By Gordon Plotkin (auth.), Alexander Kurz, Marina Lenisa, Andrzej Tarlecki (eds.)
This booklet constitutes the court cases of the 3rd foreign convention on Algebra and Coalgebra in laptop technology, CALCO 2009, shaped in 2005 by way of becoming a member of CMCS and WADT. This 12 months the convention was once held in Udine, Italy, September 7-10, 2009.
The 23 complete papers have been rigorously reviewed and chosen from forty two submissions. they're offered including 4 invited talks and workshop papers from the CALCO-tools Workshop. The convention used to be divided into the subsequent periods: algebraic results and recursive equations, idea of coalgebra, coinduction, bisimulation, stone duality, online game thought, graph transformation, and software program improvement techniques.
Read or Download Algebra and Coalgebra in Computer Science: Third International Conference, CALCO 2009, Udine, Italy, September 7-10, 2009. Proceedings PDF
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Extra info for Algebra and Coalgebra in Computer Science: Third International Conference, CALCO 2009, Udine, Italy, September 7-10, 2009. Proceedings
Ap ) in A. For example, let Σ be the signature with one binary operation symbol ∗ and with the constant symbol c. In the Σ-algebra A = TΣ Y of all Σ-trees on Y , i. , (rooted and ordered) trees so that nodes with n > 0 children are labelled in Σn and leaves are labelled in Σ0 + Y , the following system x1 ≈ x2 ∗ c x2 ≈ x1 ∗ y has as its unique solution the trees ∗ ∗ ∗ ∗ c .. y . This motivates the following .. 5. () Let A be a category with finite coproducts and let H : A → A. A flat equation morphism in an object A (of parameters) is a morphism e : X → HX +A.
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). This implies that for every leaf of the tree t labelled by an element a ∈ A we have that α(a) is the single-node tree labelled by a. But then every Σ-tree whose leaves have labels that also appear as leaf labels of t is a fixed point of μA · M α. Hence, t does not contain a leaf labelled in A. On the other hand, each tree with no such leaves is a fixed point of μA · M α. Thus there must be a unique constant in Σ and no other operation symbols, and so M is the maybe monad.
If the underlying category C is Cartesian, this structure is induced by operations ⊕ : T A × T A → T A and 0 : 1 → T A which satisfy the idempotent commutative monoid laws and distribute over Kleisli composition; we denote the induced structure on the hom-sets in the Kleisli category by ⊕ and 0 as well. A strong additive monad on a Cartesian category is an additive monad with a tensorial strength t satisfying two extra conditions: tA,B ◦ id, 0 = 0 tA,B ◦ id, f ⊕ g = tA,B ◦ id, f ⊕ tA,B ◦ id, g .