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Mastering Logic: A Comprehensive Analysis of Thinking and Reasoning Patterns

Dive deep into the essence of logic through syllogisms and critical analysis. Understand efficiency concepts and Amdahl's Law to enhance problem-solving skills. Learn to calculate speedup and apply analytical models effectively in the realm of computing.

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Mastering Logic: A Comprehensive Analysis of Thinking and Reasoning Patterns

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  1. CS 584

  2. Logic The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basis of logic is the syllogism, consisting of a major and minor premise and a conclusion.

  3. Example • Major Premise: Sixty men can do a piece of work sixty times as quickly as one man. • Minor Premise: One man can dig a post-hole in sixty seconds. • Conclusion: Sixty men can dig a post-hole in one second.

  4. Performance Analysis:"Tar Baby" • Ask the right questions • Questions to consider • What is time? • What is work? • Objectivity is the key • Take a step back from your program

  5. Performance Analysis Statements • There is always a trade-off between time and solution quality. • We should compare the quality of the answer for a given execution time. • For any performance reporting, find and clearly state the quality measure.

  6. Efficiency • Efficiency is defined as speedup/P • With superlinear speedup efficiency > 1 • Does cache make a processor work at 110%? • Why is communication not considered work but rather overhead?

  7. Speedup • Conventional speedup is defined as the reduction in execution time. • Consider running a problem on a slow parallel computer and on a faster one. • Same serial component • Speedup will be lower on the faster computer.

  8. Speedup and Amdahl's Law • Conventional speedup penalizes faster absolute speed. • Assumption that task size is constant as the computing power increases results in an exaggeration of task overhead. • Scaling the problem size reduces these distortion effects.

  9. Solution • Gustafson introduces scaled speedup. • Scale the problem size as you increase the number of processors. • Calculated in two ways • Experimentally • Analytical models

  10. Traditional Speedup C ( N ) = 1 Speedup C ( N ) P C1 is complexity (time) taken on a single processor CP is complexity (time) taken on P processors

  11. C ( PN ) = 1 Speedup C ( PN ) P Scaled Speedup C1 is complexity (time) taken on a single processor CP is complexity (time) taken on P processors

  12. Experimental Scaled Speedup • Keep the ratio N/P constant between single processor case and many processor case when testing • Example:Calculate the speedup for 8, and 16 processors. • N/P = 256 • How big should the problem be?

  13. Using analytical models • Examine the control flow of the algorithm • Find a general algebraic form for the complexity (execution time). • Fit the curve with experimental data. • If the fit is poor, find the missing terms and repeat. • Calculate the scaled speedup using formula.

  14. + C ( PN ) 2 12 ( 4 ( 128 )) 6146 = = = 1 3 . 93 + + C ( PN ) 4 12 ( 4 ( 128 ) / 4 ) 5 ( 4 ) 1560 P Example • Serial Time = 2 + 12 N seconds • Parallel Time = 4 + 12 N/P + 5P seconds • Let N/P = 128 • Scaled Speedup for 4 processors is:

  15. Traditional Speedup ideal Speedup measured Number of Processors

  16. Scaled Speedup Large Problem ideal Speedup Medium problem Small problem Number of Processors

  17. Assignment • Problems on the web • Create a model for your program • Use the model to calculate • traditional speedup • scaled speedup • Experimentally calculate the values • Compare the results.

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