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We give an algebraic model of the designs of UTP based on a variant of modal semirings, hence generalising the original relational model. This is intended to exhibit more clearly the algebraic principles behind UTP and to provide deeper insight into the general properties of designs, the program and specification operators, and refinement. Moreover, we set up a formal connection with general and total correctness of programs as discussed by a number of authors. Finally we show that the designs form a left semiring and even a Kleene and omega algebra. This is used to calculate closed expressions for the least and greatest fixedpoint semantics of the demonic while loop that are simpler than the ones obtained from standard UTP theory and previous algebraic approaches.

Abstract. We investigate the problem of finding a total order of a finite set that satisfies various local ordering constraints. Depending on the admitted constraints, we provide an efficient algorithm or prove NP-completeness. We discuss several generalisations and systematically classify the problems.

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We generalise the designs of the 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, for example, since they form a Kleene and omega algebra and a test semiring. We apply our framework to investigate symmetric linear recursion and its relation to tail-recursion. This substantially involves Kleene and omega algebra as well as additional algebraic formulations of determinacy, invariants, domain, pre-image, convergence and Noetherity. Due to the uncovered algebraic structure of UTP designs, all our general results also directly apply to UTP.