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
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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
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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
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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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