A Van Benthem/Rosen theorem for coalgebraic predicate logic

Schröder, Lutz; Pattinson, Dirk; Litak, Tadeusz

doi:10.1093/logcom/exv043

Cited by 9 publications

(18 citation statements)

References 43 publications

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“…The need for assuming that the set Λ of modalities is finite is specific to quantitative Hennessy-Milner theorems (and implicitly present also in the existing [0, 1]-valued version of the theorem [37]), and not needed in the two-valued case [45,50]. It relates to the total boundedness claim in Theorem 6.1, and features also in the van Benthem theorem, where in fact it is needed also in the two-valued case [52]; indeed, proofs of the original van Benthem theorem start by assuming, in that case w.l.o.g., that there are only finitely many propositional atoms and relational modalities. In our running examples, only the ones featuring metric transition systems are affected by this assumption; indeed, for our theorems to apply to such systems, the space of labels needs to be finite.…”

Section: Modal Approximationmentioning

confidence: 99%

“…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%

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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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Section: Modal Approximationmentioning

confidence: 99%

“…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%

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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“…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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“…Sahlqvist theory is currently a very active field of research. This field has significantly broadened its scope in recent years, extending the state of the art and the benefits of Sahlqvist theory from modal logic to a class of logics which includes, among others, intuitionistic and lattice-based (modal) logics [32,20], substructural logics [44,59,21], non-normal modal logics [28,51], hybrid logics [24], many-valued logics [45], mu-calculus [71,10,8,2,9], and coalgebraic logic [46,57]. The common ground to these results is the recognition that algebraic and order-theoretic notions play a fundamental role in the various incarnations of the Sahlqvist phenomenon.…”

Section: Introductionmentioning

confidence: 99%

Algebraic modal correspondence: Sahlqvist and beyond

Conradie

Palmigiano

Sourabh

2017

Journal of Logical and Algebraic Methods in Programming

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The present paper proposes a new introductory treatment of the very well known Sahlqvist correspondence theory for classical modal logic. The first motivation for the present treatment is pedagogical: classical Sahlqvist correspondence is presented in a uniform and modular way, and, unlike the existing textbook accounts, extends itself to a class of formulas laying outside the Sahlqvist class proper. The second motivation is methodological: the present treatment aims at highlighting the algebraic and order-theoretic nature of the correspondence mechanism. The exposition remains elementary and does not presuppose any previous knowledge or familiarity with the algebraic approach to logic. However, it provides the underlying motivation and basic intuitions for the recent developments in the Sahlqvist theory of nonclassical logics, which compose the so-called unified correspondence theory.

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A Van Benthem/Rosen theorem for coalgebraic predicate logic

Cited by 9 publications

References 43 publications

A Quantified Coalgebraic van Benthem Theorem

A Quantified Coalgebraic van Benthem Theorem

Model Comparison Games for Horn Description Logics

Algebraic modal correspondence: Sahlqvist and beyond

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