Bounded Depth Data Trees

Björklund, Henrik; Bojańczyk, Mikołaj

doi:10.1007/978-3-540-73420-8_74

Cited by 12 publications

(11 citation statements)

References 28 publications

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“…This has motivated various works on data words [3], [8], [14], [22], [25], [33], as well as on data trees [4], [6], [20], [21], [24]. The common feature of these works is the addition of equality test on the data values to the logic on trees.…”

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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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%

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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“…The extension to trees makes also sense to model XML documents with values, see e.g. [4,5,6]. In order to really speak about data, known logical formalisms for data words/trees contain a mechanism that stores a value and tests it later against other values, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Model checking memoryful linear-time logics over one-counter automata

Demri¹,

Lazi²

2010

Theoretical Computer Science

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We study complexity of the model-checking problems for LTL with registers (also known as freeze LTL and written LTL ↓ ) and for first-order logic with data equality tests (written FO(∼, <, +1)) over one-counter automata. We consider several classes of one-counter automata (mainly deterministic vs. nondeterministic) and several logical fragments (restriction on the number of registers or variables and on the use of propositional variables for control states). The logics have the ability to store a counter value and to test it later against the current counter value. We show that model checking LTL ↓ and FO(∼, <, +1) over deterministic one-counter automata is PSpace-complete with infinite and finite accepting runs. By constrast, we prove that model checking LTL ↓ in which the until operator U is restricted to the eventually F over nondeterministic one-counter automata is Σ 1 1 -complete [resp. Σ 0 1 -complete] in the infinitary [resp. finitary] case even if only one register is used and with no propositional variable. As a corollary of our proof, this also holds for FO(∼, <, +1) restricted to two variables (written FO 2 (∼, <, +1)). This makes a difference with the facts that several verification problems for one-counter automata are known to be decidable with relatively low complexity, and that finitary satisfiability for LTL ↓ and FO 2 (∼, <, +1) are decidable. Our results pave the way for model-checking memoryful (linear-time) logics over other classes of operational models, such as reversal-bounded counter machines. automata, see e.g. [16,17,18,19]. However, the storing mechanism has a long tradition (apart from its ubiquity in programming languages) since it appeared for instance in real-time logics [20] (the data are time values) and in so-called hybrid logics (the data are node addresses), see an early undecidability result with reference pointers in [21]. Meaningful restrictions for hybrid logics can also lead to decidable fragments, see e.g. [22].Our motivations. In this paper, our main motivation is to analyze the effects of adding a binding mechanism with registers to specify runs of operational models such as pushdown systems and counter automata. The registers are simple means to compare data values at different points of the execution. Indeed, runs can be naturally viewed as data words: for example, the finite alphabet is the set of control states and the infinite alphabet is the set of data values (natural numbers, stacks, etc.). To do so, we enrich an ubiquitous logical formalism for model-checking techniques, namely linear-time temporal logic LTL, with registers. Even though this was the initial motivation to introduce LTL with registers in [12], most decision problems considered in [12,13,8] are essentially oriented towards satisfiability. In this paper, we focus on the following type of model-checking problem: given a set of runs generated by an operational model, more precisely by a one-counter automaton, and a formula from LTL with registers, is there a run satisfying the given formula? In our ...

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“…Like in [3], we take that approach for technical reasons. However, each N, Σ, n R , , , λ as above can be seen as a binary tree with root n R , where left-hand and right-hand children are given by relations 1 and , respectively.…”

Section: Trees Contractions and Data Treesmentioning

confidence: 99%

“…Decidability of the latter is a difficult open question, and it is equivalent to decidability of multiplicative exponential linear logic [10]. In [3], FO 2 (+1, <, ∼) is shown decidable over finite data trees of fixed bounded depth.…”

Section: Introductionmentioning

confidence: 99%