On the Context-Freeness Problem for Vector Addition Systems

Petri nets, or equivalently vector addition systems (VAS), are widely recognized as a central model for concurrent systems. Many interesting properties are decidable for this class, such as boundedness, reachability, regularity, as well as context-freeness, which is the focus of this paper. The context-freeness problem asks whether the trace language of a given VAS is context-free. This problem was shown to be decidable by Schwer in 1992, but the proof is very complex and intricate. The resulting decision procedure relies on five technical conditions over a customized coverability graph. These five conditions are shown to be necessary, but the proof that they are sufficient is only sketched. In this paper, we revisit the context-freeness problem for VAS, and give a simpler proof of decidability. Our approach is based on witnesses of non-context-freeness, that are bounded regular languages satisfying a nesting condition. As a corollary, we obtain that the trace language of a VAS is context-free if, and only if, it has a context-free intersection with every bounded regular language.

[1]  Albert R. Meyer,et al.  The Complexity of the Finite Containment Problem for Petri Nets , 1981, JACM.

[2]  Richard M. Karp,et al.  Parallel Program Schemata , 1969, J. Comput. Syst. Sci..

[3]  Rupak Majumdar,et al.  Bounded underapproximations , 2008, Formal Methods Syst. Des..

[4]  Rupak Majumdar,et al.  A Perfect Model for Bounded Verification , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[5]  Faron Moller,et al.  Petri Nets and Regular Processes , 1999, J. Comput. Syst. Sci..

[6]  Charles Rackoff,et al.  The Covering and Boundedness Problems for Vector Addition Systems , 1978, Theor. Comput. Sci..

[7]  Peter Radford,et al.  Petri Net Theory and the Modeling of Systems , 1982 .

[8]  Seymour Ginsburg,et al.  The mathematical theory of context free languages , 1966 .

[9]  Mohamed Faouzi Atig,et al.  On Yen's Path Logic for Petri Nets , 2009, Int. J. Found. Comput. Sci..

[10]  Hsu-Chun Yen.  A Note on Fine Covers and Iterable Factors of VAS Languages , 1995, Inf. Process. Lett..

[11]  Rüdiger Valk,et al.  Petri Nets and Regular Languages , 1981, J. Comput. Syst. Sci..

[12]  Sylvain Schmitz,et al.  Model Checking Coverability Graphs of Vector Addition Systems , 2011, MFCS.

[13]  Michael Yoeli,et al.  Vector Addition Systems and Regular Languages , 1980, J. Comput. Syst. Sci..

[14]  Sylviane R. Schwer.  The Context-Freeness of the Languages Associated with Vector Addition Systems is Decidable , 1992, Theor. Comput. Sci..

[15]  S. Ginsburg,et al.  Semigroups, Presburger formulas, and languages. , 1966 .

[16]  Stéphane Demri,et al.  On selective unboundedness of VASS , 2010, J. Comput. Syst. Sci..

[17]  Mogens Nielsen,et al.  Decidability Issues for Petri Nets - a survey , 1994, Bull. EATCS.