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Widening Automata

Learn about the convergence of fixpoint computations and the approximation technique of widening automata. Explore state equivalence and exactness results for precise analysis of automata transitions.

jmcclinton
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Widening Automata

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  1. Widening Automata

  2. Verification of assertionsForward fixpoint computation • Set of initial states I • Postcondition function Post() • Error states I . . . Post(I) ? Post(Post(I)) E

  3. Verification of assertionsBackward fixpoint computation • Set of initial states I • Precondition function Pre() • Error states E E . . . Pre(E) ? Pre(Pre(E)) I

  4. Fixpoints may not converge • For infinite state systems fixpoint computations may • not converge at all • require a large number of iterations • Widening is a approximation technique that helps a fixpoint computation converge

  5. A widening operator • Idea: Instead of computing a sequence of automata A1, A2, … where Ai+1=Aipost(Ai), compute A’1, A’2, … where A‘i+1=A’i(A’ipost(A’i)) • By definition AB  AB • The goal is to find a widening operator  such that: • The sequence A’1, A’2, … converges • It converges fast • The computed fixpoint is as close as possible to the exact set of reachable states

  6. Widening Automata • Given automata A and A’ we want to compute AA’ • We say that states k and k’ are equivalent (kk’) if either • k and k’ can be reached from either initial state with the same string (unless k or k’ is a sink state) • or, the languages accepted from k and k’ are equal • or, for some state k’’, kk’’ and k’k’’ • The states of AA’ are the equivalence classes of 

  7. Example 0 1 0,1 X X 1 0 0,1 0 0 1 0 1 1 3 0 1 0,1 0 1 0 0,1 0 X 0,1 1 1 X X 0 3 0 1 2 1 0 2 0 1 0,1 3 1 0

  8. Example 0 1 0,1 0 1 0,1 X X 1 0 0,1 0 X 0,1 1 X X 1 0 0,1 0 1 0 3 0 1 1 0 2 0 1 0,1 1 0 0 X 0,1 1 0 X X = 1 0 0 1 0,1 0,1 2 3 0 1

  9. Example 0 0, X 1 1 0 X X 1 0 0 1 0,1 0 2 0 0 1 2 0 1 1 1 3 2 4 0 X 0,1 0 X 1 1 0 1 0 X X 0 0 2 1 0 1 0,1 0 3 4 0 1 2 0 1 3 1 0 1 0 1 4 X 0

  10. Example 0 X 0,1 0 0, X 1 1 0 X X 1 0 X X X 1 1 0 0 0 2 1 0 0 1 0,1 0 1 0,1 0 1 2 0 3 4 0 1 0 1 0 1 1 0 1 0 1 X 0 0 X 0,1 X 1 0,0 = 1 0 0,2 1,3 0 1

  11. An exactness result (some definitions) • An automaton (Q,,,q0,F) is called state-disjoint if for all qiqjQ, L(qi)L(qj)= • An automaton (Q1,,1,q01,F1) is called weakly equivalent to (Q2,,2,q02,F2) iff there exists f: Q1 Q2, such that: • f(q01)=q02 • f(1(q,))=2(f(q), ) for all qQ1 and  • f(q) F2 for all q F1

  12. An exactness result • If • a least fixpoint is represented by a state-disjoint automaton A • and, the first automaton As in the approximate sequence is weakly equivalent to A • then • the approximate sequence converges to the exact least fixpoint

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