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Omega Algebra, Demonic Refinement Algebra and Commands
 IN 9TH INTERNATIONAL CONFERENCE ON RELATIONAL METHODS IN COMPUTER SCIENCE AND 4TH INTERNATIONAL WORKSHOP ON APPLICATIONS OF KLEENE ALGEBRA, LECTURE
, 2006
"... Weak omega algebra and demonic refinement algebra are two ways of describing systems with finite and infinite iteration. We show that these independently introduced kinds of algebras can actually be defined in terms of each other. By defining modal operators on the underlying weak semiring, that res ..."
Abstract

Cited by 4 (3 self)
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Weak omega algebra and demonic refinement algebra are two ways of describing systems with finite and infinite iteration. We show that these independently introduced kinds of algebras can actually be defined in terms of each other. By defining modal operators on the underlying weak semiring, that result directly gives a demonic refinement algebra of commands. This yields models in which extensionality does not hold. Since in predicatetransformer models extensionality always holds, this means that the axioms of demonic refinement algebra do not characterise predicatetransformer models uniquely. The omega and the demonic refinement algebra of commands both utilise the convergence operator that is analogous to the halting predicate of modal µcalculus. We show that the convergence operator can be defined explicitly in terms of infinite iteration and domain if and only if domain coinduction for infinite iteration holds.
Normal Design Algebra
"... We generalise the designs of Unifying Theories of Programming (UTP) by defining them as matrices over semirings with ideals. This clarifies the algebraic structure of designs and considerably simplifies reasoning about them, e.g., forming a Kleene and omega algebra of designs. Moreover, we prove a g ..."
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We generalise the designs of Unifying Theories of Programming (UTP) by defining them as matrices over semirings with ideals. This clarifies the algebraic structure of designs and considerably simplifies reasoning about them, e.g., forming a Kleene and omega algebra of designs. Moreover, we prove a generalised fixpoint theorem for isotone functions on designs. We apply our framework to investigate symmetric linear recursion and its relation to tailrecursion; this substantially involves Kleene and omega algebra as well as additional algebraic formulations of determinacy, invariants, domain, preimage, convergence and noetherity. Due to the uncovered algebraic structure of UTP designs, all our general results also directly apply to UTP.