Rewriting Higher-Order Stack Trees

Higher-order pushdown systems and ground tree rewriting systems can be seen as extensions of suffix word rewriting systems. Both classes generate infinite graphs with interesting logical properties. Indeed, the model-checking problem for monadic second order logic (respectively first order logic with a reachability predicate) is decidable on such graphs. We unify both models by introducing the notion of stack trees, trees whose nodes are labelled by higher-order stacks, and define the corresponding class of higher-order ground tree rewriting systems. We show that these graphs retain the decidability properties of ground tree rewriting graphs while generalising the pushdown hierarchy of graphs.

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

[2]  Didier Caucal,et al.  On infinite transition graphs having a decidable monadic theory , 1996, Theor. Comput. Sci..

[3]  Chloe Rispal,et al.  The synchronized graphs trace the context-sensitive languages , 2003, INFINITY.

[4]  Antoine Meyer,et al.  Traces of term-automatic graphs , 2007, RAIRO Theor. Informatics Appl..

[5]  Olivier Serre,et al.  A Saturation Method for Collapsible Pushdown Systems , 2012, ICALP.

[6]  Achim Blumensath,et al.  Finite Presentations of Infinite Structures: Automata and Interpretations , 2004, Theory of Computing Systems.

[7]  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.

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

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

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

[11]  Klaus Barthelmann,et al.  When Can an Equational Simple Graph Be Generated by Hyperedge Replacement? , 1998, MFCS.

[12]  Sophie Tison,et al.  The theory of ground rewrite systems is decidable , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

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

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

[15]  Christof Löding,et al.  Transforming structures by set interpretations , 2007, Log. Methods Comput. Sci..