Results 1  10
of
23
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 ..."
Abstract

Cited by 66 (23 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 37 (13 self)
 Add to MetaCart
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.
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; ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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.
Observational Coalgebras and Complete Sets of Cooperations
, 2008
"... In this paper we introduce the notion of an observational coalgebra structure and of a complete set of cooperations. We demonstrate in various example the usefulness of these notions, in particular, we show how they give rise to coalgebraic proof and definition principles. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In this paper we introduce the notion of an observational coalgebra structure and of a complete set of cooperations. We demonstrate in various example the usefulness of these notions, in particular, we show how they give rise to coalgebraic proof and definition principles.
A general framework for sound and complete FloydHoare logics
 ACM Transactions on Computational Logic (TOCL
, 2009
"... ar ..."
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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