Parity to Safety in Polynomial Time for Pushdown and Collapsible Pushdown Systems

We give a direct polynomial-time reduction from parity games played over the configuration graphs of collapsible pushdown systems to safety games played over the same class of graphs. That a polynomial-time reduction would exist was known since both problems are complete for the same complexity class. Coming up with a direct reduction, however, has been an open problem. Our solution to the puzzle brings together a number of techniques for pushdown games and adds three new ones. This work contributes to a recent trend of liveness to safety reductions which allow the advanced state-of-the-art in safety checking to be used for more expressive specifications.

[1]  Viktor Schuppan,et al.  Liveness Checking as Safety Checking , 2002, FMICS.

[2]  C.-H. Luke Ong,et al.  On Model-Checking Trees Generated by Higher-Order Recursion Schemes , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[3]  C.-H. Luke Ong,et al.  A type-directed abstraction refinement approach to higher-order model checking , 2014, POPL.

[4]  Antoine Meyer,et al.  Winning Regions of Higher-Order Pushdown Games , 2008, 2008 23rd Annual IEEE Symposium on Logic in Computer Science.

[5]  Jochen Hoenicke,et al.  Reducing liveness to safety in first-order logic , 2017, Proc. ACM Program. Lang..

[6]  Marcin Jurdzinski,et al.  Succinct progress measures for solving parity games , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[7]  Igor Walukiewicz,et al.  Permissive strategies: from parity games to safety games , 2002, RAIRO Theor. Informatics Appl..

[8]  Andrzej S. Murawski,et al.  Collapsible Pushdown Automata and Recursion Schemes , 2008, 2008 23rd Annual IEEE Symposium on Logic in Computer Science.

[9]  Alberto Griggio,et al.  Infinite-State Liveness-to-Safety via Implicit Abstraction and Well-Founded Relations , 2016, CAV.

[10]  Helmut Veith,et al.  A short counterexample property for safety and liveness verification of fault-tolerant distributed algorithms , 2016, POPL.

[11]  Joost Engelfriet,et al.  Iterated Stack Automata and Complexity Classes , 1991, Inf. Comput..

[12]  GIACOMO LENZI.  THE MODAL μ-CALCULUS : A SURVEY , 2005 .

[13]  Javier Esparza,et al.  Reachability Analysis of Pushdown Automata: Application to Model-Checking , 1997, CONCUR.

[14]  Sanjay Jain,et al.  An ordered approach to solving parity games in quasi polynomial time and quasi linear space , 2017, SPIN.

[15]  Andreas Podelski,et al.  Proving Liveness of Parameterized Programs , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[16]  Helmut Veith,et al.  Decidability of Parameterized Verification , 2015, Synthesis Lectures on Distributed Computing Theory.

[17]  Igor Walukiewicz,et al.  The Complexity of Games on Higher Order Pushdown Automata , 2007, ArXiv.

[18]  Naoki Kobayashi,et al.  A ZDD-Based Efficient Higher-Order Model Checking Algorithm , 2014, APLAS.

[19]  C.-H. Luke Ong,et al.  TravMC2: higher-order model checking for alternating parity tree automata , 2014, SPIN.

[20]  Souhei Ito,et al.  Practical Alternating Parity Tree Automata Model Checking of Higher-Order Recursion Schemes , 2013, APLAS.

[21]  Andreas Podelski,et al.  Transition invariants , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[22]  Andreas Podelski,et al.  Transition Invariants and Transition Predicate Abstraction for Program Termination , 2011, TACAS.

[23]  Wladimir Fridman,et al.  Playing Pushdown Parity Games in a Hurry , 2012, GandALF.

[24]  Igor Walukiewicz,et al.  Pushdown Processes: Games and Model-Checking , 1996, Inf. Comput..

[25]  Pawel Urzyczyn,et al.  Higher-Order Pushdown Trees Are Easy , 2002, FoSSaCS.

[26]  Viktor Schuppan,et al.  Liveness Checking as Safety Checking for Infinite State Spaces , 2006, INFINITY.

[27]  Cristian S. Calude,et al.  Deciding parity games in quasipolynomial time , 2017, STOC.

[28]  Fabio Somenzi,et al.  Safety first: A two-stage algorithm for LTL games , 2009, 2009 Formal Methods in Computer-Aided Design.

[29]  Naoki Kobayashi,et al.  Saturation-Based Model Checking of Higher-Order Recursion Schemes , 2013, CSL.

[30]  Dietmar Berwanger,et al.  On the Power of Imperfect Information , 2008, FSTTCS.