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25
GSOS for probabilistic transition systems (Extended Abstract)
, 2002
"... We introduce probabilistic GSOS, an operator specification format for (reactive) probabilistic transition systems which arises as an adaptation of the known GSOS format for labelled (nondeterministic) transition systems. Like the standard one, the format is well behaved in the sense that on all mode ..."
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We introduce probabilistic GSOS, an operator specification format for (reactive) probabilistic transition systems which arises as an adaptation of the known GSOS format for labelled (nondeterministic) transition systems. Like the standard one, the format is well behaved in the sense that on all models bisimilarity is a congruence and the uptocontext proof principle is valid. Moreover, every specification has a final model which can be shown to offer unique solutions for guarded recursive equations. The format covers operator specifications from the literature, so that the wellbehavedness results given for those arise as instances of our general one.
Coinductive Field of Exact Real Numbers and General Corecursion
, 2006
"... In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be se ..."
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In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be seen as coalgebra maps on the coalgebra of streams and hence they will be formalised as general corecursive functions. We use the machinery of Coq proof assistant for coinductive types to present the formalisation.
Stream Differential Equations: concrete formats for coinductive definitions
, 2011
"... In this article we give an accessible introduction to stream differential equations, ie., equations that take the shape of differential equations from analysis and that are used to define infinite streams. Furthermore we discuss a syntactic format for stream differential equations that ensures that ..."
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In this article we give an accessible introduction to stream differential equations, ie., equations that take the shape of differential equations from analysis and that are used to define infinite streams. Furthermore we discuss a syntactic format for stream differential equations that ensures that any system of equations that fits into the format has a unique solution. It turns out that the stream functions that can be defined using our format are precisely the causal stream functions. Finally, we are going to discuss nonstandard stream calculus that uses basic (co)operations different from the usual head and tail operations in order to define and to reason about streams and stream functions. 1
Stream Differential Equations: Specification Formats and Solution Methods
, 2014
"... Streams, or infinite sequences, are infinite objects of a very simple type, yet they have a rich theory partly due to their ubiquity in mathematics and computer science. Stream differential equations are a coinductive method for specifying streams and stream operations, and their theory has been dev ..."
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Streams, or infinite sequences, are infinite objects of a very simple type, yet they have a rich theory partly due to their ubiquity in mathematics and computer science. Stream differential 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 classification of different formats of stream differential equations, their solution methods, and the classes of streams they can define. Moreover, we describe in detail the connection between the socalled syntactic solution method and abstract GSOS.
Foundational Extensible Corecursion
, 2014
"... This paper presents a theoretical framework for defining corecursive functions safely in a total setting, based on corecursion upto and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under wellbehaved operations, including con ..."
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This paper presents a theoretical framework for defining corecursive functions safely in a total setting, based on corecursion upto and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under wellbehaved operations, including constructors. Corecursive functions that are well behaved can be registered as such, thereby increasing the corecursor’s expressiveness. To the extensible corecursor corresponds an equally flexible coinduction principle. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The approach is foundational: The corecursor is derived from first principles, without requiring new axioms or extensions of the logic. This ensures that no inconsistencies can be introduced by omissions in a termination or productivity check.
1st Year Transfer Report
, 2006
"... This document is a summary of the work I have done during my first year whilst researching on the modularity of structural operational semantics. I discuss why modularity is important in semantics and the shortcomings of structural operational semantics in this regard. I review the related literat ..."
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This document is a summary of the work I have done during my first year whilst researching on the modularity of structural operational semantics. I discuss why modularity is important in semantics and the shortcomings of structural operational semantics in this regard. I review the related literature and explain what I have achieved so far. Also, I outline some possible directions for future work. 1
Coinduction UpTo in a Fibrational Setting ∗
"... Bisimulation upto enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows ..."
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Bisimulation upto enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain upto techniques not only for bisimilarity but for a large class of coinductive predicates modelled as coalgebras. By tuning the parameters of our framework, we obtain novel techniques for unary predicates and nominal automata, a variant of the GSOS rule format for similarity, and a new categorical treatment of weak bisimilarity. Categories and Subject Descriptors F.3 [Logics and meanings of programs]; F.4 [Mathematical logic and formal languages]