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Automata and coinduction (an exercise in coalgebra
 LNCS
, 1998
"... The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which ..."
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Cited by 63 (16 self)
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The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the present treatment of automata theory may serve as an introduction to coalgebra.
A Hierarchy of Probabilistic System Types
, 2003
"... We study various notions of probabilistic bisimulation from a coalgebraic point of view, accumulating in a hierarchy of probabilistic system types. In general, a natural transformation between two Setfunctors straightforwardly gives rise to a transformation of coalgebras for the respective functors ..."
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Cited by 37 (6 self)
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We study various notions of probabilistic bisimulation from a coalgebraic point of view, accumulating in a hierarchy of probabilistic system types. In general, a natural transformation between two Setfunctors straightforwardly gives rise to a transformation of coalgebras for the respective functors. This latter transformation preserves homomorphisms and thus bisimulations. For comparison of probabilistic system types we also need reflection of bisimulation. We build the hierarchy of probabilistic systems by exploiting the new result that the transformation also reflects bisimulation in case the natural transformation is componentwise injective and the first functor preserves weak pullbacks. Additionally, we illustrate the correspondence of concrete and coalgebraic bisimulation in the case of general Segalatype systems.
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 26 (10 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.
Final coalgebras and the HennessyMilner property
 Annals of Pure and Applied Logic
"... The existence of a final coalgebra is equivalent to the existence of a formal logic with a set (small class) of formulas that has the HennessyMilner property of distinguishing coalgebraic states up to bisimilarity. This applies to coalgebras of any functor on the category of sets for which the bisi ..."
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Cited by 1 (1 self)
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The existence of a final coalgebra is equivalent to the existence of a formal logic with a set (small class) of formulas that has the HennessyMilner property of distinguishing coalgebraic states up to bisimilarity. This applies to coalgebras of any functor on the category of sets for which the bisimilarity relation is transitive. There are cases of functors that do have logics with the HennessyMilner property, but the only such logics have a proper class of formulas. The main theorem gives a representation of states of the final coalgebra as certain satisfiable sets of formulas. The key technical fact used is that any function between coalgebras that is truthpreserving and has a simple codomain must be a coalgebraic morphism.
‡ Corresponding author.
, 2006
"... We extend ReichelJacobs coalgebraic account of specification and refinement of objects and classes in Object Oriented Programming to (generalized) binary methods. These are methods that take more than one parameter of a class type. Class types include products, sums and powerset type constructors. ..."
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We extend ReichelJacobs coalgebraic account of specification and refinement of objects and classes in Object Oriented Programming to (generalized) binary methods. These are methods that take more than one parameter of a class type. Class types include products, sums and powerset type constructors. In order to take care of class constructors, we model classes as bialgebras. We study and compare two solutions for modeling generalized binary methods, which use purely covariant functors. In the first solution, which applies when we already have a class implementation, we reduce the behaviour of a generalized binary method to that of a bunch of unary methods. These are obtained by freezing the types of the extra class parameters to constant types. If all parameter types are finitary, then the bisimilarity equivalence induced on objects by this model yields the greatest congruence with respect to method application. In the second solution, we treat binary methods as graphs instead of functions, thus turning contravariant occurrences in the functor into covariant ones. In both cases, final coalgebras are shown to exist.
Hidden Congruent Deduction
"... Proofs by coinduction are dual to proofs by induction, in that the former are based on a largest congruence, and the latter on a smallest subalgebra (e.g., see [12]). Inductive proofs require choosing a set of constructors, often called a basis; the dual notion is cobasis, and as with bases for indu ..."
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Proofs by coinduction are dual to proofs by induction, in that the former are based on a largest congruence, and the latter on a smallest subalgebra (e.g., see [12]). Inductive proofs require choosing a set of constructors, often called a basis; the dual notion is cobasis, and as with bases for induction, the right choice can result in a dramatically simplified proof. An interesting complication is that the best choice may not be part of the given signature, but rather contain operations that can be defined over it.
Covarieties of Coalgebras: Comonads and
"... Abstract. Coalgebras provide effective models of data structures and statetransition systems. A virtual covariety is a class of coalgebras closed under coproducts, images of coalgebraic morphisms, and subcoalgebras defined by split equalisers. A covariety has the stronger property of closure under ..."
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Abstract. Coalgebras provide effective models of data structures and statetransition systems. A virtual covariety is a class of coalgebras closed under coproducts, images of coalgebraic morphisms, and subcoalgebras defined by split equalisers. A covariety has the stronger property of closure under all subcoalgebras, and is behavioural if it is closed under domains of morphisms, or equivalently under images of bisimulations. There are many computationally interesting properties that define classes of these kinds. We identify conditions on the underlying category of a comonad G which ensure that there is an exact correspondence between (behavioural/virtual) covarieties of Gcoalgebras and subcomonads of G defined by comonad morphisms to G with natural categorical properties. We also relate this analysis to notions of coequationally defined classes of coalgebras. 1