Beyond Language Equivalence on Visibly Pushdown Automata

We study (bi)simulation-like preorder/equivalence checking on the class of visibly pushdown automata and its natural subclasses visibly BPA (Basic Process Algebra) and visibly one-counter automata. We describe generic methods for proving complexity upper and lower bounds for a number of studied preorders and equivalences like simulation, completed simulation, ready simulation, 2-nested simulation preorders/equivalences and bisimulation equivalence. Our main results are that all the mentioned equivalences and preorders are EXPTIME-complete on visibly pushdown automata, PSPACE-complete on visibly one-counter automata and P-complete on visibly BPA. Our PSPACE lower bound for visibly one-counter automata improves also the previously known DP-hardness results for ordinary one-counter automata and one-counter nets. Finally, we study regularity checking problems for visibly pushdown automata and show that they can be decided in polynomial time.Comment: Final version of paper, accepted by LMC

Making Self-Stabilizing any Locally Greedy Problem

We propose a way to transform synchronous distributed algorithms solving locally greedy and mendable problems into self-stabilizing algorithms in anonymous networks. Mendable problems are a generalization of greedy problems where any partial solution may be transformed -- instead of completed -- into a global solution: every time we extend the partial solution we are allowed to change the previous partial solution up to a given distance. Locally here means that to extend a solution for a node, we need to look at a constant distance from it. In order to do this, we propose the first explicit self-stabilizing algorithm computing a $(k,k-1)$-ruling set (i.e. a "maximal independent set at distance $k$"). By combining multiple time this technique, we compute a distance-$K$ coloring of the graph. With this coloring we can finally simulate \local~model algorithms running in a constant number of rounds, using the colors as unique identifiers. Our algorithms work under the Gouda daemon, which is similar to the probabilistic daemon: if an event should eventually happen, it will occur under this daemon

Regular matching problems for infinite trees

We investigate regular matching problems. The classical reference is Conway's textbook "Regular algebra and finite machines". Some of his results can be stated as follows. Let $L\subseteq(\Sigma\cup X)^*$ and $R\subseteq\Sigma^*$ be regular languages where $\Sigma$ is a set of constants and $X$ is a set of variables. Substituting every $x\in X$ by a regular subset $\sigma(x)$ of $\Sigma^*$ yields a regular set $\sigma(L)\subseteq\Sigma^*$. A substitution $\sigma$ solves a matching problem "$L\subseteq R$?" if $\sigma(L)\subseteq R$. There are finitely many maximal solutions $\sigma$; they are effectively computable and $\sigma(x)$ is regular for all $x\in X$; and every solution is included in a maximal one. Also, in the case of words "$\exists\sigma:\sigma(L)=R$?" is decidable. Apart from the last property, we generalize these results to infinite trees. We define a notion of choice function $\gamma$ which for any tree $s$ over $\Sigma\cup X$ and position $u$ of a variable $x$ selects at most one tree $\gamma(u)\in\sigma(x)$; next, we define $\gamma_\infty(s)$ as the limit of a Cauchy sequence; and the union over all $\gamma_\infty(s)$ yields $\sigma(s)$. Since our definition coincides with that for IO substitutions, we write $\sigma_{io}(L)$ instead of $\sigma(L)$. Our main result is the decidability of "$\exists\sigma:\sigma_{io}(L)\subseteq R$?" if $R$ is regular and $L$ belongs to a class of tree languages closed under intersection with regular sets. Such a special case pops up if $L$ is context-free. Note that "$\exists\sigma:\sigma_{io}(L)=R$?" is undecidable, in general in that case. However, the decidability of "$\exists\sigma:\sigma_{io}(L)=R$?" if both $L$ and $R$ are regular remains open because, in contrast to word languages, the homomorphic image of a regular tree language is not always regular if $\sigma(x)$ is regular for all $x\in X$.Comment: 18 pages. This replacement eliminates a false claim from the previous arXiv version of this paper: Item 4 of Theorem 1 did not hold for # = {=