On the orthogonality of assignments and procedures in Algol

According to folklore, Algol is an “orthogonal” extension of a simple imperative programming language with a call-by-name functional language. The former contains assignments, branching constructs, and compound statements; the latter is based on the typed λ-calculus. In an attempt to formalize the claim of “orthogonality”, we define a simple version of Algol and an extended λ-calculus. The calculus includes the full β-rule and rules for the reduction of assignment statements and commands. It has the usual properties, e.g., it satisfies a Church-Rosser and Strong Normalization Theorem. In support of the claim that the imperative and functional components are orthogonal to each other, we show that the proofs of these theorems are combinations of separate Church-Rosser and Strong Normalization theorems for each sublanguage.

An acclaimed consequence of Algol's orthogonal design is the idea that the evaluation of a program has two distinct phases. The first phase corresponds to an unrolling of the program according to the usual β and fixpoint reductions, which provide the formal counterpart to Algol's famous copy rule. The result of this phase is essentially an imperative program. The second phase executes the output of the first phase in the imperative fashion of a stack machine. Given our calculus, we can prove a Postponement Theorem and can thus formalize this phase separation.