265 research outputs found
A parameterization process as a categorical construction
The parameterization process used in the symbolic computation systems Kenzo
and EAT is studied here as a general construction in a categorical framework.
This parameterization process starts from a given specification and builds a
parameterized specification by transforming some operations into parameterized
operations, which depend on one additional variable called the parameter. Given
a model of the parameterized specification, each interpretation of the
parameter, called an argument, provides a model of the given specification.
Moreover, under some relevant terminality assumption, this correspondence
between the arguments and the models of the given specification is a bijection.
It is proved in this paper that the parameterization process is provided by a
free functor and the subsequent parameter passing process by a natural
transformation. Various categorical notions are used, mainly adjoint functors,
pushouts and lax colimits
Initial algebra for a system of right-linear functors
In 2003 we showed that right-linear systems of equations over regular expressions, when interpreted in a category of trees, have a solution when ever they enjoy a specific property that we called hierarchicity and that is instrumental to avoid critical mutual recursive definitions. In this note, we prove that a right-linear system of polynomial endofunctors on a cocartesian monoidal closed category which enjoys parameterized left list arithmeticity, has an initial algebra, provided it satisfies a property similar to hierarchicity
Van Kampen Colimits and Path Uniqueness
Fibred semantics is the foundation of the model-instance pattern of software
engineering. Software models can often be formalized as objects of presheaf
topoi, i.e, categories of objects that can be represented as algebras as well
as coalgebras, e.g., the category of directed graphs. Multimodeling requires to
construct colimits of models, decomposition is given by pullback.
Compositionality requires an exact interplay of these operations, i.e.,
diagrams must enjoy the Van Kampen property. However, checking the validity of
the Van Kampen property algorithmically based on its definition is often
impossible.
In this paper we state a necessary and sufficient yet efficiently checkable
condition for the Van Kampen property to hold in presheaf topoi. It is based on
a uniqueness property of path-like structures within the defining congruence
classes that make up the colimiting cocone of the models. We thus add to the
statement "Being Van Kampen is a Universal Property" by Heindel and
Soboci\'{n}ski the fact that the Van Kampen property reveals a presheaf-based
structural uniqueness feature
Recommended from our members
Category theory : definitions and examples
Category theory was invented as an abstract language for describing certain structures and constructions which repeatedly occur in many branches of mathematics, such as topology, algebra, and logic. In recent years, it has found several applications in computer science, e.g., algebraic specification, type theory, and programming language semantics. In this paper, we collect definitions and examples of the basic concepts in category theory: categories, functors, natural transformations, universal properties, limits, and adjoints
Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory
We describe categorical models of a circuit-based (quantum) functional
programming language. We show that enriched categories play a crucial role.
Following earlier work on QWire by Paykin et al., we consider both a simple
first-order linear language for circuits, and a more powerful host language,
such that the circuit language is embedded inside the host language. Our
categorical semantics for the host language is standard, and involves cartesian
closed categories and monads. We interpret the circuit language not in an
ordinary category, but in a category that is enriched in the host category. We
show that this structure is also related to linear/non-linear models. As an
extended example, we recall an earlier result that the category of W*-algebras
is dcpo-enriched, and we use this model to extend the circuit language with
some recursive types
Finite Presheaf categories as a nice setting for doing generic programming
The purpose of this paper is to describe how some theorems about constructions in categories can be seen as a way of doing generic programming. No prior knowledge of category theory is required to understand the paper.
We explore the class of nite presheaf categories. Each of these categories can be seen as a type or universe of structures parameterized by a diagram (actually a nite category) C. Examples of these categories are: graphs, labeled graphs, nite automata and evolutive sets.
Limits and colimits are very general ways of combining objects in categories in such a way that a new object is built and satis es a certain universal property. When con- centrating on nite presheaf categories and interpreting them as types or structures, limits and colimits can be interpreted as very general operations on types. Theorems on the construction of limits and colimits in arbitrary categories will provide a generic implementation of these operations.
Also, nite presheaf categories are toposes. Because of this, each of these categories has an internal logic. We are going to show that some theorems about the truth of sentences of this logic can be interpreted as a way an implementing a generic theorem prover.
The paper discusses non trivial theorems and de nitions from category and topos theory but the emphasis is put on their computational content and in what way they provide rich and abstract data structures and algorithms.Eje: Workshop sobre Aspectos Teoricos de la Inteligencia ArtificialRed de Universidades con Carreras en Informática (RedUNCI
A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks
Regulatory networks depict promoting or inhibiting interactions between
molecules in a biochemical system. We introduce a category-theoretic formalism
for regulatory networks, using signed graphs to model the networks and signed
functors to describe occurrences of one network in another, especially
occurrences of network motifs. With this foundation, we establish functorial
mappings between regulatory networks and other mathematical models in
biochemistry. We construct a functor from reaction networks, modeled as Petri
nets with signed links, to regulatory networks, enabling us to precisely define
when a reaction network could be a physical mechanism underlying a regulatory
network. Turning to quantitative models, we associate a regulatory network with
a Lotka-Volterra system of differential equations, defining a functor from the
category of signed graphs to a category of parameterized dynamical systems. We
extend this result from closed to open systems, demonstrating that
Lotka-Volterra dynamics respects not only inclusions and collapsings of
regulatory networks, but also the process of building up complex regulatory
networks by gluing together simpler pieces. Formally, we use the theory of
structured cospans to produce a lax double functor from the double category of
open signed graphs to that of open parameterized dynamical systems. Throughout
the paper, we ground the categorical formalism in examples inspired by systems
biology.Comment: 33 pages. Added several examples, plus minor revision
Specification, horizontal composition and parameterization of algebraic implementations
Loose specifications of abstract data types (ADTs) have many non-isomorphic algebras as models. An implementation between two loose Specifications should therefore consider many abstraction functions together with their source and target algebras. Just like specifications are stepwise refined to restrict their class of models, implementations should be stepwise refinable to restrict the class of abstraction functions. In this scenario specifications and implementations can be developed interwovenly.
We suggest to have implementation specifications analogously to loose ADT specifications: Implementations have signatures, models, axioms and sentences thus constituting an institution. Implementation specifications are the theories of this institution and refinements between implementation specifications are its theory morphisms.
In this framework, implementations between parameterized specifications and horizontal composition of implementations turn out to be special cases of the more powerful concept of parameterized implementations, which allow to instantiate an implementation by substituting a subimplementation by another implementation
Diagrammatic logic applied to a parameterization process
This paper provides an abstract definition of some kinds of logics, called
diagrammatic logics, together with a definition of morphisms and of 2-morphisms
between diagrammatic logics. The definition of the 2-category of diagrammatic
logics rely on category theory, mainly on adjunction, categories of fractions
and limit sketches. This framework is applied to the formalization of a
parameterization process. This process, which consists in adding a formal
parameter to some operations in a given specification, is presented as a
morphism of logics. Then the parameter passing process, for recovering a model
of the given specification from a model of the parameterized specification and
an actual parameter, is seen as a 2-morphism of logics
- …