Pushdown Graphs Research Articles

Pushdown reachability with constant treewidth

We consider the problem of reachability in pushdown graphs. We study the problem for pushdown graphs with constant treewidth. Even for pushdown graphs with treewidth 1, for the reachability problem we establish the following: (i) the problem is PTIME-complete, and (ii) any subcubic algorithm for the problem would contradict the k-clique conjecture and imply faster combinatorial algorithms for cliques in graphs.

Marking shortest paths on pushdown graphs does not preserve MSO decidability

In this paper we consider pushdown graphs, i.e. infinite graphs that can be described as transition graphs of deterministic real-time pushdown automata. We consider the case where some vertices are designated as being final and we build, in a breadth-first manner, a marking of edges that lead to such vertices (i.e., for every vertex that can reach a final one, we mark all out-going edges laying on some shortest path to a final vertex).Our main result is that the edge-marked version of a pushdown graph may itself no longer be a pushdown graph, as we prove that the MSO theory of this enriched graph may be undecidable.

First-Order Logic on Higher-Order Nested Pushdown Trees

We introduce a new hierarchy of higher-order nested pushdown trees generalising Alur et al.’s concept of nested pushdown trees. Nested pushdown trees are useful representations of control flows in the verification of programs with recursive calls of first-order functions. Higher-order nested pushdown trees are expansions of unfoldings of graphs generated by higher-order pushdown systems. Moreover, the class of nested pushdown trees of level n is uniformly first-order interpretable in the class of collapsible pushdown graphs of level n + 1. The relationship between the class of higher-order pushdown graphs and the class of collapsible higher-order pushdown graphs is not very well understood. We hope that the further study of the nested pushdown tree hierarchy leads to a better understanding of these two hierarchies. In this article, we are concerned with the first-order model checking problem on higher-order nested pushdown trees. We show that the first-order model checking on the first two levels of this hierarchy is decidable. Moreover, we obtain an alternating 2-EXPTIME algorithm for the class of nested pushdown trees of level 1. The proof technique involves a pseudo-local analysis of strategies in the Ehrenfeucht-Fraïssé games on two identical copies of a nested pushdown tree. Ordinary locality arguments in the spirit of Gaifman’s lemma do not apply here because nested pushdown trees tend to have small diameters. We introduce the notion of relevant ancestors which provide a sufficient description of the FO k -type of each element in a higher-order nested pushdown tree. The local analysis of these ancestors allows us to prove the existence of restricted winning strategies in the Ehrenfeucht-Fraïssé game. These strategies are then used to create a first-order model checking algorithm.

Collapsible Pushdown Graphs of Level 2 are Tree-Automatic

We show that graphs generated by collapsible pushdown systems of level 2 are tree-automatic. Even if we allow epsilon-contractions and reachability predicates (with regular constraints) for pairs of configurations, the structures remain tree-automatic whence their first-order logic theories are decidable. As a corollary we obtain the tree-automaticity of the second level of the Caucal-hierarchy.

Formats of Winning Strategies for Six Types of Pushdown Games

The solution of parity games over pushdown graphs (Walukiewicz '96) was the first step towards an effective theory of infinite-state games. It was shown that winning strategies for pushdown games can be implemented again as pushdown automata. We continue this study and investigate the connection between game presentations and winning strategies in altogether six cases of game arenas, among them realtime pushdown systems, visibly pushdown systems, and counter systems. In four cases we show by a uniform proof method that we obtain strategies implementable by the same type of pushdown machine as given in the game arena. We prove that for the two remaining cases this correspondence fails. In the conclusion we address the question of an abstract criterion that explains the results.

Games with winning conditions of high Borel complexity

We first consider infinite two-player games on pushdown graphs. In previous work, Cachat et al. [Solving pushdown games with a Σ 3 -winning condition, in: Proc. 11th Annu. Conf. of the European Association for Computer Science Logic, CSL 2002, Lecture Notes in Computer Science, Vol. 2471, Springer, Berlin, 2002, pp. 322–336] have presented a winning decidable condition that is Σ 3 -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 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 conditions is shown to be non-elementary.

Uniform Solution of Parity Games on Prefix-Recognizable Graphs

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 EXPTIME 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 configuration), 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.

Note on winning positions on pushdown games with ω-regular conditions

We consider infinite two-player games on pushdown graphs. For parity winning conditions, we show that the set of winning positions of each player is regular and we give an effective construction of an alternating automaton recognizing it. This provides a DEXPTIME procedure to decide whether a position is winning for a given player. Finally, using the same methods, we show, for any ω-regular winning condition, that the set of winning positions for a given player is regular and effective.