Symbolic Backwards-Reachability Analysis for Higher-Order Pushdown Systems

Higher-order pushdown systems (PDSs) generalise pushdown systems through the use of higher-order stacks, that is, a nested "stack of stacks" structure. These systems may be used to model higher-order programs and are closely related to the Caucal hierarchy of infinite graphs and safe higher-order recursion schemes. We consider the backwards-reachability problem over higher-order Alternating PDSs (APDSs), a generalisation of higher-order PDSs. This builds on and extends previous work on pushdown systems and context-free higher-order processes in a non-trivial manner. In particular, we show that the set of configurations from which a regular set of higher-order APDS configurations is reachable is regular and computable in n-EXPTIME. In fact, the problem is n-EXPTIME-complete. We show that this work has several applications in the verification of higher-order PDSs, such as linear-time model-checking, alternation-free mu-calculus model-checking and the computation of winning regions of reachability games.

[1]  Arnaud Carayol,et al.  The Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higher-Order Pushdown Automata , 2003, FSTTCS.

[2]  Arnaud Carayol,et al.  Regular Sets of Higher-Order Pushdown Stacks , 2005, MFCS.

[3]  Wolfgang Thomas,et al.  Solving Pushdown Games with a Sigma3 Winning Condition , 2002, CSL.

[4]  C.-H. Luke Ong,et al.  Symbolic Backwards-Reachability Analysis for Higher-Order Pushdown Systems , 2007, FoSSaCS.

[5]  Igor Walukiewicz,et al.  Unsafe Grammars and Panic Automata , 2005, ICALP.

[6]  Colin Stirling,et al.  Modal Logics and mu-Calculi: An Introduction , 2001, Handbook of Process Algebra.

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

[8]  Olivier Serre,et al.  Games with winning conditions of high Borel complexity , 2006, Theor. Comput. Sci..

[9]  Joost Engelfriet,et al.  Iterated pushdown automata and complexity classes , 1983, STOC.

[10]  David E. Muller,et al.  The Theory of Ends, Pushdown Automata, and Second-Order Logic , 1985, Theor. Comput. Sci..

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  Wolfgang Thomas,et al.  Automata on Infinite Objects , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[13]  Ernst L. Leiss,et al.  On Equations for Regular Languages, Finite Automata, and Sequential Networks , 1980, Theor. Comput. Sci..

[14]  Igor Walukiewicz,et al.  Pushdown Games with Unboundedness and Regular Conditions , 2003, FSTTCS.

[15]  Antoine Meyer,et al.  Symbolic Reachability Analysis of Higher-Order Context-Free Processes , 2004, FSTTCS.

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

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

[18]  Moshe Y. Vardi.  An Automata-Theoretic Approach to Linear Temporal Logic , 1996, Banff Higher Order Workshop.

[19]  Colin Stirling,et al.  Bisimulation, Modal Logic and Model Checking Games , 1999, Logic Journal of the IGPL.

[20]  Klaus Aehlig,et al.  Safety Is not a Restriction at Level 2 for String Languages , 2005, FoSSaCS.

[21]  Thierry Cachat,et al.  Games on pushdown graphs and extensions , 2003 .

[22]  E. Allen Emerson,et al.  An Automata Theoretic Decision Procedure for the Propositional Mu-Calculus , 1989, Inf. Comput..

[23]  Moshe Y. Vardi.  A temporal fixpoint calculus , 1988, POPL '88.

[24]  Didier Caucal.  On Infinite Terms Having a Decidable Monadic Theory , 2002, MFCS.

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

[26]  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).

[27]  Andrzej S. Murawski,et al.  Collapsible Pushdown Automata and Recursion Schemes , 2008, LICS.

[28]  Christof Löding,et al.  Visibly Pushdown Games , 2004, FSTTCS.

[29]  Hugo Gimbert,et al.  Parity and Exploration Games on Infinite Graphs , 2004, CSL.

[30]  Matthew Hague,et al.  Saturation methods for global model-checking pushdown systems , 2009 .

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