By Alexander Kurz, Marina Lenisa
This publication constitutes the complaints of the 3rd foreign convention on Algebra and Coalgebra in desktop technology, CALCO 2009, shaped in 2005 by way 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 awarded including 4 invited talks and workshop papers from the CALCO-tools Workshop. The convention used to be divided into the next periods: algebraic results and recursive equations, idea of coalgebra, coinduction, bisimulation, stone duality, online game idea, graph transformation, and software program improvement recommendations.
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Additional resources for Algebra and Coalgebra in Computer Science: Third International Conference, CALCO 2009, Udine, Italy, September 7-10, 2009, Proceedings
In this paper we consider cias in the two categories Set and SetM . To distinguish ¯ we have the ordinary cias for an endofunctor H on Set from those for a lifting H following ¯ : SetM → SetM a Kleisli-cia. 7. 8. If we spell out the definition of a cia in SetM , we see that a flat equation morphism is a map e : X → M (HX + A), and a solution of e in the algebra α : HA → M A is a map e† : X → M A such that M(λ·He† +ηA ) e† = (X −→ M (HX + A) −−−−−−−−−→ M (M HA + M A) e μX+A ·M[Minl,Minr] μA ·[α,ηA ] −−−−−−−−−−−−−→ M (HA + A) −−−−−−→ M A) This means that the algebra (A, α) as well as the recursive equation e and its solution are “effectful”.
14. Every Kleisli-cia is a λ-cia. Proof. Let α : HA → M A be a Kleisli-cia and let e : X → HX + M A be an ¯ + A as M -equation morphism. We form a flat equation morphism e¯ : X ◦ / HX follows: ¯ +εA Je ¯ + M A idHX / HX ¯ + A) . 9) that a morphism s : X ◦ / A is a solution of e¯ in the Kleisli-cia A iff it is a solution of e. Since the former exists uniquely, so does the latter; thus (A, α) is a λ-cia. 15. 14 does not hold in general. Let H = Id, let M X = X ∗ be the monad of finite lists on X, and let λ = id : M ⇒ M .
1. Stateful computations with nontermination: T A = S → (1 + A × S), where S is a fixed set of states. 2. Nondeterminism: T A = P(A), where P is the covariant power set functor. 3. Exceptions: T A = A + E, where E is a fixed set of exceptions. 4. Interactive input: T A is the smallest fixed point of γ → A + (U → γ), where U is a set of input values. 5. Interactive output: T A is the smallest fixed point of γ → A + (U × γ), where U is a set of output values. 6. Nondeterministic stateful computation: T A = S → P(A × S).