem Even if SAT does require 2Ө(n) time, the possibility of remains that many NP-complete problems of practical interest may require a lot less time 10/27/2013 RAJ KUMAR RAMPELLI 12
- 13. References 1) Richard Stearns: “http://en.wikipedia.org/wiki/Richard_Stearns_(computer_scient ist)” 2) Von Neaumann Architecture: “http://en.wikipedia.org/wiki/Von_Neumann_architecture” 3) PSPACE: “http://en.wikipedia.org/wiki/PSPACE” 4) R. E. Steams and H. B. Hunt III “Power Indices and Easier Hard Problems”, Mathematical system theory 23, (1990) 5) Karp, R.M. “Reducibility among combinatorial problems”, Complexity of Computer Computations, Plenum, N.Y. 1972 10/27/2013 RAJ KUMAR RAMPELLI 13