The Complexity of Reachability Problems for Flat Counter Machines with Periodic Loops

International audienceThis paper proves the NP-completeness of the reachability problem for the class of flat counter machines with difference bounds and, more generally, octagonal relations, labeling the transitions on the loops. The proof is based on the fact that the sequence of powers {R n } ∞ n=0 of such relations can be encoded as a periodic sequence of matrices, and that both the prefix and the period of this sequence are simply exponential in the size of the binary representation of a relation R. This result allows to characterize the complexity of the reachability problem for one of the most studied class of counter machines [6, 10], and has a potential impact on other problems in program verification

Deciding Conditional Termination

International audienceThis paper addresses the problem of conditional termination, which is that of defining the set of initial configurations from which a given program terminates. First we define the dual set, of initial configurations, from which a non-terminating execution exists, as the greatest fixpoint of the pre-image of the transition relation. This definition enables the representation of this set, whenever the closed form of the relation of the loop is definable in a logic that has quanti-fier elimination. This entails the decidability of the termination problem for such loops. Second, we present effective ways to compute the weakest precondition for non-termination for difference bounds and octagonal (non-deterministic) relations , by avoiding complex quantifier eliminations. We also investigate the existence of linear ranking functions for such loops. Finally, we study the class of linear affine relations and give a method of under-approximating the termination precondition for a non-trivial subclass of affine relations. We have performed preliminary experiments on transition systems modeling real-life systems, and have obtained encouraging results