Flat Parametric Counter Automata

Abstract

International audienceIn this paper we study the reachability problem for parametric flat counter automata, in relation with the satisfiability problem of three fragments of integer arithmetic. The equivalence between non-parametric flat counter au-tomata and Presburger arithmetic has been established previously by Comon and Jurski [5]. We simplify their proof by introducing finite state automata defined over alphabets of a special kind of graphs (zigzags). This framework allows one to express also the reachability problem for parametric automata with one control loop as the existence of solutions of a 1-parametric linear Diophantine systems. The latter problem is shown to be decidable, using a number-theoretic argument. Finally, the general reachability problem for parametric flat counter automata with more than one loops is shown to be undecidable, by reduction from Hilbert's Tenth Problem [9]

Fast Acceleration of Ultimately Periodic Relations

“…The transitive closure of a difference bounds constraint is shown to be Presburger definable first by Comon and Jurski [9]. Their proof was subsequently simplified and extended to parametric difference bounds constraints in [8]. We also showed that octagonal relations can be accelerated precisely, and that the transitive closure is also Presburger definable [6].…”

“…We also showed that octagonal relations can be accelerated precisely, and that the transitive closure is also Presburger definable [6]. The proofs of ultimate periodicity from this paper are based on some of our previous results [8,6]. For difference bounds constraints, the proof from [8] was simplified using a result from tropical semiring theory [13].…”

“…Second, from the practical side, the algorithm introduced in this paper computes the transitive closure of difference bounds and octagonal relations up to four orders of magnitude faster than our original methods from [8,6], and also scales much better in the number of variables. The experimental comparison with the FAST tool [4] for difference bounds relations shows that large relations (> 50 variables), causing FAST to run out of memory, can now be handled by our implementation in less than 8 seconds, on average.…”

“…To this end, it is important to know for which classes of transition relations it is possible to compute the transitive closure precisely and fast -the relations falling outside these classes being dealt with using suitable abstractions. To the best of our knowledge, the three main classes of integer relations for which transitive closures can be computed precisely in finite time are: (1) difference bounds constraints [9,8], (2) octagons [12,6], and (3) finite monoid affine transformations [5,10]. For these three classes, the transitive closures can be moreover defined in Presburger arithmetic.…”

Model-checking CTL* over flat Presburger counter systems

“…Hence, the abovementioned consequence appear to be incomparable with the main result from (Comon et al, 1998). Furthermore, the systems defined in Definition 10 below are more general than the ones in (Comon et al, 1998;Bozga et al, 2009) since we allow a richer language on transitions.…”