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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. ..."
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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.
An introduction to (co)algebra and (co)induction
"... Algebra is a wellestablished part of mathematics, dealing with sets with operations satisfying certain properties, like groups, rings, vector spaces, etcetera. Its results are essential throughout mathematics and other sciences. Universal algebra is a part of algebra in which algebraic structures a ..."
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Algebra is a wellestablished part of mathematics, dealing with sets with operations satisfying certain properties, like groups, rings, vector spaces, etcetera. Its results are essential throughout mathematics and other sciences. Universal algebra is a part of algebra in which algebraic structures are studied at a high
A coalgebraic view on biinfinite streams
"... Biinfinite streams arise as a natural data structure in several contexts, such as signal processing [1], symbolic dynamics [2], (balanced) representation of real/rational numbers [3] or study of sets invariant under shift transformation [4]. In this paper, we will present a coalgebraic view of the ..."
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Biinfinite streams arise as a natural data structure in several contexts, such as signal processing [1], symbolic dynamics [2], (balanced) representation of real/rational numbers [3] or study of sets invariant under shift transformation [4]. In this paper, we will present a coalgebraic view of the set of biinfinite streams which we shall denote by AZ and is formally defined as AZ = {σ  σ: Z → A} We can easily prove that AZ ∼ = (A × A)ω and therefore the set AZ is the final coalgebra for the functor FX = (A×A)×X. This reflects the fact that one can think about a biinfinite stream denoted by (..., σ−2, σ−1, σ0, σ1, σ2,...), as two infinite streams growing in parallel. σ0 σ1 σ2... σ−1 σ−2 σ−3... Using this observation, and defining a semiring structure on A × A, we could reuse the calculus developed for streams [5] to deal with the biinfinite case. However, is this the only/best way to view biinfinite streams coalgebraically? Will this reduction to the infinite case be too restrictive and not allow us to fully benefit from the structure of biinfinite streams? We shall now present another possible representation for biinfinite streams. We can see (..., σ−2, σ−1, σ0, σ1, σ2,...) as an infinite binary tree as follows: σ0 σ2 σ1 σ0 σ2 σ0 σ1 The set TA of infinite binary trees is the final coalgebra for the functor GX = X ×A×X and as showed in [6], by viewing trees as formal power series a very simple but surprisingly powerful coinductive calculus can be developed. In this framework, definitions are presented as behavioural differential equations and very compact closed formulae can be deduced for (rational) trees. For instance, in this framework the biinfinite stream (..., 0, 1, 0, 1, 0, 1, 0, 1, 0,...) would be represented by the formula (L+ R)(1 + (L − R)2)−1, where L and R represent the following constant trees.
London, United Kingdom
"... The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is wellknown but commonly elided. In this paper we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particul ..."
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The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is wellknown but commonly elided. In this paper we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particular, we consider different coalgebra structures that turn the set of infinite streams into the carrier of a final coalgebra. After that we focus on coalgebra structures that are made up using socalled cooperations. We say that a collection of cooperations is complete for a given set X if it gives rise to a coalgebra structure that turns X into the carrier set of a subcoalgebra of a final coalgebra. Any complete set of cooperations yields a coalgebraic proof and definition principle. We exploit this fact and devise a general definition scheme for constants and functions on a set X that is parametrically in the choice of the complete set of cooperations for X. Key words: Coalgebra, coinduction, infinite data structures, hidden algebra. 1