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13
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.
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 ..."
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Cited by 16 (3 self)
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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 ..."
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Cited by 12 (1 self)
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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 ..."
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Cited by 7 (4 self)
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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.
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 ..."
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Cited by 5 (4 self)
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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
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 ..."
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Cited by 5 (0 self)
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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 ..."
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Cited by 4 (1 self)
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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.
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. ..."
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Cited by 3 (2 self)
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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.
Coinductive Proofs for Basic Real Computation Tie Hou
"... Abstract. We describe two representations for real numbers, signed digit streams and Cauchy sequences. We give coinductive proofs for the correctness of functions converting between these two representations to show the adequacy of signed digit stream representation. We also show a coinductive proof ..."
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Abstract. We describe two representations for real numbers, signed digit streams and Cauchy sequences. We give coinductive proofs for the correctness of functions converting between these two representations to show the adequacy of signed digit stream representation. We also show a coinductive proof for the correctness of a corecursive program for the average function with regard to the signed digit stream representation. We implemented this proof in the interactive proof system Minlog. Thus, reliable, corecursive functions for real computation can be guaranteed, which is very helpful in formal software development for real numbers.