Laws of Programming

Theory of Programs

“…In theoretical informatics the habit has often been different: devising axiomatic theories. The most developed example is the admirable work of Hoare and colleagues [7] [8]. A notable property of these efforts is that they postulate their laws; then "of course, the mathematician should also design a model of the language, to check completeness and consistency of the laws, to provide a framework for the specifications of programs, and for proofs of correctness" [7].…”

“…The most developed example is the admirable work of Hoare and colleagues [7] [8]. A notable property of these efforts is that they postulate their laws; then "of course, the mathematician should also design a model of the language, to check completeness and consistency of the laws, to provide a framework for the specifications of programs, and for proofs of correctness" [7]. The justification for this method -postulate your ideal laws, the model will follow -is that it has, in Russell's words cited in [9], "the advantages of theft over honest toil".…”

“…Programs are just relations over sets. An informal and non-exhaustive review of the axioms of classic articles such as [7] and its extension to concurrency [8] (not considering properties specific to individual calculi), as well as [6] and [10], suggests that the properties they introduce can be derived, often straightforwardly, from the framework of this article; many indeed appear above as theorems.…”

“…They prefer to posit axioms, even if these axioms mimic the elementary properties of set operators. An example is the seminal "Laws of Programming" article [7] (with its recent extension to concurrency [8]), whose authors axiomatically introduce operators with names such as "∪" for non-deterministic choice and "⊆" for refinement. They never suggest that these could actually be the standard mathematical operators bearing the same names; but they cover several pages of Communications of the ACM with such fascinating "axioms" as P ∪ (Q ∪ R) = (P ∪ Q) ∪ R. One wonders whether the thought ever arose that if it associates like union, commutes like union, distributes like union, and typographically uses the exact symbol of union, perhaps it is union.…”

Verification conditions are code

“…The derivation of P α therefore can involve only the assignment axiom and the laws of sequence and consequence. Any sequence of assignments without intervening consequence can be merged into a single multiple assignment [23] so P α can equivalently be presented in a form whereby every pre-condition, except possibly the first, is the antecedent of a verification condition, and similarly every postcondition, except possibly the last, is the consequent of a verification condition. This, by definition, is an α-equalising proof.…”