Algebra and Coalgebra in Computer Science: Third by Alexander Kurz, Marina Lenisa

By Alexander Kurz, Marina Lenisa

This e-book constitutes the court cases of the 3rd foreign convention on Algebra and Coalgebra in computing device technological know-how, CALCO 2009, shaped in 2005 by means of becoming a member of CMCS and WADT. This yr the convention was once held in Udine, Italy, September 7-10, 2009. The 23 complete papers have been conscientiously reviewed and chosen from forty two submissions. they're offered including 4 invited talks and workshop papers from the CALCO-tools Workshop. The convention was once divided into the next classes: algebraic results and recursive equations, idea of coalgebra, coinduction, bisimulation, stone duality, online game concept, graph transformation, and software program improvement options.

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Extra resources for Algebra and Coalgebra in Computer Science: Third International Conference, CALCO 2009, Udine, Italy, September 7-10, 2009, Proceedings

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Complete Iterativity for Algebras with Effects 43 (4) For the environment monad M X = X E , α : A → AE is a λ-cia iff for each i ∈ E the map πi · α : A → A is a unary cia. 5. (5) Let M = FΣ be the monad assigning to a set X the set of all finite Σ-trees on X. If Σ consists of one constant symbol, then M is the maybe monad. In all other cases there are no unary λ-cias. In order to see this, assume that α : A → M A is a λ-cia. Then μA · M α has a unique fixed point t. Observe that the action of μA · M α is that of replacing in all trees of M A each leaf labelled by a ∈ A with α(a).

Each variable in X is mapped to the tree unfolding of its recursive definition if this unfolding is finite, and to ⊥ otherwise. 11(2). The unique solution of e : X → HΣ X+PFΣ Y assigns to a variable x the set of all possible tree unfoldings (taking into account that e(x ) ⊆ FΣ Y for some variables x ) of the recursive definition of x if all these unfoldings are finite and ∅ else. For example, for the signature with one binary operation symbol ∗ the system ∗ ∗ x ≈ x ∗ x2 x1 ≈ { , y3 } x2 ≈ { } x ≈ x1 ∗ x2 y1 y2 y3 y4 has the unique solution with e† (x) given by the set of trees with elements ∗ ∗ ∗ ∗ and y3 ∗ y1 y2 y3 y4 y3 y4 and with e (x ) = ∅.

An |E| × |E| matrix with values in X) its diagonal (considered as a map from E to X). (5) The finite-list monad M X = X ∗ has the unit given by singleton lists and the multiplication by flattening a list of lists. (6) The finite-multiset monad M = M assigns to a set X the free commutative monoid on X or, equivalently, the set of all finite multisets on X. In this paper we are interested in endofunctors on Set that have a lifting to the Kleisli category of the monad M . In fact, we shall establish that every analytic endofunctor (see [6, 7]) has a canonical lifting to the Kleisli category of every commutative monad (see [8]).

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