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Limits of Computation

by: Tamia Bernhard

Limits of Computation 22C 131

Tamia Bernhard
GPA 3.87


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This 22 page Class Notes was uploaded by Tamia Bernhard on Friday October 23, 2015. The Class Notes belongs to 22C 131 at University of Iowa taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/228055/22c-131-university-of-iowa in ComputerScienence at University of Iowa.


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Date Created: 10/23/15
Lecture Notes developed by Teodor Rus a ruscs uiowa edu The University of Iowa Department of Computer Science aThis notes have been developed by Teodor Rus They are copyrighted materials and may not be used in other course settings outside of the University of Iowa in their current form or modified form without the express written permission of one of the copyright holders During this course students are prohibited from selling notes to or being paid for taking notes by any person or commercial firm without the express written permission of one of the copyright holders Computation Theory p15 0 History 39 Current trends of computation theory 0 Problem solving methodology a Polya Four Steps MethodOIOQY 0 Problem Classification 0 probIem Solving and Areas of Computation Theory 0 Problem Solving Tools 0 ReIationship to Theory of Computation o How to Approach lt comPUtation Theory p25 The three sources of theory of computation are 1 Works performed by mathematicians such as Leibnitz and Pascal trying to mechanize calculus and logical deduction 2 Work performed by logicians such as Kurt Godel Alfred Tarski Alan Turing Alonzo Church A A Markov toward the development of a formal concept of algorithm 3 Development of computer based problem solving methodology Computation Theory p35 Gottfried Wilhelm Leibnitz German mathematician 1646 1716 Invented Calculus independent of Newton Leibnitz s notation is still used Discovered binary system Blaise Pascal French mathematician 1623 1662 Invented mechanical calculators Worked on Projective geometry and Probability Theory with Fermat Since 1654 to the end he did philosophy and theology Known to us through quotPascal languagequot Developed by Nicholas Wirth 1970 Kurt Godel Austrian American Logician 1906 BrUn Bruno Czech 1978 Princeton NJ Proved Incompleteness theorem thus challenging Hilbert s Program 1900 Alfred Tarski Polish American MathematicianLogician 1901 Warsaw Poland 1983 Berkeley Ca Invented the formal concept of truth and model theory Alan Turing English Mathematician 1912 1954 known as the father of CS Invented the formal concept of an algorithm Turing Machine Invented Enigma machine a codebreaker during Second World War Alonzo Church American Logician 1903 1995 Invented the Acalculus Known for Church s thesis everything that is computable is computable by a recursive functions TM Andrey Andreevich Markov Russian Mathematician 1903 1979 sun of AA Markov 1856 1922 Developed Russian school of constructive mathematics and logic Invented normal algorithms and proved NAZRFETMZAC Current trends Algorithm development and information computation technology IT Solving problems related to computation complexity Putting information technology on firm mathematical basis Developing domain specific languages DLS to express computations 0 Developing domain specific computer based problem solving methodology 1 Most of the current texts dedicated to the theory of computation disregard the influence computer based problem solving methodology hasd on the development of theory of computation 2 Provide please rationale for this situation 3 We will try to correct this in this offering of 220131 by choosing a textbook that can be best used to connect the theory of computation with problem solving methodology Unfortunately following tradition this class will have no programming component compmation Theory p125 Problem solving methodology 39 Problem paraphrasing GPoya Stanford 1956 to have a problem means to search consciously for some action appropriate to attain a given aim Solution to solve a problem means to find such an action See G Polya How To Solve It 1956 compmation Theory p135 Observations A problem is characterized by three principal components unknowns 0 data and 0 conditions To solve a problem means to find a way of determining the unknowns from the given data such that the conditions of the problem are satisfied compmation Theory p145 Example problem Find all real numbers 1 that for a given triple a b c of real numbers satisfy the relation 5132 951 C 0 Unknown real numbers 1 Data real numbers abc Conditions a 0 and 035132 bx c 0 Solving the problem Use the properties of the data and conditions to deduce a relation 1 fa b c a x 2 b x c O by problem statement a x 2 b x c by adding c to both terms of equality x 2 ba X ca by dividing both term by a x b2a 2 b 2 4ac4a 2 by using binomial identity x b2a sqrtb 2 4ac2a by taking square root x b sqrtb 2 4ac2a by adding b2a to equality Solutions 131 b bZ 4ac2a x2 b bZ 4ac2a compmation Theory p165 Dif culties in solving a problem In finding a way of determining unknowns this belongs to the very notion of a problem Citing Polya where there is no difficulty there is no problem It seems that what best characterizes human activity is solving problems by finding ways around or out of difficulties compmation Theory p175 Two classes of dif culties Difficulties that pertain to finding the action Difficulties that pertain to performing the ac on Example solving linear systems of equations 1 One can use Cramer s rule or Gaussian elimination method 2 One may know how and one can do it for small systems system size lt 10 3 One could do it for large systems too system size gt 1000 but perhaps she will not finish during her entire life compmation Theory pig5 Distinction between dif culties The distinction between the two classes of difficulties is in solution mechanisms Finding the way around difficulties in discovering the action seems to be a characteristic of intelligent human behavior What could one use to help this Finding the way around difficulties of performing the action can be done both creatively and mechanically What could one use to help this Mathematical methodology Polya s four steps methodology 1 Formalize the problem 2 Develop an algorithm to solve the problem 3 Perform the algorithm on the data characterizing the problem 4 Validate the solution Question is there any difference between this and computer based methodology compmation Theory p205 Computerbased methodology How do we solve problems using computers Computation Theory p215


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