Logics for unordered trees with data constraints

Abstract

We study monadic second-order logics with counting constraints (CMso) for unordered data trees. Our objective is to enhance this logic with data constraints for comparing string data values. Comparisons between data values at arbitrary positions of a data tree quickly lead to undecidability. Therefore, we restrict ourselves to comparing sibling data values of unordered trees. But even in this case CMso remains undecidable when allowing for data comparisons that can check the equality of string factors. However, for more restricted data constraints that can only check the equality of string prefixes, it becomes decidable. This decidability result is obtained by reduction to WSkS. Furthermore, we exhibit a restricted class of constraints which can be used in transitions of tree automata, resulting in a model with tractable complexity, which can be extended with structural equality tests between siblings. This efficient restriction is relevant to applications such as checking well-formedness properties of file system trees. more general graphs or structures is undecidable. In this paper, we will consider Mso over unordered data trees. This means that we annotate the elements of the data tree with strings or other data values from an infinite alphabet. Depending on which relations on data values are supported, unordered data trees subsume graphs, so that Mso becomes undecidable again.Unordered data trees are a versatile data structure that is of interest in various domains of computer science. More recently, they were used as data models of semi-structured databases, such as for NoSql databases [3] or for Xml databases [4,5,6]. Here, Mso can be used both as a query language and as a schema language. Unordered data trees also have a long history for modeling syntactic structures in computational linguistics [7] and records in programming languages [8,9,10]. The unordered data trees in this context were called feature trees and the corresponding logics feature logics [11,12]. Our own motivation is to model file systems, i.e. trees representing directories, files names, their contents etc.Yet, so far, Mso for unordered trees has been studied without data, that is over finite alphabets. The two main variants of Mso for unordered trees that were proposed are Presburger Mso and Counting Mso [4, 5, 6]. Both logics were proven decidable by reduction to corresponding notions of automata for unranked trees. The weaker logic, Counting Mso or CMso for short, is considered a canonical language for characterising recognizable sets, and is actually equivalent to Mso in the case of ordered trees or words [13]. In this present paper, we generalize CMso to unordered data trees with arbitrary ranks, and study the expressiveness of the resulting logics.The most general extension of CMso would enable comparisons of data values between arbitrary locations, but this immediately leads to undecidability. Indeed even Mso on data words with equality tests between the data values of arbitrary positions is undecidable [14] since it can be reduce...