A note on the interpretability logic of finitely axiomatized theories

Rijke, Maarten de

doi:10.1007/bf00370185

Cited by 4 publications

(4 citation statements)

References 7 publications

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“…However, this could cause issues for qualitative comparison since technically ϕ is a different value from Xϕ. Another solution would be to introduce the notion of "graded" G, G x , which operates similar to graded connectives in standard modal logic [24] and counts the number of times that the value is violated.…”

Section: Repeated Value Violationsmentioning

confidence: 99%

Ethical Planning with Multiple Temporal Values

Parker

Grandi

Lorini

et al. 2023

Social Robots in Social Institutions

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Section: Repeated Value Violationsmentioning

confidence: 99%

Ethical Planning with Multiple Temporal Values

Parker

Grandi

Lorini

et al. 2023

Social Robots in Social Institutions

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“…The literature is by now too extensive to be reviewed here; two notable results are that the finite model property was established by van der Hoek [18], and that Tobies [27] showed that the satisfiability problem for can be solved in polynomial space. Of specific interest to our investigations is the work of de Rijke [24] who introduced a notion of bisimulation that is appropriate for graded modal logic in the sense that he showed to correspond to the bisimulation-invariant fragment of first-order logic, thus transferring van Benthem’s seminal characterisation of basic modal logic to the setting with counting modalities.…”

Section: Introductionmentioning

confidence: 99%

“…This line of research started with van Benthem's seminal result, which identifies basic modal logic as the bisimulation invariant fragment of first-order logic [3]. This result has been extended in many directions; for instance, de Rijke was the first to prove a similar result for graded modal logic [24]. Van Benthem's and de Rijke's proofs are rooted in classical model theory, with a prominent role for the notion of compactness.…”

Section: Introductionmentioning

confidence: 99%

Counting to Infinity: Graded Modal Logic With an Infinity Diamond

ACOSTA¹,

Venema²

2022

The Review of Symbolic Logic

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We extend the languages of both basic and graded modal logic with the infinity diamond, a modality that expresses the existence of infinitely many successors having a certain property. In both cases we define a natural notion of bisimilarity for the resulting formalisms, that we dub $\mathtt {ML}^{\infty }$ and $\mathtt {GML}^{\infty }$ , respectively. We then characterise these logics as the bisimulation-invariant fragments of the naturally corresponding predicate logic, viz., the extension of first-order logic with the infinity quantifier. Furthermore, for both $\mathtt {ML}^{\infty }$ and $\mathtt {GML}^{\infty }$ we provide a sound and complete axiomatisation for the set of formulas that are valid in every Kripke frame, we prove a small model property with respect to a widened class of weighted models, and we establish decidability of the satisfiability problem.

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“…However, another view consists of seeing these logics as computationally well-behaved fragments of first-order logic and second-order logic (see e.g., [1] for a discussion). Some examples of well-known modal logics with a good balance between expressivity and computational complexity are graded modal logic (GML) [5,28], whose satisfiability problem is PSpacecomplete; and the temporal logics LTL, CTL and CTL * whose satisfiability problems are complete for PSpace, Exp and 2Exp, respectively [31,19,25].…”

Section: Introductionmentioning

confidence: 99%

Modal Logics and Local Quantifiers: A Zoo in the Elementary Hierarchy

Fervari

Mansutti

2022

Lecture Notes in Computer Science

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We study a family of modal logics interpreted on tree-like structures, and featuring local quantifiers $$\exists ^{k}p$$ ∃ k p that bind the proposition p to worlds that are accessible from the current one in at most k steps. We consider a first-order and a second-order semantics for the quantifiers, which enables us to relate several well-known formalisms, such as hybrid logics, $$\textsf {S5Q}$$ S 5 Q and graded modal logic. To better stress these connections, we explore fragments of our logics, called herein round-bounded fragments. Depending on whether first or second-order semantics is considered, these fragments populate the hierarchy $${2\textsc {NExp} \subset 3\textsc {NExp} \subset \cdots }$$ 2 NE X P ⊂ 3 NE X P ⊂ ⋯ or the hierarchy $${2\textsc {AExp}_{pol} \subset 3\textsc {AExp}_{pol} \subset \cdots }$$ 2 AE X P pol ⊂ 3 AE X P pol ⊂ ⋯ , respectively. For formulae up-to modal depth k, the complexity improves by one exponential.

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A note on the interpretability logic of finitely axiomatized theories

Cited by 4 publications

References 7 publications

Ethical Planning with Multiple Temporal Values

Ethical Planning with Multiple Temporal Values

Counting to Infinity: Graded Modal Logic With an Infinity Diamond

Modal Logics and Local Quantifiers: A Zoo in the Elementary Hierarchy

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