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# l4 - CS 267 Automated Verification Lecture 4 Fixpoints and...

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CS 267: Automated Verification Lecture 4: Fixpoints and Temporal Properties Instructor: Tevfik Bultan

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What is a Fixpoint (aka, Fixed Point) Given a function F : D D x D is a fixpoint of     if and only if (x) = x
Temporal Properties Fixpoints [Emerson and Clarke 80] Here are some interesting CTL equivalences: AG p = p AX AG p EG p = p EX EG p AF p = p AX AF p EF p = p EX EF p p AU q = q (p AX (p AU q)) p EU q = q (p EX (p EU q)) Note that we wrote the CTL temporal operators in terms of themselves and EX and AX operators

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Functionals Given a transition system T=(S, I, R), we will define functions from sets of states to sets of states F : 2 S 2 S For example, one such function is the EX operator (which computes the precondition of a set of states) EX : 2 S 2 S which can be defined as: EX(p) = { s | (s,s’) R and s’ p } Abuse of notation: I am using p to denote the set of states which satisfy the property p (i.e., the truth set of p)
Functionals Now, we can think of all temporal operators also as functions from sets of states to sets of states For example: AX p = ¬ EX( ¬ p) or if we use the set notation AX p = (S - EX(S - p)) Abuse of notation: I will use the set and logic notations interchangeably.

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