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Outline

Principles of model checking

2008

Abstract

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References (443)

  1. C's. (c) Change your automaton from part (a) such that between any two successive A's an odd number of symbols from the set { B, C } may occur.
  2. Same exercise as in (c), except that now an odd number of B's and an odd number of C's must occur between any two successive A symbols.
  3. Exercise 4.11. Depict an NBA for the language described by the ω-regular expression (AB + C) * ((AA + B)C) ω + (A * C) ω .
  4. Prove that for any PCTL formula there exists an equivalent PCTL formula in PNF. Exercise 10.11. Let M be a finite Markov chain over AP, s a state of M and a, b ∈ AP: Show that s |= P =1 (a U b) iff s |= ∀( (∃(a U b)) W b ).
  5. Exercise 10.12. Provide deterministic Rabin automata for the following LTL formulae: (a → ♦b), ¬ (a → ♦b), and a U ( b).
  6. Exercise 10.13. Let A 1 and A 2 be DRA over the same alphabet. Define the DRA A = A 1 ∪A 2 such that L ω (A) = L ω (A 1 ) ∪ L ω (A 2 )
  7. and size(A) = O poly(size(A 1 ), size(A 2 )) .
  8. Exercise 10.14. Consider the Markov chain M in Figure 10.22 (page 899), and let the labeling be given by L(s 2 ) = L(s 3 ) = L(s 4 ) = { a } and L(s) = ∅ for the remaining states. Question: Compute the probability Pr M (ϕ) for the LTL formula ϕ = ♦ a. (Hint: Construct a DRA A for ϕ and perform a quantitative analysis in M ⊗ A.)
  9. Exercise 10.15. Prove the following statement. For finite Markov chain M = (S, P, ι init , AP, L) and ω-regular property P over AP such that Pr M (P ) > 0, there exists a finite path fragment π = s 0 s 1 . . . s n in M with ι init (s 0 ) > 0 such that almost all paths in the cylinder set Cyl( π) fulfill
  10. P , i.e., Pr M {π ∈ Cyl( π) | trace(π) ∈ P } = P( π).
  11. Exercise 10.16. Show that there is no PCTL formula that is equivalent to the PCTL * formula P 0.5 ( a) where a is an atomic proposition. Exercise 10.17. Consider the Markov chain M, which is given by Bibliography
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