Results 11  20
of
74
Proof Principles for Datatypes with Iterated Recursion
, 1997
"... . Data types like trees which are finitely branching and of (possibly) infinite depth are described by iterating initial algebras and terminal coalgebras. We study proof principles for such data types in the context of categorical logic, following and extending the approach of [14, 15]. The technica ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
. Data types like trees which are finitely branching and of (possibly) infinite depth are described by iterating initial algebras and terminal coalgebras. We study proof principles for such data types in the context of categorical logic, following and extending the approach of [14, 15]. The technical contribution of this paper involves a description of initial algebras and terminal coalgebras in total categories of fibrations for lifted "datafunctors". These lifted functors are used to formulate our proof principles. We test these principles by proving some elementary results for four kinds of trees (with finite or infinite breadth or depth) using the proof tool pvs. 1 Introduction Algebras and coalgebras are of wellestablished importance in computer science, notably in the theory of datatypes, where especially initial algebras and terminal coalgebras play a distinguished role. Over the past decade there is more and more interest in the logic associated with initial algebras and ter...
From Settheoretic Coinduction to Coalgebraic Coinduction: some results, some problems
, 1999
"... ..."
(Show Context)
NonDeterministic 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 Miln ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
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.
Components as processes: An exercise in coalgebraic modeling
 FMOODS’2000  Formal Methods for Open ObjectOriented Distributed Systems
, 2000
"... Abstract Software components, arising, typically, in systems ’ analysis and design, are characterized by a public interface and a private encapsulated state. They persist (and evolve) in time, according to some behavioural patterns. This paper is an exercise in modeling such components as coalgebras ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
(Show Context)
Abstract Software components, arising, typically, in systems ’ analysis and design, are characterized by a public interface and a private encapsulated state. They persist (and evolve) in time, according to some behavioural patterns. This paper is an exercise in modeling such components as coalgebras for some kinds of endofunctors on ¢¡¤ £ , capturing both (interface) types and behavioural aspects. The construction of component categories, cofibred over the interface space, emerges by generalizing the usual notion of a coalgebra morphism. A collection of composition operators as well as a generic notion of bisimilarity, are discussed.
Invariants, Bisimulations and the Correctness of Coalgebraic Refinements
 Techn. Rep. CSIR9704, Comput. Sci. Inst., Univ. of Nijmegen
, 1997
"... . Coalgebraic specifications are used to formally describe the behaviour of classes in objectoriented languages. In this paper, a general notion of refinement between two such coalgebraic specifications is defined, capturing the idea that one "concrete" class specification realises the be ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
(Show Context)
. Coalgebraic specifications are used to formally describe the behaviour of classes in objectoriented languages. In this paper, a general notion of refinement between two such coalgebraic specifications is defined, capturing the idea that one "concrete" class specification realises the behaviour of the other, "abstract" class specification. Two (complete) prooftechniques are given to establish such refinements: one involving an invariant (a predicate that is closed under transitions) on the concrete class, and one involving a bisimulation (a relation that is closed under transitions) between the concrete and the abstract class. The latter can only be used if the abstract class is what we call totally specified. Parts of the underlying theory of invariants and bisimulations in a coalgebraic setting are included, involving least and greatest invariants and connections between invariants and bisimulations. Also, the proofprinciples are illustrated in examples (which are fully formalise...
Coalgebraic Reasoning about Classes in ObjectOriented Languages
 In Coalgebraic Methods in Computer Science, number 11. Electronic Notes in Computer Science
, 1998
"... This note briefly discusses how some of the ideas developed in the theory of coalgebras are used in a frontend tool called LOOP, developed jointly in Dresden and Nijmegen, for reasoning (with a backend theorem prover) about classes in objectoriented languages. It will describe reasoning both about ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
(Show Context)
This note briefly discusses how some of the ideas developed in the theory of coalgebras are used in a frontend tool called LOOP, developed jointly in Dresden and Nijmegen, for reasoning (with a backend theorem prover) about classes in objectoriented languages. It will describe reasoning both about objectoriented specifications and about JAVA implementations, via examples. 1 Introduction One of the key aspects of objectorientation is that objects, as instances of a class, have a private state, which can only be accessed and modified via the operations (usually called attributes and methods) of the class of the object. It is precisely this aspect which forms the starting point of the coalgebraic analysis of classes and objects in an objectoriented setting, see [10,5]: a class is seen as a coalgebra, and an object of a class as an element of the state space of the coalgebra. More precisely, a coalgebra is an operation of the form c: X ! T (X), where X is the carrier set (or state sp...
From Bialgebraic Semantics to Congruence Formats
, 2005
"... A general and abstract framework to defining congruence formats for various process equivalences coalgebraic approach to process equivalence, based on a notion of test suite. The resulting technique is illustrated on the example of completed trace equivalence. Rather than providing formal proofs, th ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
A general and abstract framework to defining congruence formats for various process equivalences coalgebraic approach to process equivalence, based on a notion of test suite. The resulting technique is illustrated on the example of completed trace equivalence. Rather than providing formal proofs, the paper is guiding the reader through the process of deriving a congruence format in the test suite approach.
Coalgebras For Binary Methods: Properties Of Bisimulations And Invariants
, 2001
"... Coalgebras for endofunctors C > C can be used to model classes of objectoriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors C^op x C > C . This ext ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
Coalgebras for endofunctors C > C can be used to model classes of objectoriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors C^op x C > C . This extension allows the incorporation of binary methods into coalgebraic class specifications. The paper also discusses how to define bisimulation and invariants for coalgebras of extended polynomial functors and proves many standard results.
Foundational, Compositional (Co)datatypes for HigherOrder Logic  Category Theory Applied to Theorem Proving
"... Higherorder logic (HOL) forms the basis of several popular interactive theorem provers. These follow the definitional approach, reducing highlevel specifications to logical primitives. This also applies to the support for datatype definitions. However, the internal datatype construction used in H ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
Higherorder logic (HOL) forms the basis of several popular interactive theorem provers. These follow the definitional approach, reducing highlevel specifications to logical primitives. This also applies to the support for datatype definitions. However, the internal datatype construction used in HOL4, HOL Light, and Isabelle/HOL is fundamentally noncompositional, limiting its efficiency and flexibility, and it does not cater for codatatypes. We present a fully modular framework for constructing (co)datatypes in HOL, with support for mixed mutual and nested (co)recursion. Mixed (co)recursion enables type definitions involving both datatypes and codatatypes, such as the type of finitely branching trees of possibly infinite depth. Our framework draws heavily from category theory. The key notion is that of a rich type constructor—a functor satisfying specific properties preserved by interesting categorical operations. Our ideas are formalized in Isabelle and implemented as a new definitional package, answering a longstanding user request.
The least fibred lifting and the expressivity of coalgebraic modal logic
 In Proc. CALCO 2005, volume 3629 of LNCS
, 2005
"... and relationpreserving functions. In this paper, the least (fibrewise) of such liftings, L(B), is characterized for essentially any B. The lifting has all the useful properties of the relation lifting due to Jacobs, without the usual assumption of weak pullback preservation; if B preserves weak pu ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
(Show Context)
and relationpreserving functions. In this paper, the least (fibrewise) of such liftings, L(B), is characterized for essentially any B. The lifting has all the useful properties of the relation lifting due to Jacobs, without the usual assumption of weak pullback preservation; if B preserves weak pullbacks, the two liftings coincide. Equivalence relations can be viewed as Boolean algebras of subsets (predicates, tests). This correspondence relates L(B) to the least test suite lifting T (B), which is defined in the spirit of predicate lifting as used in coalgebraic modal logic. Properties of T (B) translate to a general expressivity result for a modal logic for Bcoalgebras. In the resulting logic, modal operators of any arity can appear. 1