37 research outputs found
A decision procedure for bisimilarity of generalized regular expressions.
Contains fulltext :
84383.pdf (publisher's version ) (Closed access)SBMF '2010, 8 november 201
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
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
A coalgebraic perspective on logical interpretations
In Computer Science stepwise refinement of algebraic specifications is a well-known formal methodology for rigorous program development. This paper illustrates how techniques from Algebraic Logic, in particular that of interpretation, understood as a multifunction that preserves and reflects logical consequence, capture a number of relevant transformations in the context of software design, reuse, and adaptation, difficult to deal with in classical approaches. Examples include data encapsulation and the decomposition of operations into atomic transactions. But if interpretations open such a new research avenue in program refinement, (conceptual) tools are needed to reason about them. In this line, the paper’s main contribution is a study of the correspondence between logical interpretations and morphisms of a particular kind of coalgebras. This opens way to the use of coalgebraic constructions, such as simulation and bisimulation, in the study of interpretations between (abstract) logics.Fundação para a Ciência e a Tecnologia (FCT
Open Markov Processes and Reaction Networks
We define the concept of an `open' Markov process, a continuous-time Markov
chain equipped with specified boundary states through which probability can
flow in and out of the system. External couplings which fix the probabilities
of boundary states induce non-equilibrium steady states characterized by
non-zero probability currents flowing through the system. We show that these
non-equilibrium steady states minimize a quadratic form which we call
`dissipation.' This is closely related to Prigogine's principle of minimum
entropy production. We bound the rate of change of the entropy of a driven
non-equilibrium steady state relative to the underlying equilibrium state in
terms of the flow of probability through the boundary of the process.
We then consider open Markov processes as morphisms in a symmetric monoidal
category by splitting up their boundary states into certain sets of `inputs'
and `outputs.' Composition corresponds to gluing the outputs of one such open
Markov process onto the inputs of another so that the probability flowing out
of the first process is equal to the probability flowing into the second. We
construct a `black-box' functor characterizing the behavior of an open Markov
process in terms of the space of possible steady state probabilities and
probability currents along the boundary. The fact that this is a functor means
that the behavior of a composite open Markov process can be computed by
composing the behaviors of the open Markov processes from which it is composed.
We prove a similar black-boxing theorem for reaction networks whose dynamics
are given by the non-linear rate equation. Along the way we describe a more
general category of open dynamical systems where composition corresponds to
gluing together open dynamical systems.Comment: 140 pages, University of California Riverside PhD Dissertatio
The Algebra of Open and Interconnected Systems
Herein we develop category-theoretic tools for understanding network-style
diagrammatic languages. The archetypal network-style diagrammatic language is
that of electric circuits; other examples include signal flow graphs, Markov
processes, automata, Petri nets, chemical reaction networks, and so on. The key
feature is that the language is comprised of a number of components with
multiple (input/output) terminals, each possibly labelled with some type, that
may then be connected together along these terminals to form a larger network.
The components form hyperedges between labelled vertices, and so a diagram in
this language forms a hypergraph. We formalise the compositional structure by
introducing the notion of a hypergraph category. Network-style diagrammatic
languages and their semantics thus form hypergraph categories, and semantic
interpretation gives a hypergraph functor.
The first part of this thesis develops the theory of hypergraph categories.
In particular, we introduce the tools of decorated cospans and corelations.
Decorated cospans allow straightforward construction of hypergraph categories
from diagrammatic languages: the inputs, outputs, and their composition are
modelled by the cospans, while the 'decorations' specify the components
themselves. Not all hypergraph categories can be constructed, however, through
decorated cospans. Decorated corelations are a more powerful version that
permits construction of all hypergraph categories and hypergraph functors.
These are often useful for constructing the semantic categories of diagrammatic
languages and functors from diagrams to the semantics. To illustrate these
principles, the second part of this thesis details applications to linear
time-invariant dynamical systems and passive linear networks.Comment: 230 pages. University of Oxford DPhil Thesi
Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective
We use the framework of "props" to study electrical circuits, signal-flow
diagrams, and bond graphs. A prop is a strict symmetric monoidal category where
the objects are natural numbers, with the tensor product of objects given by
addition. In this approach, electrical circuits make up the morphisms in a
prop, as do signal-flow diagrams, and bond graphs. A network, such as an
electrical circuit, with inputs and outputs is a morphism from to
, while putting networks together in series is composition, and setting them
side by side is tensoring. Here we work out the details of this approach for
various kinds of electrical circuits, then signal-flow diagrams, and then bond
graphs. Each kind of network corresponds to a mathematically natural prop. We
also describe the "behavior" of electrical circuits, bond graphs, and
signal-flow diagrams using morphisms between props. To assign a behavior to a
network we "black box" the network, which forgets its inner workings and
records only the relation it imposes between inputs and outputs. The process of
black-boxing a network then corresponds to a morphism between props.
Interestingly, there are two different behaviors for any bond graph, related by
a natural transformation. To achieve all of this we first prove some
foundational results about props. These results let us describe any prop in
terms of generators and equations, and also define morphisms of props by naming
where the generators go and checking that relevant equations hold. Technically,
the key tools are the Rosebrugh--Sabadini--Walters result relating circuits to
special commutative Frobenius monoids, the monadic adjunction between props and
signatures, and a result saying which symmetric monoidal categories are
equivalent to props.Comment: PhD thesis, 201
Stream Differential Equations: Specification Formats and Solution Methods
Streams, or innite sequences, are innite objects of a very simple type, yet they
have a rich theory partly due to their ubiquity in mathematics and computer science.
Stream dierential equations are a coinductive method for specifying streams and stream
operations, and their theory has been developed in many papers over the past two decades.
In this paper we present a survey of the many results in this area. Our focus is on the
classication of dierent formats of stream dierential equations, their solution methods,
and the classes of streams they can dene. Moreover, we describe in detail the connection
between the so-called syntactic solution method and abstract GSOS