We show that the monadic second-order theory of an infinite tree recognized by a higher-order pushdown automaton of any level is decidable. We also show that trees recognized by pushdown automata of level n coincide with trees generated by safe higher-order grammars of level n. Our decidability result extends the result of Courcelle on algebraic (pushdown of level 1) trees and our own result on trees of level 2.

We present several characterisations of the class of prefix-recognisable

Traditionally, model checking is applied to finite-state systems and regular specifications. While researchers have successfully extended the applicability of model checking to infinite-state systems, almost all existing work still consider regular specification formalisms. There are, however, many interesting non-regular properties one would like to model check.

Walukiewicz gave in 1996 a solution for parity games on pushdown graphs: he proved the existence of pushdown strategies and determined the winner with an EX-PTIME procedure. We give a new presentation and a new algorithmic proof of these results, obtain a uniform solution for parity games (independent of their initial configu-ration), and extend the results to prefix-recognizable graphs. The winning regions of the players are proved to be effectively regular, and winning strategies are computed. 1

We show that the monadic second-order theory of any infinite tree generated by a higher-order grammar of level 2 subject to a certain syntactic restriction is decidable. By this we extend the result of Courcelle [7] that the MSO theory of a tree generated by a grammar of level 1 (algebraic) is decidable. To this end, we develop a technique of representing infinite trees by infinite lambda terms, in such a way that the MSO theory of a tree can be interpreted in the MSO theory of a lambda term.

Abstract. We consider parity games played on special pushdown graphs, namely those generated by one-counter processes. For parity games on pushdown graphs, it is known from [23] that deciding the winner is an ExpTime-complete problem. An important corollary of this result is that the µ-calculus model checking problem for pushdown processes is Exp-Time-complete. As one-counter processes are special cases of pushdown processes, it follows that deciding the winner in a parity game played on the transition graph of a one-counter process can be achieved in ExpTime. Nevertheless the proof for the ExpTime-hardness lower bound of [23] cannot be adapted to that case. Therefore, a natural question is whether the ExpTime upper bound can be improved in this special case. In this paper, we adapt techniques from [11, 4] and provide a PSpace upper bound and a DP-hard lower bound for this problem. We also give two important consequences of this result. First, we improve the best upper bound known for model-checking one-counter processes against µ-calculus. Second, we show how these games can be used to solve pushdown games with winning conditions that are Boolean combinations of a parity condition on the control states with conditions on the stack height. 1

Vol. 3 (3:1) 2007, pp. 1–23

We introduce the class of tree-interpretable structures which generalises the notion of a prefix-recognisable graph to arbitrary relational structures. We prove that every tree-interpretable structure is finitely axiomatisable in guarded second-order logic with cardinality quantifiers.

Abstract. We first consider infinite two-player games on pushdown graphs. In previous work, Cachat, Duparc and Thomas [4] have presented a winning decidable condition that is Σ3-complete in the Borel hierarchy. This was the first example of a decidable winning condition of such Borel complexity. We extend this result by giving a family of decidable winning conditions of arbitrary high finite Borel complexity. From this family, we deduce a family of decidable winning conditions of arbitrary finite Borel complexity for games played on finite graphs. The problem of deciding the winner for these winning conditions is shown to be non-elementary complete. Keywords: Pushdown Automata, Two-player Games, Borel Complexity. 1

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