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 built, 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 this enrich graph may have an undecidable MSO theory. 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.Comment: 11 pages, 2 figure

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

International audienceWe 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 É-regular winning condition, that the set of winning positions for a given player is regular and effective

Streaming Property Testing of Visibly Pushdown Languages

In the context of language recognition, we demonstrate the superiority of streaming property testers against streaming algorithms and property testers, when they are not combined. Initiated by Feigenbaum et al., a streaming property tester is a streaming algorithm recognizing a language under the property testing approximation: it must distinguish inputs of the language from those that are $\varepsilon$-far from it, while using the smallest possible memory (rather than limiting its number of input queries). Our main result is a streaming $\varepsilon$-property tester for visibly pushdown languages (VPL) with one-sided error using memory space $\mathrm{poly}((\log n) / \varepsilon)$. This constructions relies on a (non-streaming) property tester for weighted regular languages based on a previous tester by Alon et al. We provide a simple application of this tester for streaming testing special cases of instances of VPL that are already hard for both streaming algorithms and property testers. Our main algorithm is a combination of an original simulation of visibly pushdown automata using a stack with small height but possible items of linear size. In a second step, those items are replaced by small sketches. Those sketches relies on a notion of suffix-sampling we introduce. This sampling is the key idea connecting our streaming tester algorithm to property testers.Comment: 23 pages. Major modifications in the presentatio

Emptiness Of Alternating Tree Automata Using Games With Imperfect Information

We consider the emptiness problem for alternating tree automata, with two acceptance semantics: classical (all branches are accepted) and qualitative (almost all branches are accepted). For the classical semantics, the usual technique to tackle this problem relies on a Simulation Theorem which constructs an equivalent non-deterministic automaton from the original alternating one, and then checks emptiness by a reduction to a two-player perfect information game. However, for the qualitative semantics, no simulation of alternation by means of non-determinism is known. We give an alternative technique to decide the emptiness problem of alternating tree automata, that does not rely on a Simulation Theorem. Indeed, we directly reduce the emptiness problem to solving an imperfect information two-player parity game. Our new approach can successfully be applied to both semantics, and yields decidability results with optimal complexity; for the qualitative semantics, the key ingredient in the proof is a positionality result for stochastic games played over infinite graphs

How Good Is a Strategy in a Game with Nature?

International audienceWe consider games with two antagonistic players — Éloïse (modelling a program) and Abélard (modelling a byzantine environment) — and a third, unpredictable and uncontrollable player, that we call Nature. Motivated by the fact that the usual probabilistic semantics very quickly leads to undecidability when considering either infinite game graphs or imperfect information, we propose two alternative semantics that leads to decidability where the probabilistic one fails: one based on counting and one based on topology

Streaming Property Testing of Visibly Pushdown Languages

In the context of formal language recognition, we demonstrate the superiority of streaming property testers against streaming algorithms and property testers, when they are not combined. Initiated by Feigenbaum et al., a streaming property tester is a streaming algorithm recognizing a language under the property testing approximation: it must distinguish inputs of the language from those that are eps-far from it, while using the smallest possible memory (rather than limiting its number of input queries). Our main result is a streaming eps-property tester for visibly pushdown languages (V_{PL}) with memory space poly(log n /epsilon). Our construction is done in three steps. First, we simulate a visibly pushdown automaton in one pass using a stack of small height but whose items can be of linear size. In a second step, those items are replaced by small sketches. Those sketches rely on a notion of suffix-sampling we introduce. This sampling is the key idea for taking benefit of both streaming algorithms and property testers in the third step. Indeed, the last step relies on a (non-streaming) property tester for weighted regular languages based on a previous tester by Alon et al. This tester can directly be used for streaming testing special cases of instances of V_{PL} that are already hard for both streaming algorithms and property testers. We then use it to decide the correctness of completed items, given their sketches, before removing them from the stack