Results 1  10
of
16
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 26 (10 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.
Generalised Coinduction
, 2001
"... We introduce the lambdacoiteration schema for a distributive law lambda of a functor T over a functor F. Under certain conditions it can be shown to uniquely characterise functions into the carrier of a final Fcoalgebra, generalising the basic coiteration schema as given by finality. The duals of ..."
Abstract

Cited by 16 (3 self)
 Add to MetaCart
We introduce the lambdacoiteration schema for a distributive law lambda of a functor T over a functor F. Under certain conditions it can be shown to uniquely characterise functions into the carrier of a final Fcoalgebra, generalising the basic coiteration schema as given by finality. The duals of primitive recursion and courseofvalue iteration, which are known extensions of coiteration, arise as instances of our framework. One can furthermore obtain schemata justifying recursive specifications that involve operators such as addition of power series, regular operators on languages, or parallel and sequential composition of processes. Next...
Distributive laws for the coinductive solution of recursive equations
 Information and Computation
"... This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributi ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributive laws. 1
Generalizing the powerset construction, coalgebraically
, 2010
"... Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for nondeterministic automata it is ordinary bisimilarity. The powerset construction is a standard method for converting a nondeterministic automaton into an equivalent deterministic one as far as language is concerned. In this paper, we lift the powerset construction on automata to the more general framework of coalgebras with structured state spaces. Examples of applications include partial Mealy machines, (structured) Moore automata, and Rabin probabilistic automata.
Checking NFA equivalence with bisimulations up to congruence
"... Abstract—We introduce bisimulation up to congruence as a technique for proving language equivalence of nondeterministic finite automata. Exploiting this technique, we devise an optimisation of the classical algorithm by Hopcroft and Karp [12] that, instead of computing the whole determinised automa ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Abstract—We introduce bisimulation up to congruence as a technique for proving language equivalence of nondeterministic finite automata. Exploiting this technique, we devise an optimisation of the classical algorithm by Hopcroft and Karp [12] that, instead of computing the whole determinised automata, explores only a small portion of it. Although the optimised algorithm remains exponential in worst case (the problem is PSPACEcomplete), experimental results show improvements of several orders of magnitude over the standard algorithm. I.
Generalizing Substitution
, 2003
"... It is well known that, given an endofunctor H on a category C, the initial (A + H−)algebras (if existing), i.e., the algebras of (wellfounded) Hterms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Mo ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
It is well known that, given an endofunctor H on a category C, the initial (A + H−)algebras (if existing), i.e., the algebras of (wellfounded) Hterms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Moss [17] and Aczel, Adámek, Milius and Velebil [2] have shown that a similar monad, which even enjoys the additional special property of having iterations for all guarded substitution rules (complete iterativeness), arises from the inverses of the final (A + H−)coalgebras (if existing), i.e., the algebras of nonwellfounded Hterms. We show that, upon an appropriate generalization of the notion of substitution, the same can more generally be said about the initial T ′ (A, −)algebras resp. the inverses of the final T ′ (A, −)coalgebras for any endobifunctor T ′ on any category C such that the functors T ′ (−,X) uniformly carry a monad structure.
Coinduction for Exact Real Number Computation
, 2007
"... This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the corr ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the correctness of various corecursive exact real number algorithms. The basic framework is the classical theory of coinductive sets as greatest fixed points of monotone operators and hence is different from (though related to) the type theoretic approach by Ciaffaglione and Gianantonio. Key words: Exact real number computation, coinduction, corecursion, signed digit streams. 1
Relating Two Approaches to Coinductive Solution of Recursive Equations
 Milius (Eds.), Proceedings of the 7th Workshop on Coalgebraic Methods in Computer Science, CMCS’04 (Barcelona, March 2004), Electron. Notes in Theoret. Comput. Sci
, 2004
"... This paper shows that the approach of [2,12] for obtaining coinductive solutions of equations on infinite terms is a special case of a more general recent approach of [4] using distributive laws. ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
This paper shows that the approach of [2,12] for obtaining coinductive solutions of equations on infinite terms is a special case of a more general recent approach of [4] using distributive laws.
Coalgebraic BisimulationUpTo
, 2013
"... Bisimulationupto enhances the bisimulation proof method for process equivalence. We present its generalization from labelled transition systems to arbitrary coalgebras, and show that for a large class of systems, enhancements such as bisimulation up to bisimilarity, up to equivalence and up to con ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Bisimulationupto enhances the bisimulation proof method for process equivalence. We present its generalization from labelled transition systems to arbitrary coalgebras, and show that for a large class of systems, enhancements such as bisimulation up to bisimilarity, up to equivalence and up to context are sound proof techniques. This allows for simplified bisimulation proofs for many different types of statebased systems.