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Symbolic Model Checking Revision Slides Dr. Eng. Amr T. Abdel-Hamid NETW 703 Winter 2012 Network Protocols Slides based on slides of: Jim Kurose, Keith Ross, “Computer Networking: A Top Down Approach Featuring the Internet”, 2nd edition, Addison-Wesley, July 2002. Jiangchuan (JC) Liu, Assistant Professor, SFU & others

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Dr. Amr Talaat Netw 703 Network Protocols Functionals Now, we can think of all temporal operators also as functions fr om sets of states to sets of states For example: or if we use the set notation AX p = (S - EX(S - p)) LogicSet p q p q p q p q p S – p False TrueS

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Dr. Amr Talaat Netw 703 Network Protocols Fixpoint Characterizations Fixpoint CharacterizationEquivalences AG p = y. p AX y AG p = p AX AG p EG p = y. p EX y EG p = p EX EG p AF p = y. p AX y AF p = p AX AF p EF p = y. p EX y EF p = p EX EF p A(pUq) = y. q A (p X (y)) A(pUq)=q (p AX (p AU q)) E(pUq) = y. q E (p X (y)) E(pUq) = q (p EX (p EU q))

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Dr. Amr Talaat Netw 703 Network Protocols EF Fixpoint Computation EF p = y. p EX y is the limit of the sequence: , p EX , p EX(p EX ), p EX(p EX(p EX )),... which is equivalent to , p, p EX p, p EX (p EX (p) ),...

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Dr. Amr Talaat Netw 703 Network Protocols EF Fixpoint Computation s2s1s4s3 p p Start 1 st iteration p EX = {s1,s4} EX( )= {s1,s4} ={s1,s4} 2 nd iteration p EX(p EX ) = {s1,s4} EX({s1,s4})= {s1,s4} {s3}={s1,s3,s4} 3 rd iteration p EX(p EX(p EX )) = {s1,s4} EX({s1,s3,s4})= {s1,s4} {s2,s3,s4}={s1,s2,s3,s4} 4 th iteration p EX(p EX(p EX(p EX ))) = {s1,s4} EX({s1,s2,s3,s4})= {s1,s4} {s1,s2,s3,s4} = {s1,s2,s3,s4}

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Dr. Amr Talaat Netw 703 Network Protocols EF Fixpoint Computation p EF(p)states that can reach p p EX(p) EX(EX(p))... EF(p) states that can reach p p EX(p) EX(EX(p)) ... EF(p)

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Dr. Amr Talaat Netw 703 Network Protocols Greatest Fixpoint Given a monotonic function F, its greatest fixpoint is the least upp er bound (lub) of all the extensive elements: y. F y = { y | F y y } The greatest fixpoint y. F y is the limit of the following sequenc e (assuming F is -continuous): S, F S, F 2 S, F 3 S,... If S is finite, then we can compute the greatest fixpoint using the above sequence

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Dr. Amr Talaat Netw 703 Network Protocols EG Fixpoint Computation Similarly, EG p = y. p EX y is the limit of the sequence: S, p EX S, p EX(p EX S), p EX(p EX (p EX S)),... which is equivalent to S, p, p EX p, p EX (p EX (p) ),...

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Dr. Amr Talaat Netw 703 Network Protocols EG Fixpoint Computation s2s1s4s3 pp p Start S = {s1,s2,s3,s4} 1 st iteration p EX S = {s1,s3,s4} EX({s1,s2,s3,s4})= {s1,s3,s4} {s1,s2,s3,s4}={s1,s3,s4} 2 nd iteration p EX(p EX S) = {s1,s3,s4} EX({s1,s3,s4})= {s1,s3,s4} {s2,s3,s4}={s3,s4} 3 rd iteration p EX(p EX(p EX S)) = {s1,s3,s4} EX({s3,s4})= {s1,s3,s4} {s2,s3,s4}={s3,s4}

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Dr. Amr Talaat Netw 703 Network Protocols EG Fixpoint Computation EG(p) EG(p) states that can avoid reaching pp EX(p) EX(EX(p))... EG(p) states that can avoid reaching p p EX(p) EX(EX(p)) ...

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Dr. Amr Talaat Netw 703 Network Protocols Example 11/80 1 2 3 4 5 6 a,b c b,c a dc For the FSM below, formally check the following properties, using Fixpoint Theorm: AG(a ∨c ∨b) AF(a b) If failed show the subset of the design the property holds for as well as the counter example S = {1,2,3,4,5,6}, AP = {a,b,c,d}, R = {(1,2), (1,3),(2,3), (3,4), (4,4), (4,5), (5,2), (2,6), (6,1)} L(1) = {a,b}, L(2) = {c}, L(3) = {b,c}, L(4) = {a}, L(5) = {c}, L(6) = {d}

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Dr. Amr Talaat Netw 703 Network Protocols Example (cont.) Remember that: H(a ∪ b) = H(a) ∪ H(b) ∪ H(c) ={1,4} ∪ {2,3,5} ∪ {1,3} = {1,2,3,4,5} AG(a ∨ c ∨ b) = AG p = y. p AX y = y. p AX y AX p = EX( p) I0 S = {1,2,3,4,5,6} I1 {1,2,3,4,5} ∩ S = {1,2,3,4,5} ∩ {1,3,4,5,6} = {1,2,3,4,5} I2 {1,2,3,4,5} ∩ AX(1,2,3,4,5) = {1,2,3,4,5} ∩ {1,3,4,5,6} = {1,3,4,5} This is because that : AX(1,2,3,4,5) = EX( (1,2,3,4,5)) = EX(6) = (2) = S- {2 } = {1,3,4,5,6} I3 {1,2,3,4,5} ∩ AX(1,3,4,5) = {1,3,4,5} This is because that : AX(1,3,4,5) = EX( (1,3,4,5)) = **** I3 = I2 H(AG(a ∨ b ∨ c)) = {1,3,4,5} The property does not hold, except for the above states, and it is clear that s tates {2,6} can be considered as counter examples. state 6 does not contain neither a,c,b and state 2 does not have a proceedin g one on one of its pathes path (2,6) 12/80

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Dr. Amr Talaat Netw 703 Network Protocols Example (AF(ab)) 13/80

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