Structured program theorem

The structured program theorem, also called Böhm-Jacopini theorem, is a result in programming language theory. It states that a class of control flow graphs (historically called charts in this context) can compute any computable function if it combines subprograms in only three specific ways (control structures). These are

The structured chart subject to these constraints may however use additional variables in the form of bits (stored in an extra integer variable in the original proof) in order to keep track of information that the original program represents by the program location. The construction was based on Böhm's programming language P′′.

The theorem is typically credited to a 1966 paper by Corrado Böhm and Giuseppe Jacopini.David Harel wrote in 1980 that the Böhm–Jacopini paper enjoyed "universal popularity", particularly with proponents of structured programming. Harel also noted that "due to its rather technical style [the 1966 Böhm–Jacopini paper] is apparently more often cited than read in detail" and, after reviewing a large number of papers published up to 1980, Harel argued that the contents of the Böhm–Jacopini proof was usually misrepresented as a folk theorem that essentially contains a simpler result, a result which itself can be traced to the inception of modern computing theory in the papers of von Neumann and Kleene.

Harel also writes that the more generic name was proposed by H.D. Mills as "The Structure Theorem" in the early 1970s.

This version of the theorem replaces all the original program's control flow with a single global while loop that simulates a program counter going over all possible labels (flowchart boxes) in the original non-structured program. Harel traced the origin of this folk theorem to two papers marking the beginning of computing. One is the 1946 description of the von Neumann architecture, which explains how a program counter operates in terms of a while loop. Harel notes that the single loop used by the folk version of the structured programming theorem basically just provides operational semantics for the execution of a flowchart on a von Neumann computer. Another, even older source that Harel traced the folk version of the theorem is Stephen Kleene's normal form theorem from 1936.

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