70 research outputs found
Monadic Second-Order Logic and Bisimulation Invariance for Coalgebras
Generalizing standard monadic second-order logic for Kripke models, we
introduce monadic second-order logic interpreted over coalgebras for an
arbitrary set functor. Similar to well-known results for monadic second-order
logic over trees, we provide a translation of this logic into a class of
automata, relative to the class of coalgebras that admit a tree-like supporting
Kripke frame. We then consider invariance under behavioral equivalence of
formulas; more in particular, we investigate whether the coalgebraic
mu-calculus is the bisimulation-invariant fragment of monadic second-order
logic. Building on recent results by the third author we show that in order to
provide such a coalgebraic generalization of the Janin-Walukiewicz Theorem, it
suffices to find what we call an adequate uniform construction for the functor.
As applications of this result we obtain a partly new proof of the
Janin-Walukiewicz Theorem, and bisimulation invariance results for the bag
functor (graded modal logic) and all exponential polynomial functors.
Finally, we consider in some detail the monotone neighborhood functor, which
provides coalgebraic semantics for monotone modal logic. It turns out that
there is no adequate uniform construction for this functor, whence the
automata-theoretic approach towards bisimulation invariance does not apply
directly. This problem can be overcome if we consider global bisimulations
between neighborhood models: one of our main technical results provides a
characterization of the monotone modal mu-calculus extended with the global
modalities, as the fragment of monadic second-order logic for the monotone
neighborhood functor that is invariant for global bisimulations
An expressive completeness theorem for coalgebraic modal mu-calculi
Generalizing standard monadic second-order logic for Kripke models, we
introduce monadic second-order logic interpreted over coalgebras for an
arbitrary set functor. We then consider invariance under behavioral equivalence
of MSO-formulas. More specifically, we investigate whether the coalgebraic
mu-calculus is the bisimulation-invariant fragment of the monadic second-order
language for a given functor. Using automatatheoretic techniques and building
on recent results by the third author, we show that in order to provide such a
characterization result it suffices to find what we call an adequate uniform
construction for the coalgebraic type functor. As direct applications of this
result we obtain a partly new proof of the Janin-Walukiewicz Theorem for the
modal mu-calculus, avoiding the use of syntactic normal forms, and bisimulation
invariance results for the bag functor (graded modal logic) and all exponential
polynomial functors (including the "game functor"). As a more involved
application, involving additional non-trivial ideas, we also derive a
characterization theorem for the monotone modal mu-calculus, with respect to a
natural monadic second-order language for monotone neighborhood models.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0721
Expressiveness of the modal mu-calculus on monotone neighborhood structures
We characterize the expressive power of the modal mu-calculus on monotone
neighborhood structures, in the style of the Janin-Walukiewicz theorem for the
standard modal mu-calculus. For this purpose we consider a monadic second-order
logic for monotone neighborhood structures. Our main result shows that the
monotone modal mu-calculus corresponds exactly to the fragment of this
second-order language that is invariant for neighborhood bisimulations
Disjunctive bases: normal forms and model theory for modal logics
We present the concept of a disjunctive basis as a generic framework for
normal forms in modal logic based on coalgebra. Disjunctive bases were defined
in previous work on completeness for modal fixpoint logics, where they played a
central role in the proof of a generic completeness theorem for coalgebraic
mu-calculi. Believing the concept has a much wider significance, here we
investigate it more thoroughly in its own right. We show that the presence of a
disjunctive basis at the "one-step" level entails a number of good properties
for a coalgebraic mu-calculus, in particular, a simulation theorem showing that
every alternating automaton can be transformed into an equivalent
nondeterministic one. Based on this, we prove a Lyndon theorem for the full
fixpoint logic, its fixpoint-free fragment and its one-step fragment, a Uniform
Interpolation result, for both the full mu-calculus and its fixpoint-free
fragment, and a Janin-Walukiewicz-style characterization theorem for the
mu-calculus under slightly stronger assumptions.
We also raise the questions, when a disjunctive basis exists, and how
disjunctive bases are related to Moss' coalgebraic "nabla" modalities. Nabla
formulas provide disjunctive bases for many coalgebraic modal logics, but there
are cases where disjunctive bases give useful normal forms even when nabla
formulas fail to do so, our prime example being graded modal logic. We also
show that disjunctive bases are preserved by forming sums, products and
compositions of coalgebraic modal logics, providing tools for modular
construction of modal logics admitting disjunctive bases. Finally, we consider
the problem of giving a category-theoretic formulation of disjunctive bases,
and provide a partial solution
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
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