The Affine Hull of a Binary Automaton is Computable in Polynomial Time

Abstract

We present the class of binary automaton, a new representation for the subsets of Nm that naturally extends the NDD ([Wolper, P. and B. Boigelot, An automata-theoretic approach to Presburger arithmetic constraints, in: Proc. 2nd Int. Symp. Static Analysis (SAS'95), Glasgow, UK, Sep. 1995, Lecture Notes in Computer Science 983 (1995), pp. 21–32], [Boudet, A. and H. Comon, Diophantine equations, Presburger arithmetic and finite automata, in: Proc. 21st Int. Coll. on Trees in Algebra and Programming (CAAP'96), Linköping, Sweden, Apr. 1996, Lecture Notes in Computer Science 1059 (1996), pp. 30–43]). We prove that the affine hull of the set of vectors represented by a binary automaton is computable in polynomial time. As application, we show that the set of place invariants [Ciardo, G., Petri nets with marking-dependent arc cardinality: Properties and analysis, in: Proc. 15th Int. Conf. Application and Theory of Petri Nets (ICATPN'94), Zaragoza, Spain, June 1994, Lecture Notes in Computer Science 815 (1994), pp. 179–198] of a counter system (an extension of the Broadcast Protocols [Emerson, E. A. and K. S. Namjoshi, On model checking for non-deterministic infinite-state systems, in: Proc. 13th IEEE Symp. Logic in Computer Science (LICS'98), Indianapolis, IN, USA, June 1998 (1998), pp. 70–80], [Delzanno, G., Verification of consistency protocols via infinite-state symbolic model checking: A case study, in: Proc. IFIP Joint Int. Conf. Formal Description Techniques & Protocol Specification, Testing, and Verification (FORTE-PSTV'00), Pisa, Italy, Oct. 2000, IFIP Conference Proceedings 183 (2000), pp. 171–186], [Delzanno, G., Constraint-based verification of parameterized cache coherence protocols, Formal Methods in System Design. To appear], the Reset/Transfer Petri Nets [Dufourd, C., A. Finkel and P. Schnoebelen, Reset nets between decidability and undecidability, in: Proc. 25th Int. Coll. Automata, Languages, and Programming (ICALP'98), Aalborg, Denmark, July 1998, Lecture Notes in Computer Science 1443 (1998), pp. 103–115], [Ciardo, G., Petri nets with marking-dependent arc cardinality: Properties and analysis, in: Proc. 15th Int. Conf. Application and Theory of Petri Nets (ICATPN'94), Zaragoza, Spain, June 1994, Lecture Notes in Computer Science 815 (1994), pp. 179–198] and the linear systems [Finkel, A. and J. Leroux, How to compose Presburger-accelerations: Applications to broadcast protocols, in: Proc. 22nd Conf. Found. of Software Technology and Theor. Comp. Sci. (FST&TCS'2002), Kanpur, India, Dec. 2002, Lecture Notes in Computer Science 2556 (2002), pp. 145–156]), is computable in polynomial time.