29 research outputs found
Bisimulation in Inquisitive Modal Logic
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:1707.0825
A Characterization Theorem for a Modal Description Logic
Modal description logics feature modalities that capture dependence of
knowledge on parameters such as time, place, or the information state of
agents. E.g., the logic S5-ALC combines the standard description logic ALC with
an S5-modality that can be understood as an epistemic operator or as
representing (undirected) change. This logic embeds into a corresponding modal
first-order logic S5-FOL. We prove a modal characterization theorem for this
embedding, in analogy to results by van Benthem and Rosen relating ALC to
standard first-order logic: We show that S5-ALC with only local roles is, both
over finite and over unrestricted models, precisely the bisimulation invariant
fragment of S5-FOL, thus giving an exact description of the expressive power of
S5-ALC with only local roles
Graded modal logic and counting bisimulation
This note sketches the extension of the basic characterisation theorems as
the bisimulation-invariant fragment of first-order logic to modal logic with
graded modalities and matching adaptation of bisimulation. We focus on showing
expressive completeness of graded multi-modal logic for those first-order
properties of pointed Kripke structures that are preserved under counting
bisimulation equivalence among all or among just all finite pointed Kripke
structures
Recurrent graph neural networks and their connections to bisimulation and logic
The success of Graph Neural Networks (GNNs) in practice
has motivated extensive research on their theoretical properties. This includes recent results that characterise node classifiers expressible by GNNs in terms of first order logic.
Most of the analysis, however, has been focused on GNNs
with fixed number of message-passing iterations (i.e., layers), which cannot realise many simple classifiers such as
reachability of a node with a given label. In this paper, we
start to fill this gap and study the foundations of GNNs that
can perform more than a fixed number of message-passing
iterations. We first formalise two generalisations of the basic
GNNs: recurrent GNNs (RecGNNs), which repeatedly apply
message-passing iterations until the node classifications become stable, and graph-size GNNs (GSGNNs), which exploit
a built-in function of the input graph size to decide the number of message-passings. We then formally prove that GNN
classifiers are strictly less expressive than RecGNN ones, and
RecGNN classifiers are strictly less expressive than GSGNN
ones. To get this result, we identify novel semantic characterisations of the three formalisms in terms of suitable variants
of bisimulation, which we believe have their own value for
our understanding of GNNs. Finally, we prove syntactic logical characterisations of RecGNNs and GSGNNs analogous to
the logical characterisation of plain GNNs, where we connect
the two formalisms to monadic monotone fixpoint logic—a
generalisation of first-order logic that supports recursion
A model category for modal logic
We define Quillen model structures on a family of presheaf toposes arising
from tree unravellings of Kripke models, leading to a homotopy theory for modal
logic. Modal preservation theorems and the Hennessy-Milner property are
revisited from a homotopical perspective.Comment: 25 page
Inquisitive bisimulation
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
Weak MSO: Automata and Expressiveness Modulo Bisimilarity
We prove that the bisimulation-invariant fragment of weak monadic
second-order logic (WMSO) is equivalent to the fragment of the modal
-calculus where the application of the least fixpoint operator is restricted to formulas that are continuous in . Our
proof is automata-theoretic in nature; in particular, we introduce a class of
automata characterizing the expressive power of WMSO over tree models of
arbitrary branching degree. The transition map of these automata is defined in
terms of a logic that is the extension of first-order
logic with a generalized quantifier , where means that there are infinitely many objects satisfying . An
important part of our work consists of a model-theoretic analysis of
.Comment: Technical Report, 57 page