IT'S TIME TO

RECONSIDER TIME

-Richard Edwin Stearns

RAJ KUMAR RAMPELLI

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Outline● Introduction

● Interests

● Complexity

● Deterministic Time

● Complexity Model

● Hardness concepts

● NP-complete

● PSPACE

● Power Index

● Observations

● References

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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]

Born 5 July 1936

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R.E.Stearns Interests

● Game Theory

● J. von Neumann and Morgenstern, Theory of Games and Economic Behaviour.

● Competition

● Computer Science

● Von Neumann stored program computer model -- Father of computer science[2]

● Computation

John von Neumann

How well do our models reflect the salient features of the object or situation we wish to describe?04/12/2023 RAJ KUMAR RAMPELLI 4

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

● 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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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

● DTIME(T(n)) = DTIME(c * T(n)) for all c>0

● O(T(n)) not T(n)

● Result: n-> [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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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

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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 NP-complete

● Polynomial reduction of any instance of Y into an instance of X having the same answer

● SAT: Satisfiability problem for Boolean formula

● PSPACE hardness

● Based on the concept of PSPACE completeness

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PSPACESet of all decision problems which can be solved by a Turing machine using a polynomial amount of space[3].

● PSPACE-hardness > NP-hardness

● No reason in the sense that PSPACE prob require more time

● Might require exponential time even if it unexpectedly turns out that the NP-complete problems can be solved in polynomial time

● PSPACE-Complete problems: Hardest problems in PSPACE

● 1st problem in PSPACE completeness

● QSAT: Deciding if a quantified CNF formula is trueFigure: A representation of the relation among

complexity classes, which are subsets of each other.

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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.

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RAJ KUMAR RAMPELLI

EXPSPACE Subdivided by power Index

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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

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References

1) Richard Stearns: “http://en.wikipedia.org/wiki/Richard_Stearns_(computer_scientist)”

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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