Abstract

I show how both Hoare triples and Plotkin triples can be replaced by an abstract algebra, in which all the primitives are pairs and singletons, i.e., constants, binary operators and binary predicates. The algebra deals separately but equally simply with both sequential and concurrent composition of programs; and many further program primitives and structuring operators like choice and disjoint concurrency can be introduced individually, one at a time. The relevant proof rules for Hoare Triples, and the relevant Plotkin triples for operational semantics, are derived simply and separately for each binary operator.

Familiar algebraic properties – associativity, commutivity, monotonicity, idempotence are widely reused. The interactions between operators can be described just two at a time by distributivity laws. These include mutual distribution, a weak form of the exchange law of two-category theory. This leads to an elementary presentation of separation logic and of Kleene algebra.

Thus we extend to the mathematics of programming the old saying about marriage: ‘two’s fun, three’s none’. Down with the triple!