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 # = {=

Regular matching problems for infinite trees

We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an alphabet of variables and $\sigma$ is a substitution such that $\sigma(x)$ is a set of trees in $T(\Sigma\cup H)\setminus H$ for all $x\in \mathcal{X}$. Here, $H$ denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if $L$ and $R$ are regular, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma(L)\subseteq R$?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when $L,R$ are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if $L\subseteq T(\Sigma\cup\mathcal{X})$ and $R\subseteq T(\Sigma)$ are regular tree languages over finite or infinite trees, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma_{\mathrm{io}}(L)\subseteq R$?" is decidable. Here, the subscript "$\mathrm{io}$" in $\sigma_{\mathrm{io}}(L)$ refers to "inside-out". Moreover, there are only finitely many maximal solutions $\sigma$, the maximal solutions are regular substitutions and effectively computable. The corresponding question for the outside-in extension $\sigma_{\mathrm{oi}}$ remains open, even in the restricted setting of finite trees

A Local Computation Approximation Scheme to Maximum Matching

We present a polylogarithmic local computation matching algorithm which guarantees a (1-\eps)-approximation to the maximum matching in graphs of bounded degree.Comment: Appears in Approx 201

Locally Optimal Load Balancing

This work studies distributed algorithms for locally optimal load-balancing: We are given a graph of maximum degree $\Delta$, and each node has up to $L$ units of load. The task is to distribute the load more evenly so that the loads of adjacent nodes differ by at most $1$. If the graph is a path ($\Delta = 2$), it is easy to solve the fractional version of the problem in $O(L)$ communication rounds, independently of the number of nodes. We show that this is tight, and we show that it is possible to solve also the discrete version of the problem in $O(L)$ rounds in paths. For the general case ($\Delta > 2$), we show that fractional load balancing can be solved in $\operatorname{poly}(L,\Delta)$ rounds and discrete load balancing in $f(L,\Delta)$ rounds for some function $f$, independently of the number of nodes.Comment: 19 pages, 11 figure

A survey of max-type recursive distributional equations

In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X =^d g((\xi_i,X_i),i\geq 1). Here (\xi_i) and g(\cdot) are given and the X_i are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(\cdot) is essentially a ``maximum'' or ``minimum'' function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process X_i, are the X_i measurable functions of the innovations process (\xi_i)?Comment: Published at http://dx.doi.org/10.1214/105051605000000142 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Deterministic Automata for Unordered Trees

Automata for unordered unranked trees are relevant for defining schemas and queries for data trees in Json or Xml format. While the existing notions are well-investigated concerning expressiveness, they all lack a proper notion of determinism, which makes it difficult to distinguish subclasses of automata for which problems such as inclusion, equivalence, and minimization can be solved efficiently. In this paper, we propose and investigate different notions of "horizontal determinism", starting from automata for unranked trees in which the horizontal evaluation is performed by finite state automata. We show that a restriction to confluent horizontal evaluation leads to polynomial-time emptiness and universality, but still suffers from coNP-completeness of the emptiness of binary intersections. Finally, efficient algorithms can be obtained by imposing an order of horizontal evaluation globally for all automata in the class. Depending on the choice of the order, we obtain different classes of automata, each of which has the same expressiveness as CMso.Comment: In Proceedings GandALF 2014, arXiv:1408.556

Infinite games with finite knowledge gaps

Infinite games where several players seek to coordinate under imperfect information are deemed to be undecidable, unless the information is hierarchically ordered among the players. We identify a class of games for which joint winning strategies can be constructed effectively without restricting the direction of information flow. Instead, our condition requires that the players attain common knowledge about the actual state of the game over and over again along every play. We show that it is decidable whether a given game satisfies the condition, and prove tight complexity bounds for the strategy synthesis problem under $\omega$-regular winning conditions given by parity automata.Comment: 39 pages; 2nd revision; submitted to Information and Computatio