A generalization of Kleene's theorem and nondeterministic structured programming

Abstract

We define a completely ordered (c.o.) monoid to be a set equipped with monoid and complete lattice structures such that the product is continuous in each argument. The languages over an alphabet form a c.o. monoid. We give a generalization of Kleene's theorem from languages to elements of an arbitrary c.o. monoid. In the case of the c.o. monoid of all binary relations on a set we obtain the computational equivalence between flowchart-like nondeterministic programs and structured nondeterministic programs constructed using the generalization of the rational operations on languages.