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Behavioural Differential Equations: A Coinductive Calculus of Streams, Automata, and Power Series
, 2000
"... Streams, (automata and) languages, and formal power series are viewed coalgebraically. In summary, this amounts to supplying these sets with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduct ..."
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Cited by 52 (17 self)
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Streams, (automata and) languages, and formal power series are viewed coalgebraically. In summary, this amounts to supplying these sets with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductive definitions take the shape of what we have called behavioural differential equations, after Brzozowski's notion of input derivative. A calculus is developed for coinductive reasoning about all of the afore mentioned structures, closely resembling (and at times generalising) calculus from classical analysis. 2000 Mathematics Subject Classification: 68Q10, 68Q55, 68Q85 1998 ACM Computing Classification System: F.1, F.3 Keywords & Phrases: Coalgebra, automaton, finality, coinduction, stream, formal language, formal power series, differential equation, input derivative, behaviour, semiring, maxplus algebra 1 Contents 1 Introductio...
A coinductive calculus of streams
, 2005
"... We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analo ..."
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Cited by 27 (9 self)
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We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.
Towards Semantics of SelfAdaptive Software
, 2000
"... When people perform computations, they routinely monitor their results, and try to adapt and improve their algorithms when a need arises. The idea of selfadaptive software is to implement this common facility of human mind within the framework of the standard logical methods of software engineering ..."
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Cited by 7 (0 self)
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When people perform computations, they routinely monitor their results, and try to adapt and improve their algorithms when a need arises. The idea of selfadaptive software is to implement this common facility of human mind within the framework of the standard logical methods of software engineering. The ubiquitous practice of testing, debugging and improving programs at the design time should be automated, and established as a continuing run time routine. Technically, the task thus requires combining functionalities of automated software development tools and of runtime environments. Such combinations lead not just to challenging engineering problems, but also to novel theoretical questions. Formal methods are needed, and the standard techniques do not suffice. As a first contribution in this direction, we present a basic mathematical framework suitable for describing selfadaptive software at a high level of semantical abstraction. A static view leads to a structure akin...
Coinductive Counting With Weighted Automata
, 2002
"... A general methodology is developed to compute the solution of a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite weighted automaton; (2) the automaton is reduced by means of the quantitative notion of stream bisimulation; ..."
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Cited by 4 (0 self)
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A general methodology is developed to compute the solution of a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite weighted automaton; (2) the automaton is reduced by means of the quantitative notion of stream bisimulation; (3) the reduced automaton is used to compute an expression (in terms of stream constants and operators) that represents the stream of all counts.
Induction, Coinduction, and Adjoints
, 2002
"... We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and supp ..."
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Cited by 3 (2 self)
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We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and suppose that an initial algebra F (X) of the functor H(Y ) = X + F (Y ) exists; then a right adjoint G(X) to F (X) exists if and only if a nal coalgebra G(X) of the functor K(Y ) = X G(Y ) exists. Motivated by the problem of understanding the structures that arise from initial algebras, we show the following: if F is a left adjoint with a certain commutativity property, then an initial algebra of H(Y ) = X + F (Y ) generates a subcategory of functors with inductive types where the functorial composition is constrained to be a Cartesian product.
The Music of Streams
 Information Processing Letters
, 2000
"... Data streams make preeminent instruments for playing certain classical themes from analysis. Complex networks of processes, e#ortlessly orchestrated by lazy evaluation, can enumerate terms of formal power series ad infinitum. Expressed in a language like Haskell, working programs for powerseries op ..."
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Cited by 2 (0 self)
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Data streams make preeminent instruments for playing certain classical themes from analysis. Complex networks of processes, e#ortlessly orchestrated by lazy evaluation, can enumerate terms of formal power series ad infinitum. Expressed in a language like Haskell, working programs for powerseries operations are tiny gems, because the natural programming style for data streams fits the mathematics so welloften better than timehonored summation notation. The cleverest copyist is the one whose music is performed with the most ease without the performer guessing why.  Jean Jacques Rousseau, Dictionary of Music 1 Overture Like persistent folk tunes, the themes I intend to play here have been arranged for various ensembles over many years. Some performances have been angular, and some melodic, but all share the staying power of good music. The themes stick in mind, to be enjoyed again and again as each performance exposes new surfaces and depths. My subject is "powerstream compositio...
Hoare Logic in the Abstract
"... Abstract. We present an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. We first identify a particular class of functors – which we call ‘verification functors ’ – between traced symmetric monoidal categories and subcategories ..."
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Abstract. We present an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. We first identify a particular class of functors – which we call ‘verification functors ’ – between traced symmetric monoidal categories and subcategories of Preord (the category of preordered sets and monotone mappings). We then give an abstract definition of Hoare triples, parametrised by a verification functor, and prove a single soundness and completeness theorem for such triples. In the particular case of the traced symmetric monoidal category of while programs we get back Hoare’s original rules. We discuss how our framework handles extensions of the Hoare logic for while programs, e.g. the extension with pointer manipulations via separation logic. Finally, we give an example of how our theory can be used in the development of new Hoare logics: we present a new sound and complete set of Hoarelogiclike rules for the verification of linear dynamical systems, modelled via stream circuits. 1
Elements of stream calculus
 In MFPS 2001, ENTCS 45
, 2001
"... CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a themeoriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. ..."
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Cited by 1 (1 self)
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CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a themeoriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.
Coinductive Proofs for Streams in PVS ⋆
"... Abstract. WepresentanimplementationinthetheoremproverPVS of coinductive stream calculus. Stream calculus can be used to model signal flow graphs, and thus provides a nice mathematical foundation for reasoning about properties of signal flow graphs, which are again used to model a variety of systems ..."
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Abstract. WepresentanimplementationinthetheoremproverPVS of coinductive stream calculus. Stream calculus can be used to model signal flow graphs, and thus provides a nice mathematical foundation for reasoning about properties of signal flow graphs, which are again used to model a variety of systems such as digital signal processing. We show how proofs by coinduction are used to prove equality of streams, and present a strategy to do this automatically. 1
A PVS Implementation of Stream Calculus
, 2005
"... We present early work on a PVS implementation of a model of simple control as signal flow graphs to enable formal verification of input/output behaviour of the control system. As has been shown by Rutten, Signal flow graphs can be described using Escardo's coinductive stream calculus, which incl ..."
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We present early work on a PVS implementation of a model of simple control as signal flow graphs to enable formal verification of input/output behaviour of the control system. As has been shown by Rutten, Signal flow graphs can be described using Escardo's coinductive stream calculus, which includes a definition of di#erentiation for streams over the real numbers and the use of di#erential equations. The basics of coinductive stream calculus has been implemented in PVS.