78 research outputs found
Relation lifting, with an application to the many-valued cover modality
We introduce basic notions and results about relation liftings on categories
enriched in a commutative quantale. We derive two necessary and sufficient
conditions for a 2-functor T to admit a functorial relation lifting: one is the
existence of a distributive law of T over the "powerset monad" on categories,
one is the preservation by T of "exactness" of certain squares. Both
characterisations are generalisations of the "classical" results known for set
functors: the first characterisation generalises the existence of a
distributive law over the genuine powerset monad, the second generalises
preservation of weak pullbacks. The results presented in this paper enable us
to compute predicate liftings of endofunctors of, for example, generalised
(ultra)metric spaces. We illustrate this by studying the coalgebraic cover
modality in this setting.Comment: 48 pages, accepted for publication in LMC
Linear Time Logics - A Coalgebraic Perspective
We describe a general approach to deriving linear time logics for a wide
variety of state-based, quantitative systems, by modelling the latter as
coalgebras whose type incorporates both branching behaviour and linear
behaviour. Concretely, we define logics whose syntax is determined by the
choice of linear behaviour and whose domain of truth values is determined by
the choice of branching, and we provide two equivalent semantics for them: a
step-wise semantics amenable to automata-based verification, and a path-based
semantics akin to those of standard linear time logics. We also provide a
semantic characterisation of the associated notion of logical equivalence, and
relate it to previously-defined maximal trace semantics for such systems.
Instances of our logics support reasoning about the possibility, likelihood or
minimal cost of exhibiting a given linear time property. We conclude with a
generalisation of the logics, dual in spirit to logics with discounting, which
increases their practical appeal in the context of resource-aware computation
by incorporating a notion of offsetting.Comment: Major revision of previous version: Sections 4 and 5 generalise the
results in the previous version, with new proofs; Section 6 contains new
result
A Note on the Completeness of Many-Valued Coalgebraic Modal Logic
In this paper, we investigate the many-valued version of coalgebraic modal
logic through predicate lifting approach. Coalgebras, understood as generic
transition systems, can serve as semantic structures for various kinds of modal
logics. A well-known result in coalgebraic modal logic is that its completeness
can be determined at the one-step level. We generalize the result to the
finitely many-valued case by using the canonical model construction method. We
prove the result for coalgebraic modal logics based on three different
many-valued algebraic structures, including the finitely-valued {\L}ukasiewicz
algebra, the commutative integral Full-Lambek algebra (FL-algebra)
expanded with canonical constants and Baaz Delta, and the FL-algebra
expanded with valuation operations.Comment: 17 pages, preprint for journal submissio
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
Characteristic Logics for Behavioural Metrics via Fuzzy Lax Extensions
Behavioural distances provide a fine-grained measure of equivalence in systems involving quantitative data, such as probabilistic, fuzzy, or metric systems. Like in the classical setting of crisp bisimulation-type equivalences, the wide variation found in system types creates a need for generic methods that apply to many system types at once. Approaches of this kind are emerging within the paradigm of universal coalgebra, based either on lifting pseudometrics along set functors or on lifting general real-valued (fuzzy) relations along functors by means of fuzzy lax extensions. An immediate benefit of the latter is that they allow bounding behavioural distance by means of fuzzy bisimulations that need not themselves be (pseudo-)metrics, in analogy to classical bisimulations (which need not be equivalence relations). The known instances of generic pseudometric liftings, specifically the generic Kantorovich and Wasserstein liftings, both can be extended to yield fuzzy lax extensions, using the fact that both are effectively given by a choice of quantitative modalities. Our central result then shows that in fact all fuzzy lax extensions are Kantorovich extensions for a suitable set of quantitative modalities, the so-called Moss modalities. For non-expansive fuzzy lax extensions, this allows for the extraction of quantitative modal logics that characterize behavioural distance, i.e. satisfy a quantitative version of the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a quantitative version of Moss\u27 coalgebraic logic
Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions
In systems involving quantitative data, such as probabilistic, fuzzy, or
metric systems, behavioural distances provide a more fine-grained comparison of
states than two-valued notions of behavioural equivalence or behaviour
inclusion. Like in the two-valued case, the wide variation found in system
types creates a need for generic methods that apply to many system types at
once. Approaches of this kind are emerging within the paradigm of universal
coalgebra, based either on lifting pseudometrics along set functors or on
lifting general real-valued (fuzzy) relations along functors by means of fuzzy
lax extensions. An immediate benefit of the latter is that they allow bounding
behavioural distance by means of fuzzy (bi-)simulations that need not
themselves be hemi- or pseudometrics; this is analogous to classical
simulations and bisimulations, which need not be preorders or equivalence
relations, respectively. The known generic pseudometric liftings, specifically
the generic Kantorovich and Wasserstein liftings, both can be extended to yield
fuzzy lax extensions, using the fact that both are effectively given by a
choice of quantitative modalities. Our central result then shows that in fact
all fuzzy lax extensions are Kantorovich extensions for a suitable set of
quantitative modalities, the so-called Moss modalities. For nonexpansive fuzzy
lax extensions, this allows for the extraction of quantitative modal logics
that characterize behavioural distance, i.e. satisfy a quantitative version of
the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a
quantitative version of Moss' coalgebraic logic. All our results explicitly
hold also for asymmetric distances (hemimetrics), i.e. notions of quantitative
simulation
Many-valued coalgebraic logic over semi-primal varieties
We study many-valued coalgebraic logics with semi-primal algebras of
truth-degrees. We provide a systematic way to lift endofunctors defined on the
variety of Boolean algebras to endofunctors on the variety generated by a
semi-primal algebra. We show that this can be extended to a technique to lift
classical coalgebraic logics to many-valued ones, and that (one-step)
completeness and expressivity are preserved under this lifting. For specific
classes of endofunctors, we also describe how to obtain an axiomatization of
the lifted many-valued logic directly from an axiomatization of the original
classical one. In particular, we apply all of these techniques to classical
modal logic
Many-Valued Coalgebraic Logic: From Boolean Algebras to Primal Varieties
We study many-valued coalgebraic logics with primal algebras of truth-degrees. We describe a way to lift algebraic semantics of classical coalgebraic logics, given by an endofunctor on the variety of Boolean algebras, to this many-valued setting, and we show that many important properties of the original logic are inherited by its lifting. Then, we deal with the problem of obtaining a concrete axiomatic presentation of the variety of algebras for this lifted logic, given that we know one for the original one. We solve this problem for a class of presentations which behaves well with respect to a lattice structure on the algebra of truth-degrees
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