Higher-Order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set of elements for which a formula holds), and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems, we provide an effective solution. This is obtained, thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

[1]  Christof Löding,et al.  MSO on the Infinite Binary Tree: Choice and Order , 2007, CSL.

[2]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs IX: Machines and their Behaviours , 1995, Theor. Comput. Sci..

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

[4]  Saharon Shelah,et al.  Rabin's uniformization problem , 1983, Journal of Symbolic Logic.

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

[6]  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).

[7]  Klaus Aehlig.  A Finite Semantics of Simply-Typed Lambda Terms for Infinite Runs of Automata , 2006, CSL.

[8]  Axel Haddad.  Model Checking and Functional Program Transformations , 2013, FSTTCS.

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

[10]  Moshe Y. Vardi,et al.  Global Model-Checking of Infinite-State Systems , 2004, CAV.

[11]  A. Church,et al.  Some properties of conversion , 1936 .

[12]  C.-H. Luke Ong,et al.  A Type System Equivalent to the Modal Mu-Calculus Model Checking of Higher-Order Recursion Schemes , 2009, 2009 24th Annual IEEE Symposium on Logic In Computer Science.

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

[14]  Andrzej S. Murawski,et al.  Collapsible Pushdown Parity Games , 2020, ArXiv.

[15]  Pawel Parys.  Recursion Schemes and the WMSO+U Logic , 2018, STACS.

[16]  Paul-André Melliès,et al.  Finitary Semantics of Linear Logic and Higher-Order Model-Checking , 2015, MFCS.

[17]  Bruno Courcelle,et al.  Monadic Second-Order Definable Graph Transductions: A Survey , 1994, Theor. Comput. Sci..

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

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

[20]  Alfred V. Aho,et al.  Translations on a Context-Free Grammar , 1971, Inf. Control..

[21]  André Arnold,et al.  Rudiments of Mu-calculus , 2001 .

[22]  Igor Walukiewicz,et al.  On the Expressive Completeness of the Propositional mu-Calculus with Respect to Monadic Second Order Logic , 1996, CONCUR.

[23]  Alfred V. Aho,et al.  Translations on a context free grammar , 1969, STOC.

[24]  C.-H. Luke Ong,et al.  Recursion Schemes and Logical Reflection , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[25]  Arnaud Carayol.  Automates infinis, logiques et langages , 2006 .

[26]  Jan-Pascal van Best,et al.  Trips on Trees , 1999, Acta Cybern..

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

[28]  C.-H. Luke Ong,et al.  On Global Model Checking Trees Generated by Higher-Order Recursion Schemes , 2009, FoSSaCS.

[29]  Axel Haddad.  Shape-Preserving Transformations of Higher-Order Recursion Schemes , 2013 .

[30]  Werner Damm,et al.  An Automata-Theoretical Characterization of the OI-Hierarchy , 1986, Inf. Control..

[31]  Pawel Parys.  On the Significance of the Collapse Operation , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[32]  C.-H. Luke Ong,et al.  On Full Abstraction for PCF: I, II, and III , 2000, Inf. Comput..

[33]  Axel Haddad.  IO vs OI in Higher-Order Recursion Schemes , 2012, FICS.

[34]  Naoki Kobayashi.  Types and higher-order recursion schemes for verification of higher-order programs , 2009, POPL '09.

[35]  Thomas Wilke,et al.  Alternating tree automata, parity games, and modal {$\mu$}-calculus , 2001 .

[36]  Wolfgang Thomas,et al.  Languages, Automata, and Logic , 1997, Handbook of Formal Languages.

[37]  M. Rabin.  Decidability of second-order theories and automata on infinite trees. , 1969 .

[38]  Olivier Serre,et al.  Collapsible Pushdown Automata and Labeled Recursion Schemes: Equivalence, Safety and Effective Selection , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

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

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

[41]  Patricia Bouyer,et al.  Automata, Logics and Games for Infinite Trees , 2019 .

[42]  Arnaud Carayol,et al.  Positional Strategies for Higher-Order Pushdown Parity Games , 2008, MFCS.

[43]  Pawel Urzyczyn,et al.  Deciding Monadic Theories of Hyperalgebraic Trees , 2001, TLCA.

[44]  I. Walukiewicz.  A landscape with games in the background , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[45]  Yuri Gurevich,et al.  Trees, automata, and games , 1982, STOC '82.

[46]  M. Rabin.  Decidability of second-order theories and automata on infinite trees , 1968 .