By Alexander Kurz, Marina Lenisa
This ebook constitutes the court cases of the 3rd overseas convention on Algebra and Coalgebra in computing device technological know-how, CALCO 2009, shaped in 2005 through becoming a member of CMCS and WADT. This yr 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 offered including 4 invited talks and workshop papers from the CALCO-tools Workshop. The convention was once divided into the subsequent periods: algebraic results and recursive equations, thought of coalgebra, coinduction, bisimulation, stone duality, video game concept, graph transformation, and software program improvement ideas.
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Additional info for Algebra and Coalgebra in Computer Science: Third International Conference, CALCO 2009, Udine, Italy, September 7-10, 2009, Proceedings
Pik and r = qi1 . . qik for some sequence of indices i1 , . . , ik , as follows. If we treat symbols of Σ as morphisms with the typing 1 → T 1, s can be defined by the term n s = do x ← init x ← i=1 ret(do pi ; c; qi−1 ) in n ret(do pi ; x; qi−1 )∗ ; x i=1 Then, the fact that the PCP instance does have a solution can be expressed by the inequality ret s ≤ do x ← r; ret(s + x) (6) where r is a metaprogram presenting the nondeterministic collection of terms ret(do l; c; l−1 ), defined by the term r = init x ← ret c in ret(do a; x; a) + ret(do b; x; b)∗ .
An , if do a1 ; . . ; an ≤ p, then do a1 ; . . ; an ≤ q and vice versa. This means precisely that ι(p) = ι(q) holds over the algebra of regular events. Hence, by the completeness result of , the last equation must be provable in the corresponding calculus for Kleene monads. We are done if we show how to turn this proof into a proof in MCE. To this end, define an operator κX taking any Kleene algebra term to a term of the MCE of type X → T X over the signature Σ by – – – – – κX (ˆ a) = a ˆ(z), a ˆ∈Σ κX (∅) = ∅, κX (p + q) = κX (p) + κX (q), κX (p · q) = do z ← κX (q); κX (p), κX (p∗ ) = ret z + init z ← κX (p) in κX (p)∗ .
A typical example of this kind is init x ← p in(do x ← q; ret a)∗ , which is equivalent to p + do x ← (init x ← q in q[a/x]∗ ); ret a. But in fact, it is not even semi-decidable to check whether a program admits a regular implementation. Theorem 15. 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.
Algebra and Coalgebra in Computer Science: Third International Conference, CALCO 2009, Udine, Italy, September 7-10, 2009, Proceedings by Alexander Kurz, Marina Lenisa