Two-Variable Logic with Two Order Relations

Schwentick, Thomas; Zeume, Thomas

doi:10.2168/lmcs-8(1:15)2012

Cited by 22 publications

(43 citation statements)

References 16 publications

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“…A few works such as [31], [35], [36] are on words, while in most applications we need to consider trees. Moreover, these works are incomparable to some interesting existing formalisms [17], [6], [2], [12], [24], [14], [29] known to be able to capture various interesting scenarios common in practice.…”

Section: Introductionmentioning

confidence: 99%

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An Automata Model for Trees with Ordered Data Values

Tan

2012

2012 27th Annual IEEE Symposium on Logic in Computer Science

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Data trees are trees in which each node, besides carrying a label from a finite alphabet, also carries a data value from an infinite domain. They have been used as an abstraction model for reasoning tasks on XML and verification. However, most existing approaches consider the case where only equality test can be performed on the data values.In this paper we study data trees in which the data values come from a linearly ordered domain, and in addition to equality test, we can test whether the data value in a node is greater than the one in another node. We introduce an automata model for them which we call ordered-data tree automata (ODTA), provide its logical characterisation, and prove that its emptiness problem is decidable in 3-NEXPTIME. We also show that the twovariable logic on unranked trees, studied by Bojanczyk, Muscholl, Schwentick and Segoufin in 2009, corresponds precisely to a special subclass of this automata model.Then we define a slightly weaker version of ODTA, which we call weak ODTA, and provide its logical characterisation. The complexity of the emptiness problem drops to NP. However, a number of existing formalisms and models studied in the literature can be captured already by weak ODTA. We also show that the definition of ODTA can be easily modified, to the case where the data values come from a tree-like partially ordered domain, such as strings.

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Section: Introductionmentioning

confidence: 99%

“…In [35] it is shown that the satisfaction problem of the logic FO 2 (<, ≺) on words is decidable. This logic is incomparable with our model.…”

Section: Introductionmentioning

confidence: 99%

An Automata Model for Trees with Ordered Data Values

Tan

2012

2012 27th Annual IEEE Symposium on Logic in Computer Science

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“…Satisfiability and finite satisfiability of C 2 with a single equivalence relation was shown to be NEXPTIME-complete, while C 2 with two equivalence relations is undecidable [26]. Decidability of extensions of F O 2 with successor relations, order relations, and equivalence relations without counting were studied [13], [16], [17], [20], [21], [30].…”

Section: The Two-variable Fragment and Alcqiomentioning

confidence: 99%

Extending ALCQIO with Trees

Kotek

imkus

Veith

et al. 2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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We study the description logic ALCQIO, which extends the standard description logic ALC with nominals, inverses and counting quantifiers. ALCQIO is a fragment of first order logic and thus cannot define trees. We consider the satisfiability problem of ALCQIO over finite structures in which k relations are interpreted as forests of directed trees with unbounded outdegrees.We show that the finite satisfiability problem of ALCQIO with forests is polynomial-time reducible to finite satisfiability of ALCQIO. As a consequence, we get that finite satisfiability is NEXPTIME-complete. Description logics with transitive closure constructors or fixed points have been studied before, but we give the first decidability result of the finite satisfiability problem for a description logic that contains nominals, inverse roles, and counting quantifiers and can define trees.

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“…Most of them turn out to have undecidable satisfiability problem (see, e.g., [10,16]). On the positive side, FO 2 remains decidable when extended by counting quantifiers [9,26,27], counting quantifiers and two forests accessible by their successor relations [5] (finite satisfiability only), a linear order [25] (or, in the case of finite satisfiability, even two linear orders, subject to some further restrictions on the signatures, see [29]), one or two equivalence relations [18,19], one transitive relation [31] (general satisfiability only, the case of finite models is open).…”

Section: Related Workmentioning

confidence: 99%

Decidability of weak logics with deterministic transitive closure

Charatonik

Kieroński

Mazowiecki

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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The deterministic transitive closure operator, added to languages containing even only two variables, allows to express many natural properties of a binary relation, including being a linear order, a tree, a forest or a partial function. This makes it a potentially attractive ingredient of computer science formalisms. In this paper we consider the extension of the two-variable fragment of first-order logic by the deterministic transitive closure of a single binary relation, and prove that the satisfiability and finite satisfiability problems for the obtained logic are decidable and EXPSPACE-complete. This contrasts with the undecidability of two-variable logic with the deterministic transitive closures of several binary relations, known before. We also consider the class of universal first-order formulas in prenex form. Its various extensions by deterministic closure operations were earlier considered by other authors, leading to both decidability and undecidability results. We examine this scenario in more details.

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Two-Variable Logic with Two Order Relations

Cited by 22 publications

References 16 publications

An Automata Model for Trees with Ordered Data Values

An Automata Model for Trees with Ordered Data Values

Extending ALCQIO with Trees

Decidability of weak logics with deterministic transitive closure

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