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By Alexander Kurz, Marina Lenisa
ISBN-10: 3642037402
ISBN-13: 9783642037405
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.
Read Online 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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Additional info for Algebra and Coalgebra in Computer Science: Third International Conference, CALCO 2009, Udine, Italy, September 7-10, 2009, Proceedings
Example text
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 [9], 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
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