Porous Invariants

We introduce the notion of porous invariants for multipath (or branching/nondeterministic) affine loops over the integers; these invariants are not necessarily convex, and can in fact contain infinitely many ‘holes’. Nevertheless, we show that in many cases such invariants can be automatically synthesised, and moreover can be used to settle (non-)reachability questions for various interesting classes of affine loops and target sets.

[1]  Nicolas Halbwachs,et al.  Automatic discovery of linear restraints among variables of a program , 1978, POPL.

[2]  A. Chistov,et al.  Algorithm of polynomial complexity for factoring polynomials and finding the components of varieties in subexponential time , 1986 .

[3]  Tero Harju,et al.  Undecidability of infinite post correspondence problem for instances of Size 9 , 2006, RAIRO Theor. Informatics Appl..

[4]  Alain Finkel,et al.  Reachability in Register Machines with Polynomial Updates , 2013, MFCS.

[5]  Michael Karr,et al.  Affine relationships among variables of a program , 1976, Acta Informatica.

[6]  Joël Ouaknine,et al.  Decision Problems for Linear Recurrence Sequences , 2012, RP.

[7]  Richard J. Lipton,et al.  Polynomial-time algorithm for the orbit problem , 1986, JACM.

[8]  Joël Ouaknine,et al.  Decision Problems for Linear Recurrence Sequences , 2012, SCSS.

[9]  Nathanael Fijalkow,et al.  On the Monniaux Problem in Abstract Interpretation , 2019, SAS.

[10]  James Worrell,et al.  O-Minimal Invariants for Discrete-Time Dynamical Systems , 2022, ACM Transactions on Computational Logic.

[11]  L. Kronecker,et al.  Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. , 1857 .

[12]  Qinghui Liu,et al.  Undecidability of infinite post correspondence problem for instances of size 8 , 2012, RAIRO Theor. Informatics Appl..

[13]  Marius Bozga,et al.  Fast Acceleration of Ultimately Periodic Relations , 2010, CAV.

[14]  Jochen Hoenicke,et al.  Termination Analysis by Learning Terminating Programs , 2014, CAV.

[15]  David Monniaux.  On the decidability of the existence of polyhedral invariants in transition systems , 2018, Acta Informatica.

[16]  Joël Ouaknine,et al.  Abstraction and Counterexample-Guided Refinement in Model Checking of Hybrid Systems , 2003, Int. J. Found. Comput. Sci..

[17]  Daniel Fremont.  The Reachability Problem for Affine Functions on the Integers , 2013, ArXiv.

[18]  Joël Ouaknine,et al.  Polynomial Invariants for Affine Programs , 2018, LICS.

[19]  Jérôme Leroux.  The General Vector Addition System Reachability Problem by Presburger Inductive Invariants , 2009, 2009 24th Annual IEEE Symposium on Logic In Computer Science.

[20]  Jérôme Leroux.  Vector Addition System Reachability Problem: A Short Self-contained Proof , 2011, LATA.

[21]  S. Ginsburg,et al.  Bounded -like languages , 1964 .

[22]  Wen-Guey Tzeng,et al.  A Polynomial-Time Algorithm for the Equivalence of Probabilistic Automata , 1992, SIAM J. Comput..

[23]  Jürgen Giesl,et al.  Analyzing Program Termination and Complexity Automatically with AProVE , 2016, Journal of Automated Reasoning.

[24]  Zachary Kincaid,et al.  Closed forms for numerical loops , 2019, Proc. ACM Program. Lang..

[25]  S. Ginsburg,et al.  BOUNDED ALGOL-LIKE LANGUAGES^) , 1964 .