# l10 - CS 267 Automated Verification Lecture 10 LTL Buchi...

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CS 267: Automated Verification Lecture 10: LTL Buchi Automata Translation, Nested Depth First Search Instructor: Tevfik Bultan

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LTL We are going to discuss LTL to Buchi automata translation First let’s recall LTL semantics I will also add a new operator called R (release) to make the translation to Buchi automata easier
LTL Semantics Given an execution path x and LTL properties p and q x |= p iff L(x 0 , p) = True, where p AP x |= ¬ p iff not x |= p x |= p q iff x |= p and x |= q x |= p q iff x |= p or x |= q x |= X p iff x 1 |= p x |= G p iff for all i 0, x i |= p x |= F p iff there exists an i 0 such that x i |= p x |= p U q iff there exists an i 0 such that x i |= q and for all 0 j < i, x j |= p x |= p R q iff for all j 0, if for all 0 i < j, x i |= p then x j |= q

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LTL Equivalences Given an LTL formula convert it to positive normal form: Negations are only applied to atomic propositions (there is no negation outside of a temporal operator) Use the following equivalences to translate the LTL formulas to positive normal form: ¬ (p U q) ¬ p R ¬ q ¬ (p R q) ¬ p U ¬ q ¬ (X p) X ¬ p ¬ (p R q) ¬ p U ¬ q F p true U p G p false R p
LTL Buchi Automata Translation [Gerth, Peled, Vardi, Wolper 95] Each state of the automata will store a set of properties that should be satisfied on paths starting at that state These properties will be stored in lists Old and New where Old means already processed and New means still needs to be processed Each state will also store a set of properties which should be satisfied on paths starting at the next states of that state These properties will be stored in the list Next The incoming transitions for a state will be stored in the list Incoming

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LTL Buchi Automata Translation We will start with a node which has the input LTL property in its New list We will process the formulas in the New list of each node one by one When we have f U g in the New list we will use f U g g (f X ( f U g)) When we have f R g in the New list we will use f R g g (f X (f R g))
LTL Buchi Automata Translation When we process a formula from a node we will either replace the node with a new node or we will replace it with two new nodes (i.e., we will split it to two nodes) When a node q is replaced by a node q’ we will have (Old(q) New(q) X Next(q)) (Old(q’) New(q’) X Next(q’)) When a node q is split into two nodes q 1 and q 2 we will have (Old(q) New(q) X Next(q)) ( (Old(q 1 ) New(q 1 ) X Next(q 1 )) (Old(q 2 ) New(q 2 ) X Next(q 2 )) ) Means conjunction of all the formulas in these lists

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Translate(f) { expand([Incoming:={init}, Old:= , New:={f}, Next:= ], ) } Expand(q, NodeList) { If New(q) is empty then if there exists a node r in NodeList s.t. Old(r) = Old(q) and Next(r) = Next(q)
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• Fall '09
• bultan
• Depth-first search, connected component, LTL Buchi Automata Translation, buchi automata translation, LTL Buchi Automata

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