Spring2009ModelChecking

Spring2009ModelChecking - CSE 541 - Logic in Computer...

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Unformatted text preview: CSE 541 - Logic in Computer Science Solutions for Selected Exercises on Model Checking Exercise 3.7.1 . Let S be the set { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } and H 1 , H 2 , and H 3 be functions of type P ( S ) → P ( S ) defined by: H 1 ( Y ) = Y- { 1 , 4 , 7 } H 2 ( Y ) = { 2 , 5 , 9 } - Y H 3 ( Y ) = { 1 , 2 , 3 , 4 , 5 } ∩ ( { 2 , 4 , 8 } ∪ Y ) for all Y ⊆ S . (a) First note that, for all sets X , Y , and Z , if X ⊆ Y then X ∪ Z ⊆ Y ∪ Z and X ∩ Z ⊆ Y ∩ Z . Consequently, H 1 and H 3 are monotone functions (as both can be defined via set union and intersection, e.g., H 1 ( Y ) = Y ∩ { 2 , 3 , 5 , 6 , 7 , 9 , 10 } ). The function H 2 is not monotone. For instance, ∅ ⊆ { 2 , 5 , 9 } , but H 2 ( ∅ ) = { 2 , 5 , 9 } 6⊆ ∅ = H 2 ( { 2 , 5 , 9 } . (b) Since H 3 is monotone, we can use the Fixed Point Theorem to infer that H 10 3 ( ∅ ) and H 10 3 ( S ) are least and greatest fixed points, respectively, of H 3 . The caluclation of these fixed points yields...
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Spring2009ModelChecking - CSE 541 - Logic in Computer...

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