tailieunhanh - CONCUR 2004 – Concurrency Theory- P11

CONCUR 2004 – Concurrency Theory- P11: The purpose of the CONCUR conferences is to bring together researchers, developers and students in order to advance the theory of concurrency and promote its applications. Interest in this topic is continually growing, as a consequence of the importance and ubiquity of concurrent systems and their applications, and of the scientific relevance of their foundations. | 286 E. Clarke et al. Analogous to the bounds on 2-connection topologies it can be shown that each -connection topology has at most 2k processes and that there are at most 3fc fe-i 2fe distinct -connection topologies. By an argument analogous to that of the previous section we obtain the following corollary Corollary 2. Let q x be a k-indexed quantifier-free LTL X property. Then PG p i iff PGG p sitei sitez . sitek . The notion of fc-topology is also defined completely analogously Definition 5. Given a network graph G S C the k-topology of G is given by Tk G I i Sk all indices in i are distinct . Consequently we obtain a model checking procedure from the following theorem similar to the case of 2-indices Theorem 2. The following are equivalent i PG x ii There exists a connection topology T e Tk G such that PT j sitez . site . As mentioned before 7 t G 3 fc-1 2fc. Specifications with General Quantifier Prefixes In this section we will show how to obtain reductions for -indexed specifications with first order prefixes. Let us for simplicity consider the 2-indexed formula Vx 3y. p x y . Over a network graph G S C S n it is clear that is equivalent to Ai j n Vi j n ip i j . A naive application of Corollary 2 would therefore require n2 calls to the model checker which may be expensive for practical values of n. In practice however we can bound the number of model checker calls by T2 G since this is the maximum number of different connection topologies. We conclude that the n2 model checker calls must contain repetitions. In the program we can make sure that at most 36 calls to the model checker are needed. We obtain the following algorithm 1 Determine 72 G . 2 For each T e T2 G 3 model check PT p site-i sitef 4 g T 1 iff model checking successful and 0 otherwise 5 Output Al i n Vl n5 G iJ - By simplifying the formula in line 5 we may further increase performance. The algorithm can be adapted for k indices in the obvious way. To state the main theorem of this .

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