Two variants of pebble tree-walking automata on trees are considered that were introduced in the literature. It is shown that for each number of pebbles, the two models have the same expressive power both in the deterministic case and in the nondeterministic case. Furthermore, nondeterministic (resp. deterministic) treewalking automata with n + 1 pebbles can recognize more languages than those with n pebbles. Moreover, there is a regular tree language that is not recognized by any treewalking automaton with pebbles. As a consequence, FO+posTC is strictly included in MSO over trees.Work supported by the French-German cooperation programme PROCOPE, KBN Grant 4 T11C 042 25, and the EU-TMR network GAMES.Expressive power of pebble automata. The main result of this paper is that pebble automata do not recognize all regular tree languages.This result is refined by showing that the hierarchy for pebble automata based on the number of pebbles is strict for both nondeterministic and deterministic pebble automata.This settles open questions raised in [4,5]. Furthermore, for each n, there is a language recognized by a nondeterministic tree-walking automaton but not by a deterministic n-pebble automaton. This improves the result in [2] that tree-walking automata (pebble automata with no pebbles) can not always be determinized.It is still an open problem to know whether DPA is strictly included in PA or not.Strong pebble automata. In [5], strong pebble automata were introduced as a model which corresponds to natural logics on trees. It was stated as an open question whether this model is stronger than the original one. We were surprised that this is actually not the case.Theorem 1.4. For each n ≥ 0, sPA n = PA n and sDPA n = DPA n .This proof is effective, but the state space increases n-fold exponentially. In a recent paper [9], it was shown that DPA n is closed under complement but the closure under complement of sDPA n was left open. Nevertheless, it was shown that the complement of a language in sDPA n is in sDPA 3n . From Theorem 1.4 we get the following stronger result:Corollary 1.5. For each n ≥ 0, sDPA n is closed under complement.Consequences for logics. In [5], the expressive power of strong pebble automata has been characterized in terms of logics. It was shown that FO+DTC=sDPA and FO+posTC=sPA.Here, FO+DTC is the extension of first-order logic with unary deterministic transitive closure operators and FO+posTC is the extension with positive unary transitive closure operators. By combining these results with ours and the fact that the regular tree languages are captured by monadic second-order logic (MSO), we immediately obtain the following result.Corollary 1.6. FO+posTC MSO. Whether FO+TC MSO and FO+DTC FO+posTC remains open.In Section 2 we give precise definitions of pebble automata and develop some related terminology. In Section 3 we prove some basic facts about the behavior of pebble automata on trees, in particular we show a kind of universality of n-pebble automata: for each n-pebble automaton A, there is an ...

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We consider various kinds of deterministic tree-walking automata, with and without pebbles, over ranked and unranked trees. For each such kind of automata we show that there is an equivalent one which never loops. The main consequence of this result is the closure under complementation of the various types of automata we consider with a focus on the number of pebbles used in order to complement the automata.

We consider tree-walking automata using k pebbles. The pebbles are either strong (can be lifted from anywhere) or weak (can be lifted only when the automaton is on it). For each k, we give the precise complexities of the problems of emptiness and inclusion of tree-walking automata using k pebbles.