8 Conclusion 7.4 Scenario B: Incremental Model Checking

A simple method for undecidability proofs and some applications. We have shown how to use the Feferman{Vaught Theorem and its generalization to Monadic Second Order Logic in the context of model checking of transition systems. In MR95] we have analyzed the same method also for Transitive Closure and Fixed Point Logic. In the case of-sums the method yields considerable complexity reductions for model checking of MSOL{formulas in the case of incremental model checking. We concentrated here on model checking of properties of transition systems. But the method is much more general and in future work we shall analyze other applications as well. Acknowledgements We are would like to thank S.Shelah, who gave us his manuscript She92]. The rst author had stimulating discussions with Y. Hirshfeld and E. Clarke in the early stages of this research. In later stages, B. Courcelle, A. Arnold, D.Niwi nski and O. Grumberg helped us in nding more examples to which we could apply our method. References AN92] A. Arnold and D. Niwi nski. Fixed point characterization of weak monadic logic deenable sets of trees. 15 In this case T new = P p (T (n); T (l)), where P p denotes polynom of degree p, and T old = T (l n). The question to answer is: when f(n l) > P p (f(n); f(l)). According to our assumptions we obtain: e g(nl) > a p (e pg(n) + e pg(l)). Assume that n = l. Then g(n 2) > p g(n) + ln2 + ln(a p). Assume that g(x) = ln 2 (x), then f(x) = x ln(x). In this case we obtain that: ln 2 (n 2) > p ln 2 (n) + ln2 + ln(a p) or ln 2 (n) > ln(2ap) 2?p. On the other hand, if we use some FOL-deenable logic, then Model Checking procedure is polynomial in A and in I and each A { too. In this case we do not obtain time gain. Assume that our aim is as follows: we need to change several times (let us denote the number of the times by l) some xed component of A called A {. Assume that each time we need to check whether A satisses some xed formula. We use the notations as above. Let T old be time to solve the present problem by the usually applied way. It should be clear that T old = …