590 research outputs found
An Institution of Modal Logics for Coalgebras
This paper presents a modular framework for the specification of certain inductively-defined coalgebraic types. Modal logics for coalgebras of polynomial endofunctors on the category of sets have been studied in [M. Rößiger, Coalgebras and modal logic, in: H. Reichel (Ed.), Coalgebraic Methods in Computer Science, Electronic Notes in Theoretical Computer Science, vol. 33, Elsevier Science, 2000, pp. 299–320; B. Jacobs, Many-sorted coalgebraic modal logic: a model-theoretic study, Theoretical Informatics and Applications 35(1) (2001) 31–59]. These logics are here generalised to endofunctors on categories of sorted sets, in order to allow collections of inter-related types to be specified simultaneously. The inductive nature of the coalgebraic types considered is then used to formalise semantic relationships between different types, and to define translations between the associated logics. The resulting logical framework is shown to be an institution, whose specifications and specification morphisms admit final and respectively cofree models
Generic Trace Semantics via Coinduction
Trace semantics has been defined for various kinds of state-based systems,
notably with different forms of branching such as non-determinism vs.
probability. In this paper we claim to identify one underlying mathematical
structure behind these "trace semantics," namely coinduction in a Kleisli
category. This claim is based on our technical result that, under a suitably
order-enriched setting, a final coalgebra in a Kleisli category is given by an
initial algebra in the category Sets. Formerly the theory of coalgebras has
been employed mostly in Sets where coinduction yields a finer process semantics
of bisimilarity. Therefore this paper extends the application field of
coalgebras, providing a new instance of the principle "process semantics via
coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page
General Recursion via Coinductive Types
A fertile field of research in theoretical computer science investigates the
representation of general recursive functions in intensional type theories.
Among the most successful approaches are: the use of wellfounded relations,
implementation of operational semantics, formalization of domain theory, and
inductive definition of domain predicates. Here, a different solution is
proposed: exploiting coinductive types to model infinite computations. To every
type A we associate a type of partial elements Partial(A), coinductively
generated by two constructors: the first, return(a) just returns an element
a:A; the second, step(x), adds a computation step to a recursive element
x:Partial(A). We show how this simple device is sufficient to formalize all
recursive functions between two given types. It allows the definition of fixed
points of finitary, that is, continuous, operators. We will compare this
approach to different ones from the literature. Finally, we mention that the
formalization, with appropriate structural maps, defines a strong monad.Comment: 28 page
Indexed induction and coinduction, fibrationally.
This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobs’ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE)
On coalgebras with internal moves
In the first part of the paper we recall the coalgebraic approach to handling
the so-called invisible transitions that appear in different state-based
systems semantics. We claim that these transitions are always part of the unit
of a certain monad. Hence, coalgebras with internal moves are exactly
coalgebras over a monadic type. The rest of the paper is devoted to supporting
our claim by studying two important behavioural equivalences for state-based
systems with internal moves, namely: weak bisimulation and trace semantics.
We continue our research on weak bisimulations for coalgebras over order
enriched monads. The key notions used in this paper and proposed by us in our
previous work are the notions of an order saturation monad and a saturator. A
saturator operator can be intuitively understood as a reflexive, transitive
closure operator. There are two approaches towards defining saturators for
coalgebras with internal moves. Here, we give necessary conditions for them to
yield the same notion of weak bisimulation.
Finally, we propose a definition of trace semantics for coalgebras with
silent moves via a uniform fixed point operator. We compare strong and weak
bisimilation together with trace semantics for coalgebras with internal steps.Comment: Article: 23 pages, Appendix: 3 page
A single complete relational rule for coalgebraic refinement
A transition system can be presented either as a binary relation or as a coalgebra for the powerset functor, each representation being obtained from the other by transposition. More generally, a coalgebra for a functor F generalises transition systems in the sense that a shape for transitions is determined by F, typically encoding a signature of methods and observers. This paper explores such a duality to frame in purely relational terms coalgebraic refinement, showing that relational (data) refinement of transition relations, in its two variants, downward and upward (functional) simulations, is equivalent to coalgebraic refinement based on backward and forward morphisms, respectively. Going deeper, it is also shown that downward simulation provides a complete relational rule to prove coalgebraic refinement. With such a single rule the paper defines a pre-ordered calculus for refinement of coalgebras, with bisimilarity as the induced equivalence. The calculus is monotonic with respect to the main relational operators and arbitrary relator F, therefore providing a framework for structural reasoning about refinement
A Kleene theorem for polynomial coalgebras
For polynomial functors G, we show how to generalize the classical notion of regular expression to G-coalgebras. We introduce a language of expressions for describing elements of the final G-coalgebra and, analogously to Kleene’s theorem, we show the correspondence between expressions and finite G-coalgebras
A coalgebraic semantic framework for component-based development in UML
This paper introduces a generic semantic framework for component-based development, expressed in the unified modelling language UML. The principles of a coalgebraic semantics for class, object and statechart diagrams as well as for use cases, are developed. It is also discussed how to formalize the refinement steps in the development process based upon a suitable notion of behavior refinement. In this way, a formal basis for component-based development in UML is studied, which allows the construction of more complex and specific systems from independent components.FCT -Fuel Cell Technologies Program(POSI/ICHS/44304/2002
Non-Deterministic Kleene Coalgebras
In this paper, we present a systematic way of deriving (1) languages of
(generalised) regular expressions, and (2) sound and complete axiomatizations
thereof, for a wide variety of systems. This generalizes both the results of
Kleene (on regular languages and deterministic finite automata) and Milner (on
regular behaviours and finite labelled transition systems), and includes many
other systems such as Mealy and Moore machines
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