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.

1. IT'S TIME TO
RECONSIDER TIME
-Richard Edwin Stearns
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2. Outline
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Introduction
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Interests
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Complexity
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Deterministic Time
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Complexity Model
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Hardness concepts
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NP-complete
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PSPACE
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Power Index
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Observations
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References
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3. 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]
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Born 5 July 1936
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4. 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?
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5. Complexity
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“Computational Complexity”
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Richard Edwin Stearns and Hartmanis
On the computational complexity of algorithms paper at the fifth Annual
Smposium1964
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Abstract view of complexity classes
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Deterministic time: To define complexity classes
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Definition: DTIME(T(n)) is the set of all languages L for which there is a
multitape Turing machine suth that the machine
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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
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6. 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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[T(n) * log(T(n))]/U(n) = 0
DTIME(U(n)) contains the language which is not in DTIME(T(n))
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7. Hardness Concepts
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Hard to show that there is no good method at all for solving a particular
problem
Stephen Cook introduced the concpets in 1971
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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”
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NP: Non-deterministic Polynomial time
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Set of all decision problems
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8. NP-complete
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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
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Take a problem Y already known to NPcomplete
Polynomial reduction of any instance of Y
into an instance of X having the same
answer
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SAT: Satisfiability problem for Boolean formula
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PSPACE hardness
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Based on the concept of PSPACE
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completeness
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9. 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.
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No reason in the sense that
PSPACE prob require more time
QSAT: Deciding if a quantified
CNF formula is true
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10. 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.
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11. EXPSPACE Subdivided by power Index
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12. Observations
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All NP-complete problems are not equally hard
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All PSPACE-complete problems are not equally hard
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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
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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
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