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Summary: The Monadic Second Order Theory of Trees
Given by Arbitrary Level-Two Recursion
Schemes Is Decidable
Klaus Aehlig ? , Jolie G. de Miranda, and C.-H. Luke Ong
Oxford University Computing Laboratory
Abstract. A tree automaton can simulate the successful runs of a word
or tree automaton working on the word or tree denoted by a level-2
lambda-tree. In particular the monadic second order theory of trees given
by arbitrary, rather than only by safe, recursion schemes of level 2 is
decidable. This solves the level-2 case of an open problem by Knapik,
Niwinski and Urzyczyn.
1 Introduction and Related Work
Since Rabin [11] showed the decidability of the monadic second order theory of
the binary tree this result has been applied and extended to various mathematical
structures, including algebraic trees [4] and a hierarchy of graphs [3] obtained by
iterated unfolding and inverse rational mappings from nite graphs. The interest
in these kinds of structures arose in recent years in the context of veri cation of
in nite state systems [9, 13].
Recently Knapik, Niwinski and Urzyczyn [6] showed that the monadic second
order (MSO) theory of any in nite tree generated by a level-2 grammar satisfying
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