96 research outputs found
Bisimulation for Weakly Expressive Coalgebraic Modal Logics
Research on the expressiveness of coalgebraic modal logics with respect to semantic equivalence notions has so far focused mainly on finding logics that are able to distinguish states that are not behaviourally equivalent (such logics are said to be expressive). In other words, the notion of behavioural equivalence is taken as the starting point, and the expressiveness of the logic is evaluated against it.
However, for some applications, modal logics that are not expressive are of independent interest. Such an example is given by contingency logic.
We can now turn the question of expressiveness around and ask, given a modal logic, what is a suitable notion of semantic equivalence? In this paper, we propose a notion of Lambda-bisimulation which is parametric in a collection
Lambda of predicate liftings. We study the basic properties of Lambda-bisimilarity, and prove as our main result a Hennessy-Milner style theorem, which shows that (for finitary functors) Lambda-bisimilarity exactly matches the expressiveness of the coalgebraic modal logic arising from Lambda
A (Co)algebraic Approach to Hennessy-Milner Theorems for Weakly Expressive Logics
Coalgebraic modal logic, as in [9, 6], is a framework in which modal logics for specifying coalgebras can be developed parametric in the signature of the modal language and the coalgebra type functor T. Given a base logic (usually classical propositional logic), modalities are interpreted via so-called predicate liftings for the functor T. These are natural transformations that turn a predicate over the state space X into a predicate over TX. Given that T-coalgebras come with general notions of T-bisimilarity [11] and behavioral equivalence [7], coalgebraic modal logics are designed to respect those. In particular, if two states are behaviourally equivalent then they satisfy the same formulas. If the converse holds, then the logic is said to be expressive. and we have a generalisation of the classic Hennessy-Milner theorem [5] which states that over the class of image-fjnite Kripke models, two states are Kripke bisimilar if and only if they satisfy the same formulas in Hennessy-Milner logic
Expressive Logics for Coinductive Predicates
The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata
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
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
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
Coalgebraic Fuzzy geometric logic
The paper aims to develop a framework for coalgebraic fuzzy geometric logic
by adding modalities to the language of fuzzy geometric logic. Using the
methods of coalgebra, the modal operators are introduced in the language of
fuzzy geometric logic. To define the modal operators, we introduce a notion of
fuzzy-open predicate lifting. Based on coalgebras for an endofunctor on the
category of fuzzy topological spaces and fuzzy continuous
maps, we build models for the coalgebraic fuzzy geometric logic. Bisimulations
for the defined models are discussed in this work
PSPACE Bounds for Rank-1 Modal Logics
For lack of general algorithmic methods that apply to wide classes of logics,
establishing a complexity bound for a given modal logic is often a laborious
task. The present work is a step towards a general theory of the complexity of
modal logics. Our main result is that all rank-1 logics enjoy a shallow model
property and thus are, under mild assumptions on the format of their
axiomatisation, in PSPACE. This leads to a unified derivation of tight
PSPACE-bounds for a number of logics including K, KD, coalition logic, graded
modal logic, majority logic, and probabilistic modal logic. Our generic
algorithm moreover finds tableau proofs that witness pleasant proof-theoretic
properties including a weak subformula property. This generality is made
possible by a coalgebraic semantics, which conveniently abstracts from the
details of a given model class and thus allows covering a broad range of logics
in a uniform way
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