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Inquisitive modal logic, InqML, in its epistemic incarnation, extends standard epistemic logic to capture not just the information that agents have, but also the questions that they are interested in. We use the natural notion of bisimulation equivalence in the setting of InqML, as introduced in [Ciardelli/Otto: JSL 2021], to characterise the expressiveness of InqML as the bisimulation invariant fragment of first-order logic over natural classes of two-sorted first-order structures that arise as relational encodings of inquisitive epistemic (S5-like) models. The non-elementary nature of these classes crucially requires non-classical model-theoretic methods for the analysis of first-order expressiveness, irrespective of whether we aim for characterisations in the sense of classical or of finite model theory., Comment: this is a second companion paper to arXiv:1803.03483 (after Journal of Symbolic Logic 86(1), pp.77-109, 2021. doi:10.1017/jsl.2020.77). This revised version corrects a technical mistake in the main part of previous versions and attempts to improve presentation, largely in response to anonymous referee reports which we gratefully received

We present a natural standard translation of inquisitive modal logic InqML into first-order logic over the natural two-sorted relational representations of the intended models, which captures the built-in higher-order features of InqML. This translation is based on a graded notion of flatness that ties the inherent second-order, team-semantic features of InqML over information states to subsets or tuples of bounded size. A natural notion of pseudo-models, which relaxes the non-elementary constraints on the intended models, gives rise to an elementary, purely model-theoretic proof of the compactness property for InqML. Moreover, we prove a Hennessy-Milner theorem for InqML, which crucially uses $\omega$-saturated pseudo-models and the new standard translation. As corollaries we also obtain van Benthem style characterisation theorems., Comment: 23 pages; version 2: revised and expanded (new section, Section 5); version 3: revised (essential corrections in Section 5)

Inquisitive modal logic InqML is a generalisation of standard Kripke-style modal logic. In its epistemic incarnation, it extends standard epistemic logic to capture not just the information that agents have, but also the questions that they are interested in. Technically, InqML fits within the family of logics based on team semantics. From a model-theoretic perspective, it takes us a step in the direction of monadic second-order logic, as inquisitive modal operators involve quantification over sets of worlds. We introduce and investigate the natural notion of bisimulation equivalence in the setting of InqML. We compare the expressiveness of InqML and first-order logic in the context of relational structures with two sorts, one for worlds and one for information states. We characterise inquisitive modal logic, as well as its multi-agent epistemic S5-like variant, as the bisimulation invariant fragment of first-order logic over various natural classes of two-sorted structures. These results crucially require non-classical methods in studying bisimulation and first-order expressiveness over non-elementary classes of structures, irrespective of whether we aim for characterisations in the sense of classical or of finite model theory., Comment: This revised version has been prepared for publication in JSL; it covers, in greater detail, the general theory expounded in (v1) and is to be complemented by a companion paper that deals with the epistemic setting (Section 9 in (v1), which is not covered in the current revised version (v2)). arXiv admin note: text overlap with arXiv:1707.08742

Inquisitive modal logic, InqML, is a generalisation of standard Kripke-style modal logic. In its epistemic incarnation, it extends standard epistemic logic to capture not just the information that agents have, but also the questions that they are interested in. Technically, InqML fits within the family of logics based on team semantics. From a model-theoretic perspective, it takes us a step in the direction of monadic second-order logic, as inquisitive modal operators involve quantification over sets of worlds. We introduce and investigate the natural notion of bisimulation equivalence in the setting of InqML. We compare the expressiveness of InqML and first-order logic, and characterise inquisitive modal logic as the bisimulation invariant fragments of first-order logic over various classes of two-sorted relational structures. These results crucially require non-classical methods in studying bisimulations and first-order expressiveness over non-elementary classes., Comment: In Proceedings TARK 2017, arXiv:1707.08250

We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (sub-groupoid), and only counts transitions between colour classes (cosets). These groupoids are employed towards a generic construction method for finite hypergraphs that realise specified overlap patterns and avoid small cyclic configurations. The constructions are based on reduced products with groupoids generated by the elementary local extension steps, and can be made to preserve the symmetries of the given overlap pattern. In particular, we obtain highly symmetric, finite hypergraph coverings without short cycles. The groupoids and their application in reduced products are sufficiently generic to be applicable to other constructions that are specified in terms of local glueing operations and require global finite closure., Comment: Explicit completion of H in HxI (Section 2) is unstable (incompatible with restrictions), hence does not support inductive construction towards Prop. 2.17 based on Lem 2.16 as claimed. For corresponding technical result, now see arxiv:1806.08664; for discussion of main applications first announced here, now see arxiv:1709.00031

We give a new, simplified and detailed account of the correspondence between levels of the Sherali-Adams relaxation of graph isomorphism and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler-Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Tinhofer (1986, 1991) and Ramana, Scheinerman and Ullman (1994), is re-interpreted as the base level of Sherali-Adams and generalised to higher levels in this sense by Atserias and Maneva (2012) and Malkin (2014), who prove that the two resulting hierarchies interleave. In carrying this analysis further, we here give (a) a precise characterisation of the level-k Sherali-Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali-Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict. We also investigate the variation based on boolean arithmetic instead of real/rational arithmetic and obtain analogous correspondences and separations for plain k-pebble equivalence (without counting). Our results are driven by considerably simplified accounts of the underlying combinatorics and linear algebra.