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march26 - CIS540 Spring2014 http/www.seas.upenn.edu/~cis540...

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CIS 540 Principles of Embedded Computation Spring 2014  http://www.seas.upenn.edu/~cis540/ Instructor: Rajeev Alur [email protected]
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Tableau Construction Overview CIS 540 Principles of Embedded Computation; Spring 2014;  Lecture March 26 e & ~f e & ~f , , N , N , e, ~f, , , N , f, , , e, Automaton/tableau state: Collection of “relevant” LTL formulas  Intended meaning: All the formulas in a state must hold on every infinite path starting at a state Local consistency rules ensure that for every “non-atomic” formula  , the state contains  additional formulas ensuring that   holds Transition rules ensure that (1) every atomic formula holds at current time, and (2) all Next- formulas are propagated to next state
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Tableau Construction Example CIS 540 Principles of Embedded Computation; Spring 2014;  Lecture March 26 A f, N A f, f f e e e Consider    (Eventually e) Until (Always f) N , E e , e N , E e, N E e A f, N A f, f, E e, N E e A f, N A f, f, E e, e e & f f f
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Tableau Construction Example CIS 540 Principles of Embedded Computation; Spring 2014;  Lecture March 26 A f, N A f, f f e e e N , E e , e N , E e, N E e A f, N A f, f, E e, N E e A f, N A f, f, E e, e e & f f f Ensuring fulfillment of E e: Accepting set: {q 0 , q 1 , q 4 } q 0 q 1 q 4 q 2 q 3 Ensuring fulfillment of (E e) U (A f): Accepting set: {q 0 , q 3 , q 4 }
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Formal Construction CIS 540 Principles of Embedded Computation; Spring 2014;  Lecture March 26 Given an LTL-formula  , define Sub( ), called the closure of formula These are all the formulas that are relevant to evaluation of  Contains all the syntactic subformulas of  If it contains  Always  , should also contain  Next Always  If it contains  Eventually  , should also contain  Next Eventually  If it contains    , should also contain  Next (   ) What is Sub(  Always Eventually e & Next f ) Always Eventually e & Next f, Always Eventually e, Next f, f,  Eventually e, Next Always Eventually e, e, Next Eventually e  } Note: Number of formulas in Sub( ) is linear in length of  Note: It suffices to consider only Next and Until operators (why?) 
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Tableau States CIS 540 Principles of Embedded Computation; Spring 2014;  Lecture March 26 A state of the desired automaton is a subset of Sub( ) that satisfies some consistency rules Should not contain both a formula   and its negation   Contains   &    exactly when it contains both   and   Contains   |    exactly when it contains at least one of   and  If it contains  Always   then must have both   and  Next Always 
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