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A Theoretical Investigation of Generalized Voters for Redundant Systems Class: CS791F - Fall 2005 Professor : Dr. Bojan Cukic Student: Yue Jiang

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A Theoretical Investigation of Generalized Voters for Redundant Systems Introduction Different kinds of generalized voters Comparison of generalized voters Conclusions

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1 Introduction Objective: introduction to and analysis of voting techniques in fault tolerant systems Related work: (1) majority voting; (2) adaptive or weighted voter; (3) median selection method This paper: 1) Formalized majority voter 2) Generalized median voter 3) Formalized plurality voter 4) Weighted averaging voter

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2 Generalized voters Formalized majority voter - select majority Generalized median voter - select median Formalized plurality voter - partition the set of output based on metric equality and select the output from largest group Weighted averaging technique - combines the output in a weighted average Assumption: N-versions software, N is odd

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2 Generalized voters-Formalized majority voter Definition: If more than half of the version outputs agree, this common output becomes the output of the N-version structure. Agree is not the same, i.e. output is real value. A threshold, ε, is needed.

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2 Generalized voters-Formalized majority voter x 1 = x 2 = x 3 = x 4 = x 5 = ε = |x 1 - x 3 | = |x 1 - x 4 | = |x 2 - x 5 | = ( x 1, x 3, x 4 ) ( x 2, x 5 ) Example 1 Result: x 1 or x 3 or x 4

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2 Generalized voters-Formalized majority voter x 1 = (2.1350, , ) x 2 = (2.1340, , ) x 3 = (2.1376, , ) ε = d 2 (x 1, x 2 ) = ( ) 2 + [ ( )] 2 + ( ) 2 d(x 1, x 2 ) = > ε d(x 1, x 3 ) = > ε d(x 2, x 3 ) = < ε Example 2 Result: x 2 or x 3

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2 Generalized voters-Formalized majority voter x 1 = (0, 0, 0, 1, 0, 0, 0, 0) x 2 = (0, 1, 0, 0, 0, 0, 0, 0) x 3 = (0, 0, 0, 1, 0, 0, 0, 0) ε = d(x 1, x 2 ) =2 > ε d(x 1, x 3 ) =0 = ε d(x 2, x 3 ) =2 > ε Example 3 Result: x 1 or x 3

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2 Generalized voters-Formalized majority voter x 1 = x 2 = x 3 = x 4 = x 5 = 4954 ε = 0 {x 1, x 2, x 4 } {x 3, x 5 } Example 4 Result: x 1 or x 2 or x 4 d(x 1, x 2 )= | x 1 - x 2 | mod d(x 1, x 2 ) =|x 1 - x 2 | mod = 0 = ε d(x 1, x 3 ) = |x 1 – x 3 | mod = > ε d(x 1, x 4 ) = |x 1 – x 4 | mod = 0 = ε d(x 1, x 5 ) = |x 1 – x 5 | mod = > ε d(x 2, x 3 ) = > ε d(x 2, x 4 ) = 0 = ε d(x 2, x 5 ) = > ε d(x 3, x 4 ) = > ε d(x 3, x 5 ) = 0 = ε d(x 4, x 5 ) = > ε

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2 Generalized voters-Generalized median voter Select a median value from the set of N outputs.

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2 Generalized voters-Generalized median voter x 1 = x 2 = x 3 = x 4 = x 5 = ε = |x 1 - x 2 | = |x 1 - x 3 | = |x 1 - x 4 | = |x 1 - x 5 | = |x 2 - x 3 | = |x 2 - x 4 | = |x 2 - x 5 | = |x 3 - x 4 | = |x 3 - x 5 | = (!) |x 4 - x 5 | = ( x 1, x 2, x 4 ) Example 5 (1) Result: x 4 |x 1 - x 2 | = (!) |x 1 - x 4 | = |x 2 - x 4 | = => x4x4

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2 Generalized voters-Generalized median voter x 1 = (2.1350, , ) x 2 = (2.1340, , ) x 3 = (2.1376, , ) d(x 1, x 2 ) = d(x 1, x 3 ) = (!) d(x 2, x 3 ) = Result : x 2 Example 6 (2)

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2 Generalized voters-Generalized median voter Example 3 x 1 = (0, 0, 0, 1, 0, 0, 0, 0) x 2 = (0, 1, 0, 0, 0, 0, 0, 0) x 3 = (0, 0, 0, 1, 0, 0, 0, 0) ε = d(x 1, x 2 ) =2 (!) d(x 1, x 3 ) =0 d(x 2, x 3 ) =2 Example 7 (3) Majority Result: x 1 or x 3 d(x 1, x 2 ) =2 > ε d(x 1, x 3 ) =0 = ε d(x 2, x 3 ) =2 > ε Median Result: x 1 or x 3

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2 Generalized voters-Generalized median voter x 1 = x 2 = x 3 = x 4 = x 5 = 4954 ε = 0 Example 8 (4) Median Result: x 1 or x 2 or x 4 d(x 1, x 2 ) = d(x 1, x 4 ) = d(x 2, x 4 ) = d(x 3, x 5 ) =0 d(x 1, x 3 ) = d(x 1, x 5 ) = d(x 2, x 3 ) =d(x 2, x 5 ) = d(x 3, x 4 ) = d(x 4, x 5 ) = Majority Result: x 1 or x 2 or x 4

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2 Generalized voters - formalized plurality voter Construct a partition V 1, …, V k of A where for each i the set V i is maximal with respect to the property that for any x, y in V i d(x,y)<= ε If there exist a set V a from V 1, …, V k, such that |V a | > | V i | for any V i <> V a, randomly select an element from V a is the voter output.

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Suppose N versions of software with outputs in x produce the outputs x 1, x 2, … x N. Let w 1, w 2, … w N denote the weight. Then Define a new element of x by 2 Generalized voters – weighted averaging voter i=1 N W 1 =1 i=1 N W 1 X i X=

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2 Generalized voters – weighted averaging voter Weight w i can be a priori knowledge Weight w i can be calculated dynamically, i.e. by and where s= a is a fixed constant for scaling.

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A Theoretical Investigation of Generalized Voters for Redundant Systems Introduction Different kinds of generalized voters Comparison of generalized voters Conclusions

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3 Formalized majority vs. formalized plurality 1. Majority: result > half; plurality not necessary, relative large. 2. Majority is a special kind of plurality.

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3 Formalized majority vs. formalized plurality x 1 = x 2 = x 3 = x 4 = x 5 = x 6 = x 7 = ε =0.01 Formalized majority {x 1, x 2, x 5, x 7 } >= (N+1)/2 {x 4, x 6 } {x 3 } Formalized plurality {x 1, x 4, x 5 } {x 2, x 7 } {x 3 } {x 6 }

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3 Formalized majority vs. generalized median The output produced by the formalized majority voting algorithm always contain the output of generalized median voter

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3 Formalized majority vs. generalized median x 1 = x 2 = x 3 = x 4 = x 5 = ε = Example 1-5 Majority Result: x 1 or x 3 or x 4 x3x3 Median result:

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3 Formalized majority vs. formalized median x 1 = (2.1350, , ) x 2 = (2.1340, , ) x 3 = (2.1376, , ) ε = Example 2-6 Majority result: x 2 or x 3 Median result: x 2 d(x 1, x 2 ) = > ε d(x 1, x 3 ) = > ε d(x 2, x 3 ) = < ε

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3 Formalized majority vs. formalized median x 1 = (0, 0, 0, 1, 0, 0, 0, 0) x 2 = (0, 1, 0, 0, 0, 0, 0, 0) x 3 = (0, 0, 0, 1, 0, 0, 0, 0) ε = Example Majority Result: x 1 or x 3 Median result: x 1 or x 3 d(x 1, x 2 ) =2 > ε d(x 1, x 3 ) =0 = ε d(x 2, x 3 ) =2 > ε

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3 Formalized majority vs. formalized median x 1 = x 2 = x 3 = x 4 = x 5 = 4954 ε = 0 Example Majority Result: x 1 or x 2 or x 4 the same as Median voter d(x 1, x 2 )= | x 1 - x 2 | mod d(x 1, x 2 ) =|x 1 - x 2 | mod = 0 = ε d(x 1, x 3 ) = |x 1 – x 3 | mod = > ε d(x 1, x 4 ) = |x 1 – x 4 | mod = 0 = ε d(x 1, x 5 ) = |x 1 – x 5 | mod = > ε d(x 2, x 3 ) = > ε d(x 2, x 4 ) = 0 = ε d(x 2, x 5 ) = > ε d(x 3, x 4 ) = > ε d(x 3, x 5 ) = 0 = ε d(x 4, x 5 ) = > ε

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3 Formalized majority vs. formalized median x 1 = 101 x 2 = 102 x 3 = 103 x 4 = 104 x 5 = 105 ε = 1 Majority and plurality: cannot make a decision. Median result: x 3

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4 Conclusion Formalized majority voter - select majority Generalized median voter - select median – result is contained in the formalized majority voter Formalized plurality voter - select relative larger output Weighted averaging technique - dynamically combines the output in a weighted average

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