On Two Dually Nondeterministic Refinement Algebras

Abstract

Abstract.A dually nondeterministic refinement algebra with a negation operator is proposed. The algebra facilitates reasoning about totalcorrectness preserving program transformations and nondeterministic programs. The negation operator is used to express enabledness and termination operators through a useful explicit definition. As a small application, a property of action systems is proved employing the algebra. A dually nondeterministic refinement algebra without the negation operator is also discussed.

A Sketch of a Dynamic Epistemic Semiring

“…How more elaborate models than the relational one presented in this paper (such as hypertheories) relate to our algebra is also an interesting question that we shall return to. Finally, many of the recent abstract-algebraic approaches have proved practical in applications, see for example [27,17,23]; in an epistemic context, modal semirings have recently been applied by Möller [18]. It is a promising thought that the dynamic epistemic semiring would prove useful in applications involving epistemic notions.…”

“…The first statement (additivity) is proved by (p + q) = { (27)} (p + q)(p + q) = {; distributive over +} (p + q)p + (p + q)q = { (23) and commutativity of +} p + q, the second one (isotony) is straightforward from the first.…”

Refinement Algebra for Probabilistic Programs

“…5. On the other hand, the new weak iteration axioms (11) and (13) are essential for the last data refinement rule of Sect. 6.…”

“…For non-probabilistic demonic refinement algebra, a termination operator was defined by Solin and von Wright [12,11]. When applied to a non-probabilistic program, this termination operator returns an assertion that skips in those states from which the program will terminate, otherwise it aborts.…”

“…One of the earliest such abstract algebras is Kleene algebra with tests, which was employed by Kozen for proving transformation rules of loops [4]. Recently, McIver, Cohen and Morgan used probabilistic Kleene algebra for protocol verification [5] and Solin and von Wright used refinement algebras for program reasoning in a total-correctness environment [14,15,12,11].…”