By Ernie Cohen (auth.), Renate A. Schmidt (eds.)

The ebook constitutes the joint refereed lawsuits of the ninth overseas convention on Relational tools in desktop technological know-how, RelMiCS 2006, and the 4th overseas Workshop on purposes of Kleene Algebras, AKA 2006, held in Manchester, united kingdom in August/September 2006.

The 25 revised complete papers offered including invited papers and the summary of an invited speak have been conscientiously reviewed and chosen from forty four submissions. The papers are dedicated to the idea of relation algebras, Kleene algebras, and similar formalisms in addition to to their different purposes in software program engineering, databases, and synthetic intelligence. a different concentration is on formal equipment, logics of courses, and hyperlinks to neighboring disciplines.

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Amer. Math. Soc. 272(2) (1982) 501–526 8. : Relation Algebras. Volume 150 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam (2006) 9. : Contributions to the theory of models. III. Nederl. Akad. Wetensch. Proc. Ser. A. 58 (1955) 56–64 = Indagationes Math. 17, 56–64 (1955) 10. : Some suﬃcient conditions for the representability of relation algebras. Algebra Universalis 8(2) (1978) 162–172 11. : Varieties of relation algebras. Algebra Universalis 15(3) (1982) 273– 298 Finite Symmetric Integral Relation Algebras with No 3-Cycles 29 12.

8. The transformer model of programs is the subspace consisting of strict, positively conjunctive and continuous transformers. X models the appropriate laws is of course standard and well documented [D76, H92, N89]. The transformer semantics is given in Fig. 8. But for implicit consistency with the relational semantics we prefer to deduce it— as much as is possible—from the Galois connection between the relational and transformer models (in the next section). For now, we record: Theorem (transformer model).

W. Sanders Lemma (coercions). b = skip ✁ b ✄ magic is injective and satisﬁes 1. true = skip ; 2. c) and, in particular, coercions commute; 3. b | b ∈ B } and, in particular, coer is antitone; 4. false = magic . ¬b o9 B ) = A ✁ b ✄ B . ¬b o9 magic) . We must be careful not to allow such simply duality to raise our hopes concerning the degree to which there is a transformer-like dual on the space of (relational) computations. 3 Models In this section we recall the relational and transformer models of computations (and programs in particular) and the Galois connection between them.