Model theory of XPath on data trees. Part II: Binary bisimulation and definability

Abriola, Sergio; Descotte, María Emilia; Figueira, Santiago

doi:10.1016/j.ic.2017.01.002

Cited by 10 publications

(9 citation statements)

References 11 publications

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“…The best transport plan is to transport 1 2 − ϵ from x 1 to 1 (where the distance is ϵ), ϵ from x 1 to 2 (distance 1 2 ) and 1 2 from x 2 to 2 (distance 0). So, summing up, we have d W 3 (x, ) = ( 1 2 − ϵ) · ϵ + ϵ · 1 2 + 1 2 · 0 = ϵ − ϵ 2 .…”

Section: Behavioural Distances and Gamesmentioning

confidence: 99%

A van Benthem Theorem for Fuzzy Modal Logic

Wild

Schröder

Pattinson

et al. 2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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In probabilistic transition systems, behavioural metrics provide a more fine-grained and stable measure of system equivalence than crisp notions of bisimilarity. They correlate strongly to quantitative probabilistic logics, and in fact the distance induced by a probabilistic modal logic taking values in the real unit interval has been shown to coincide with behavioural distance. For probabilistic systems, probabilistic modal logic thus plays an analogous role to that of Hennessy-Milner logic on classical labelled transition systems. In the quantitative setting, invariance of modal logic under bisimilarity becomes non-expansivity of formula evaluation w.r.t. behavioural distance. In the present paper, we provide a characterization of the expressive power of probabilistic modal logic based on this observation: We prove a probabilistic analogue of the classical van Benthem theorem, which states that modal logic is precisely the bisimulation-invariant fragment of first-order logic. Specifically, we show that quantitative probabilistic modal logic lies dense in the bisimulation-invariant fragment, in the indicated sense of non-expansive formula evaluation, of quantitative probabilistic first-order logic; more precisely, bisimulation-invariant first-order formulas are approximable by modal formulas of bounded rank.For a description logic perspective on the same result, see [46].

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Section: Behavioural Distances and Gamesmentioning

confidence: 99%

A van Benthem Theorem for Fuzzy Modal Logic

Wild

Schröder

Pattinson

et al. 2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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“…Also, there is a continuation (a 'Part II'): a work by Abriola, Descotte, and Figueira (2015), after the preliminary version by Abriola, Descotte, and Figueira (2014), where they address further model theoretical questions such as definability and separation, and bisimulation notions for pairs of nodes (instead of single nodes) to capture the idea of "indistinguishability by means of path expressions" (instead of node expressions).…”

Section: Related Workmentioning

confidence: 99%

Model Theory of XPath on Data Trees. Part I: Bisimulation and Characterization

Figueira

Areces

2015

jair

Self Cite

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We investigate model theoretic properties of XPath with data (in)equality tests over the class of data trees, i.e., the class of trees where each node contains a label from a finite alphabet and a data value from an infinite domain.We provide notions of (bi)simulations for XPath logics containing the child, parent, ancestor and descendant axes to navigate the tree. We show that these notions precisely characterize the equivalence relation associated with each logic. We study formula complexity measures consisting of the number of nested axes and nested subformulas in a formula; these notions are akin to the notion of quantifier rank in first-order logic. We show characterization results for fine grained notions of equivalence and (bi)simulation that take into account these complexity measures. We also prove that positive fragments of these logics correspond to the formulas preserved under (non-symmetric) simulations. We show that the logic including the child axis is equivalent to the fragment of first-order logic invariant under the corresponding notion of bisimulation. If upward navigation is allowed the characterization fails but a weaker result can still be established. These results hold both over the class of possibly infinite data trees and over the class of finite data trees.Besides their intrinsic theoretical value, we argue that bisimulations are useful tools to prove (non)expressivity results for the logics studied here, and we substantiate this claim with examples.

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“…In the two-valued setting, there has been increased recent interest in variants and generalizations of this result (e.g. [54,14,52,22,55,1])…”

Section: Introductionmentioning

confidence: 99%

“…Related Work There is a substantial body of work on two-valued modal characterization theorems, e.g. for logics with frame conditions [14], coalgebraic modal logics [52], fragments of XPath [12,22,1], neighbourhood logic [28], modal logic with team semantics [38], modal µ-calculi (within monadic second order logics) [35,19], PDL (within weak chain logic) [11], modal first-order logics [6,54], and two-dimensional modal logics with an S5-modality [55]. We are not aware of quantitative modal characterization theorems other than the mentioned ones for fuzzy and probabilistic modal logics [57,58].…”

Section: Introductionmentioning

confidence: 99%

A Quantified Coalgebraic van Benthem Theorem

Wild

Schröder

2021

Lecture Notes in Computer Science

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The classical van Benthem theorem characterizes modal logic as the bisimulation-invariant fragment of first-order logic; put differently, modal logic is as expressive as full first-order logic on bisimulation-invariant properties. This result has recently been extended to two flavours of quantitative modal logic, viz. fuzzy modal logic and probabilistic modal logic. In both cases, the quantitative van Benthem theorem states that every formula in the respective quantitative variant of first-order logic that is bisimulation-invariant, in the sense of being nonexpansive w.r.t. behavioural distance, can be approximated by quantitative modal formulae of bounded rank. In the present paper, we unify and generalize these results in three directions: We lift them to full coalgebraic generality, thus covering a wide range of system types including, besides fuzzy and probabilistic transition systems as in the existing examples, e.g. also metric transition systems; and we generalize from real-valued to quantale-valued behavioural distances, e.g. nondeterministic behavioural distances on metric transition systems; and we remove the symmetry assumption on behavioural distances, thus covering also quantitative notions of simulation.

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Model theory of XPath on data trees. Part II: Binary bisimulation and definability

Cited by 10 publications

References 11 publications

A van Benthem Theorem for Fuzzy Modal Logic

A van Benthem Theorem for Fuzzy Modal Logic

Model Theory of XPath on Data Trees. Part I: Bisimulation and Characterization

A Quantified Coalgebraic van Benthem Theorem

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