2,018 research outputs found
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
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
Uniform Interpolation for Coalgebraic Fixpoint Logic
We use the connection between automata and logic to prove that a wide class
of coalgebraic fixpoint logics enjoys uniform interpolation. To this aim, first
we generalize one of the central results in coalgebraic automata theory, namely
closure under projection, which is known to hold for weak-pullback preserving
functors, to a more general class of functors, i.e.; functors with
quasi-functorial lax extensions. Then we will show that closure under
projection implies definability of the bisimulation quantifier in the language
of coalgebraic fixpoint logic, and finally we prove the uniform interpolation
theorem
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
Strongly Complete Logics for Coalgebras
Coalgebras for a functor model different types of transition systems in a
uniform way. This paper focuses on a uniform account of finitary logics for
set-based coalgebras. In particular, a general construction of a logic from an
arbitrary set-functor is given and proven to be strongly complete under
additional assumptions. We proceed in three parts. Part I argues that sifted
colimit preserving functors are those functors that preserve universal
algebraic structure. Our main theorem here states that a functor preserves
sifted colimits if and only if it has a finitary presentation by operations and
equations. Moreover, the presentation of the category of algebras for the
functor is obtained compositionally from the presentations of the underlying
category and of the functor. Part II investigates algebras for a functor over
ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical
extensions of Boolean algebras with operators to this setting. Part III shows,
based on Part I, how to associate a finitary logic to any finite-sets
preserving functor T. Based on Part II we prove the logic to be strongly
complete under a reasonable condition on T
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
Coalgebraic Geometric Logic: Basic Theory
Using the theory of coalgebra, we introduce a uniform framework for adding
modalities to the language of propositional geometric logic. Models for this
logic are based on coalgebras for an endofunctor on some full subcategory of
the category of topological spaces and continuous functions. We investigate
derivation systems, soundness and completeness for such geometric modal logics,
and we we specify a method of lifting an endofunctor on Set, accompanied by a
collection of predicate liftings, to an endofunctor on the category of
topological spaces, again accompanied by a collection of (open) predicate
liftings. Furthermore, we compare the notions of modal equivalence, behavioural
equivalence and bisimulation on the resulting class of models, and we provide a
final object for the corresponding category
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
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