March23Lecture_Undone

# March23Lecture_Undone - CIS 540 Principles of Embedded...

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CIS 540 Principles of Embedded Computation Spring 2017 Instructor: Rajeev Alur [email protected]

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Buchi Automaton M Definition V: set of Boolean input variables Finite set Q of states Set Init of initial states Set F of accepting states Set of edges/transitions, where each edge is of the form q – Guard q’ where Guard is a Boolean-valued condition over input vars V Given an input trace = v 1 , v 2 , v 3 , … over V, an accepting run/execution of M over is an infinite sequence of states q 0 , q 1 , q 2 , … such that 1. State q 0 is initial 2. For each i, there exists an edge q i -Guard q i+1 such that input v i satisfies Guard 3. There are infinitely many positions i such that state q i is in F The automaton M accepts the input trace if there exists an accepting run of M over CIS 540 Spring 2017; Lecture March 23
Buchi Automata: More Examples e Eventually e ~ e e Eventually e CIS 540 Spring 2017; Lecture March 23

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Buchi Automata Examples a e c b f Eventually [e & Next Eventually f] Eventually e | Eventually f a e c f b CIS 540 Spring 2017; Lecture March 23
Nondeterministic Buchi Automaton e Persistently e Can we construct an equivalent deterministic Buchi automaton ? No! Nondeterminism is sometimes necessary! CIS 540 Spring 2017; Lecture March 23

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Omega-Regular Languages The language of a Buchi automaton is the set of traces it accepts Such languages are called omega-regular Well-developed theory of omega-regular languages Analogous the classical theory of regular languages (i.e. languages of finite strings of input characters accepted by finite automata) Relevance to us: Given an LTL formula , there is an algorithm to construct a Buchi automaton M that accepts exactly those traces that satisfy the formula CIS 540 Spring 2017; Lecture March 23
System Safety Monitors Monitor Is there an execution of the System for which the Monitor can enter an error state? Monitor is designed so that such an execution indicates a bug!

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• Fall '09
• ALUR
• U.S. state, Quantification, Universal quantification, Existential quantification, Automata theory, Buchi automaton

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