Verimag

Technical Reports

Marius Bozga, Radu Iosif and Yassine Lakhnech
Flat Parametric Counter Automata (2005)

TR-2005-15.ps

Keywords: counter automata, diophantine systems

Abstract: In 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 automata and Presburger arithmetic has been established previously by Comon and Jurski \cite{comon-jurski98}. 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 {\em 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 \cite{hilbert}.

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