Algebra and Coalgebra in Computer Science: Third by Gordon Plotkin (auth.), Alexander Kurz, Marina Lenisa,

By Gordon Plotkin (auth.), Alexander Kurz, Marina Lenisa, Andrzej Tarlecki (eds.)

This booklet constitutes the lawsuits of the 3rd overseas convention on Algebra and Coalgebra in machine technology, CALCO 2009, shaped in 2005 by means of becoming a member of CMCS and WADT. This 12 months the convention used to be held in Udine, Italy, September 7-10, 2009.

The 23 complete papers have been conscientiously reviewed and chosen from forty two submissions. they're provided including 4 invited talks and workshop papers from the CALCO-tools Workshop. The convention was once divided into the subsequent classes: algebraic results and recursive equations, conception of coalgebra, coinduction, bisimulation, stone duality, online game thought, graph transformation, and software program improvement techniques.

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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

Example text

E. 5 Worked Example: Stack Reverse in the MCE In spite of the negative result proved in Theorem 12 we believe the calculus MCE to be a reasonable framework for proving program equivalence. In fact, we have encoded the calculus in the Isabelle/HOL prover. We now present an example that we have axiomatised as an Isabelle theory and successfully verified1: the double-reverse theorem stating that reversing a list twice yields the original list. More specifically, we have proved this theorem for two implementations of reverse, the single-stack example from the introduction and a further implementation using two stacks.

Xn , a1 . . 1) where X = {x1 , . . , xn } is a set of variables, a1 , . . , ak are elements of the algebra (called parameters) and the ti are Σ-terms that are not single variables x ∈ X, has a unique solution. For example, the algebra TΣ Y of all (not necessarily finite) Σ-trees on the set Y is iterative. The support by the German Research Foundation (DFG) under the project AD 207/1-1 is acknowledged. A. Kurz, M. Lenisa, and A. ): CALCO 2009, LNCS 5728, pp. 34–48, 2009. c Springer-Verlag Berlin Heidelberg 2009 Complete Iterativity for Algebras with Effects 35 The notion of iterative algebra was extended and generalised in [1]; there, iterative algebras are studied for finitary endofunctors on Set (or, more generally, on a locally finitely presentable category).

Vn )) = i → σ(v1 (i), . . , vn (i)) for σ ∈ Σn and v1 , . . , vn : E → X and i ∈ E. More generally, there exists a canonical distributive law of every endofunctor H over M as follows: observe that X E ∼ = X E E i∈E X with projections πi : X → X for each i ∈ E. Define λX : H(X ) → (HX)E as the unique morphism such that πiHX · λX = HπiX for every i ∈ E. It is easy to prove that λ is a distributive law of H over M . 9 the identity transformation λ = id : M ⇒ M (which, in fact, is a distributive law for any monad).

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