160 research outputs found
A modular approach to defining and characterising notions of simulation
We propose a modular approach to defining notions of simulation, and modal logics which characterise them. We use coalgebras to model state-based systems, relators to define notions of simulation for such systems, and inductive techniques to define the syntax and semantics of modal logics for coalgebras. We show that the expressiveness of an inductively defined logic for coalgebras w.r.t. a notion of simulation follows from an expressivity condition involving one step in the definition of the logic, and the relator inducing that notion of simulation. Moreover, we show that notions of simulation and associated characterising logics for increasingly complex system types can be derived by lifting the operations used to combine system types, to a relational level as well as to a logical level. We use these results to obtain Baltag’s logic for coalgebraic simulation, as well as notions of simulation and associated logics for a large class of non-deterministic and probabilistic systems
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
Bisimilar States in Uncertain Structures
We provide a categorical notion called uncertain bisimilarity, which allows to reason about bisimilarity in combination with a lack of knowledge about the involved systems. Such uncertainty arises naturally in automata learning algorithms, where one investigates whether two observed behaviours come from the same internal state of a black-box system that can not be transparently inspected. We model this uncertainty as a set functor equipped with a partial order which describes possible future developments of the learning game. On such a functor, we provide a lifting-based definition of uncertain bisimilarity and verify basic properties. Beside its applications to Mealy machines, a natural model for automata learning, our framework also instantiates to an existing compatibility relation on suspension automata, which are used in model-based testing. We show that uncertain bisimilarity is a necessary but not sufficient condition for two states being implementable by the same state in the black-box system. We remedy the lack of sufficiency by a characterization of uncertain bisimilarity in terms of coalgebraic simulations
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
Constructive Final Semantics of Finite Bags
Finitely-branching and unlabelled dynamical systems are typically modelled as coalgebras for the finite powerset functor. If states are reachable in multiple ways, coalgebras for the finite bag functor provide a more faithful representation. The final coalgebra of this functor is employed as a denotational domain for the evaluation of such systems. Elements of the final coalgebra are non-wellfounded trees with finite unordered branching, representing the evolution of systems starting from a given initial state.
This paper is dedicated to the construction of the final coalgebra of the finite bag functor in homotopy type theory (HoTT). We first compare various equivalent definitions of finite bags employing higher inductive types, both as sets and as groupoids (in the sense of HoTT). We then analyze a few well-known, classical set-theoretic constructions of final coalgebras in our constructive setting. We show that, in the case of set-based definitions of finite bags, some constructions are intrinsically classical, in the sense that they are equivalent to some weak form of excluded middle. Nevertheless, a type satisfying the universal property of the final coalgebra can be constructed in HoTT employing the groupoid-based definition of finite bags. We conclude by discussing generalizations of our constructions to the wider class of analytic functors
Bisimilar States in Uncertain Structures
We provide a categorical notion called uncertain bisimilarity, which allows
to reason about bisimilarity in combination with a lack of knowledge about the
involved systems. Such uncertainty arises naturally in automata learning
algorithms, where one investigates whether two observed behaviours come from
the same internal state of a black-box system that can not be transparently
inspected. We model this uncertainty as a set functor equipped with a partial
order which describes possible future developments of the learning game. On
such a functor, we provide a lifting-based definition of uncertain bisimilarity
and verify basic properties. Beside its applications to Mealy machines, a
natural model for automata learning, our framework also instantiates to an
existing compatibility relation on suspension automata, which are used in
model-based testing. We show that uncertain bisimilarity is a necessary but not
sufficient condition for two states being implementable by the same state in
the black-box system. To remedy the failure of the one direction, we
characterize uncertain bisimilarity in terms of coalgebraic simulations
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