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COUNTING TO INFINITY: GRADED MODAL LOGIC WITH AN INFINITY DIAMOND
Published online by Cambridge University Press: 08 July 2022
Abstract
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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- © The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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