108 research outputs found
Separation and Renaming in Nominal Sets
Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which involve arbitrary substitutions rather than permutations, through a categorical adjunction. In particular, the left adjoint relates the separated product of nominal sets to the Cartesian product of nominal renaming sets. Based on these results, we define the new notion of separated nominal automata. We show that these automata can be exponentially smaller than classical nominal automata, if the semantics is closed under substitutions
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
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
Coalgebra Learning via Duality
Automata learning is a popular technique for inferring minimal automata
through membership and equivalence queries. In this paper, we generalise
learning to the theory of coalgebras. The approach relies on the use of logical
formulas as tests, based on a dual adjunction between states and logical
theories. This allows us to learn, e.g., labelled transition systems, using
Hennessy-Milner logic. Our main contribution is an abstract learning algorithm,
together with a proof of correctness and termination
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
Presenting Distributive Laws
Distributive laws of a monad T over a functor F are categorical tools for
specifying algebra-coalgebra interaction. They proved to be important for
solving systems of corecursive equations, for the specification of well-behaved
structural operational semantics and, more recently, also for enhancements of
the bisimulation proof method. If T is a free monad, then such distributive
laws correspond to simple natural transformations. However, when T is not free
it can be rather difficult to prove the defining axioms of a distributive law.
In this paper we describe how to obtain a distributive law for a monad with an
equational presentation from a distributive law for the underlying free monad.
We apply this result to show the equivalence between two different
representations of context-free languages
Bisimilarity of Open Terms in Stream GSOS
Stream GSOS is a specification format for operations and calculi on infinite
sequences. The notion of bisimilarity provides a canonical proof technique for
equivalence of closed terms in such specifications. In this paper, we focus on
open terms, which may contain variables, and which are equivalent whenever they
denote the same stream for every possible instantiation of the variables. Our
main contribution is to capture equivalence of open terms as bisimilarity on
certain Mealy machines, providing a concrete proof technique. Moreover, we
introduce an enhancement of this technique, called bisimulation up-to
substitutions, and show how to combine it with other up-to techniques to obtain
a powerful method for proving equivalence of open terms
Up-to Techniques for Branching Bisimilarity
Ever since the introduction of behavioral equivalences on processes one has
been searching for efficient proof techniques that accompany those
equivalences. Both strong bisimilarity and weak bisimilarity are accompanied by
an arsenal of up-to techniques: enhancements of their proof methods. For
branching bisimilarity, these results have not been established yet. We show
that a powerful proof technique is sound for branching bisimilarity by
combining the three techniques of up to union, up to expansion and up to
context for Bloom's BB cool format. We then make an initial proposal for
casting the correctness proof of the up to context technique in an abstract
coalgebraic setting, covering branching but also {\eta}, delay and weak
bisimilarity
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