Automata for unordered trees

Boiret, Adrien; Hugot, Vincent; Niehren, Joachim; Treinen, Ralf

doi:10.1016/j.ic.2016.07.012

Cited by 2 publications

(10 citation statements)

References 29 publications

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“…We choose to parametrize the counting constraints as test in the arities. One of the advantage of this choice is that it ties into one of our previous publications [17] where we create automata classes parametrized by their arity constraints.…”

Section: Counting Mso With Comparisons Of Sibling Data Values: Cmso(θ)mentioning

confidence: 99%

“…To have well-defined complexity bounds, it is usually more advisable to work with tree automata rather than in tree logics. In a previous publication [17], we explored the comparative complexities of Mso logic and automata on unordered trees, both enriched with various types of arity constraints. Notably, in Proposition 43, it is shown how the complexity of the emptiness problem for such automata is linked to the satisfiability problem for an arity constraint.…”

Section: A More Efficient Restrictionmentioning

confidence: 99%

“…Unfortunately, the classes considered in [17] do not permit data comparisons between siblings. At the other end of the spectrum, CC(Θ WSkS ) has no known complexity results that an class of automata could benefit from.…”

Section: A More Efficient Restrictionmentioning

confidence: 99%

“…To efficiently decide satisfiability of a formula of CC sufL , we use a small-model argument, then we nondeterministically guess a solution. We shall later use this logic in a class of bottom-up automata named aut(Θ L ) (see [17]). Intuitively, they are bottom-up automata on data trees, with rules of the form γ → q; to evaluate a tree in q, they test a counting constraint γ, in this case of CC sufL , on the children of the tree's root.…”

Section: A More Efficient Restrictionmentioning

confidence: 99%

“…We previously studied the complexity of decision problems for automata using various such formalisms as guards for their bottom-up transitions in [17]. The focus was on devising good notions of deterministic machines capable of executing such counting operations, sufficiently expressive while allowing for efficient algorithms.…”

Section: Introductionmentioning

confidence: 99%

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Logics for unordered trees with data constraints

Boiret

Hugot

Niehren³

et al. 2019

Journal of Computer and System Sciences

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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...

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Section: Counting Mso With Comparisons Of Sibling Data Values: Cmso(θ)mentioning

confidence: 99%

Section: A More Efficient Restrictionmentioning

confidence: 99%

Section: A More Efficient Restrictionmentioning

confidence: 99%

Section: A More Efficient Restrictionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Logics for unordered trees with data constraints

Boiret

Hugot

Niehren³

et al. 2019

Journal of Computer and System Sciences

Self Cite

View full text Add to dashboard Cite

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Presburger Büchi Tree Automata with Applications to Logics with Expressive Counting

Bednarczyk¹,

Fiuk²

2022

Logic, Language, Information, and Computation

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Order-invariant first-order logic is an extension of first-order logic (FO) where formulae can make use of a linear order on the structures, under the proviso that they are order-invariant, i.e. that their truth value is the same for all linear orders. We continue the study of the two-variable fragment of order-invariant first-order logic initiated by Zeume and Harwath, and study its complexity and expressive power. We first establish coNExpTime-completeness for the problem of deciding if a given two-variable formula is order-invariant, which tightens and significantly simplifies the coN2ExpTime proof by Zeume and Harwath. Second, we address the question of whether every property expressible in order-invariant two-variable logic is also expressible in first-order logic without the use of a linear order. While we were not able to provide a satisfactory answer to the question, we suspect that the answer is "no". To justify our claim, we present a class of finite tree-like structures (of unbounded degree) in which a relaxed variant of order-invariant two-variable FO expresses properties that are not definable in plain FO. On the other hand, we show that if one restricts their attention to classes of structures of bounded degree, then the expressive power of order-invariant two-variable FO is contained within FO.

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Automata for unordered trees

Cited by 2 publications

References 29 publications

Logics for unordered trees with data constraints

Logics for unordered trees with data constraints

Presburger Büchi Tree Automata with Applications to Logics with Expressive Counting

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