Reasoning about Loops in Total and General Correctness

Abstract

Abstract. We introduce a calculus for reasoning about programs in total correctness which blends UTP designs with von Wright's notion of a demonic refinement algebra. We demonstrate its utility in verifying the familiar loop-invariant rule for refining a total-correctness specification by a while loop. Total correctness equates non-termination with completely chaotic behaviour, with the consequence that any situation which admits non-termination must also admit arbitrary terminating behaviour. General correctness is more discriminating in allowing nontermination to be specified together with more particular terminating behaviour. We therefore introduce an analogous calculus for reasoning about programs in general correctness which blends UTP prescriptions with a demonic refinement algebra. We formulate a loop-invariant rule for refining a general-correctness specification by a while loop, and we use our general-correctness calculus to verify the new rule.

UTP by Example: Designs

“…This includes the hardware description languages Handel-C [55,56] and Verilog [78]; the multi-paradigm languages Circus [52,13,74,53,66] and CML [73,70]; Safety-Critical Java [21,19,24,54,20]; and Simulink [17]. A wide variety of programming theories have been formalised in UTP, including confidentiality [6,7]; general correctness [27,29,36,28]; theories of testing [15,67,16]; hybrid systems; and theories of undefinedness [71,5]. These are complemented by a collection of meta-theory, including work on higher-order UTP [75]; UTP and temporal-logic model checking [2]; and CSP as a retract of CCS [41].…”