A van Benthem Theorem for Fuzzy Modal Logic

Wild, P.; Schröder, Lutz; Pattinson, Dirk; König, Barbara

doi:10.1145/3209108.3209180

Cited by 15 publications

(16 citation statements)

References 65 publications

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“…Fuzzy logic is a form of multi-valued logic later popularized as logic of vagueness by Zadeh as was reviewed by [12]. It is based on replacing the standard set of Boolean truth values with a dif f erent lattice, most often, like in the present paper, the unit interval.…”

Section: Related Workmentioning

confidence: 97%

Trilattice-Based Access Control Models: How to Secure Current Computer Network

Kucherov

Bogulskaya

2018

MATEC Web Conf.

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Designing security, from the hardware level, is essential to ensure the integrity of the intelligent cyber-physical infrastructure that is the Industrial Internet of Things (IIoT). If intelligent cyber-physical infrastructure fails to do the right things because it is insecure and vulnerable, then there will be negative social consequences [1]. Security is, in a sense, the access control to IIoT systems, which increasingly relies on the ability to compose different policies. Therefore, the advantage in any framework for compiling policies is that it is intuitive, formal, expressive, application-independent, as well as expandable to create domain-specific instances. Recently, such a scheme was proposed based on Belnap logic FOUR2 [2]. Four values of the Belnap bilattice have been interpreted as grant, deny, conflict, or unspecified with respect to access-control policy. Belnap’s four-valued logic has found a variety of applications in various fields, such as deductive database theory, distributed logic programming, and other areas. However, it turns out that the truth order in FOUR2 is a truth-and-falsity order at the same time [3]. The smallest lattice, where the orders of truth and falsity are independent of each other, which is especially important for security policy, is that of Shramko-Wansing’s SIXTEEN3. This generalization is well-motivated and leads from the bilattice FOUR2 with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN3 with an information ordering together with a truth ordering and a (distinct) falsity ordering.Based onSIXTEEN3 and new Boolean predicates to control access [4], we define an expressive access-control policy language, having composition statements based on the statements of Schramko-Wansing’s logic. Natural orderings on politics are obtained by independent lifting the orders of truth and falsity of trilattice, which results in a query language in which conflict freedom analysis can be developed. The reduction of formal verification of queries to that on predicates over access requests enables to carry out policy analysis. We evaluate our approach through examples of control access model policy.

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Section: Related Workmentioning

confidence: 97%

Trilattice-Based Access Control Models: How to Secure Current Computer Network

Kucherov

Bogulskaya

2018

MATEC Web Conf.

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“…To our best knowledge, the only previously known instances of this result in the real-valued setting are the ones for [0, 1]-valued fuzzy modal logic [57] and for quantitative probabilistic modal logic [58]. In the two-valued setting, we cover a previous coalgebraic van Benthem result [52] by instantiating to V = 2, and in fact obtain an additional asymmetric version, characterizing fragments that are preserved under simulation.…”

Section: Locality and Modal Characterizationmentioning

confidence: 99%

“…in quantitative van Benthem theorems. In previous work with Pattinson and König, we have established such theorems for quantitative modal logics of fuzzy [57] and probabilistic [58] transition systems. In the quantitative setting, these theorems take the form of approximability properties, and state that every behaviourally nonexpansive quantitative firstorder property is approximable by quantitative modal formulae of bounded rank.…”

Section: Introductionmentioning

confidence: 99%

“…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]. Prior to the quantitative Hennessy-Milner theorems mentioned above [10,37], Hennessy-Milner theorems have been established for two-valued logics and two-valued bisimilarity over quantitative systems, e.g.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Bisimulations have been studied for the guarded fragment and many variations [33], [38], [46]. Bisimulations have also been studied recently for coalgebraic modal logics [47], [48] and fuzzy modal logics [49].…”

Section: Introductionmentioning

confidence: 99%

Model Comparison Games for Horn Description Logics

Jung

Papacchini

Wolter

et al. 2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Horn description logics are syntactically defined fragments of standard description logics that fall within the Horn fragment of first-order logic and for which ontologymediated query answering is in PTIME for data complexity. They were independently introduced in modal logic to capture the intersection of Horn first-order logic with modal logic. In this paper, we introduce model comparison games for the basic Horn description logic hornALC (corresponding to the basic Horn modal logic) and use them to obtain an Ehrenfeucht-Fraïssé type definability result and a van Benthem style expressive completeness result for hornALC. We also establish a finite model theory version of the latter. The Ehrenfeucht-Fraïssé type definability result is used to show that checking hornALC indistinguishability of models is EXPTIME-complete, which is in sharp contrast to ALC indistinguishability (i.e., bisimulation equivalence) checkable in PTIME. In addition, we explore the behavior of Horn fragments of more expressive description and modal logics by defining a Horn guarded fragment of first-order logic and introducing model comparison games for it. 1 The results obtained in this paper could have been presented as a contribution to modal rather than description logic. The only reason why we have chosen the DL environment is that the impact of Horn fragments in description logic has so far been much more significant than in modal logic.

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A van Benthem Theorem for Fuzzy Modal Logic

Cited by 15 publications

References 65 publications

Trilattice-Based Access Control Models: How to Secure Current Computer Network

Trilattice-Based Access Control Models: How to Secure Current Computer Network

A Quantified Coalgebraic van Benthem Theorem

Model Comparison Games for Horn Description Logics

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