Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 210
Appendix C. The Dijkstra loop
E.W. Dijkstra [C.1] (see also [C.2]) introduced and justified a multibranch generalization of the
conventional WHILE loop in his theory of systematic derivation of imperative programs. This loop proves
convenient for expression and — most importantly — for verification of the algorithms which are usually
expressed in terms of embedded loops, thus significantly reducing the effort of debugging.
In the language Oberon-07 presented in [C.3], the syntax of this loop is defined as follows:
WhileStatement
= WHILE logical expression DO
operator sequence
{ELSIF logical expression DO
operator sequence}
END.
If any of the logical expressions (guards) is evaluated to TRUE then the corresponding operator sequence is
executed. The evaluation of the guards and the execution of the corresponding operator sequences is
repeated until all guards evaluate to FALSE. So, the postcondition for this loop is the conjunction of
negations of all guards.
Example:
WHILE m > n DO m := m - n
ELSIF n > m DO n := n - m
END
Postcondition: ~(m > n) & ~(n > m), which is equivalent to n = m.
The loop invariant must hold at the beginning and at the end of each branch.
Roughly speaking, an n-branch Dijkstra loop corresponds to constructions with n usual loops that are
somehow embedded one into another.
The advantage of the Dijkstra loop is due to the fact that all the logic of an arbitrarily complex loop is
expressed at the same level and is radically clarified, so that algorithms with various cases of embedded
interacting ("braided") loops receive a completely uniform treatement.
An efficient way to construct such a loop consists in enumerating all possible situations that can emerge,
describing them by the corresponding guards, and for each guard — independently of the others —
constructing operations that advance the algorithm towards its goal, togethers with the operations that
restore the invariant. The enumeration of the guards is stopped when the disjunction (OR) of all guards
covers the assumed precondition of the loop. It is useful to remember that the task of construction of a
correct loop is simplified if one postpones worrying about the order of evaluation of the guards and
execution of the branches, optimizations of evaluation of the guards etc., until the loop is correctly
constructed. Such lower-level optimizations are rarely significant, and their implementation is greatly
simplified when the correctness of a complicated loop has already been ensured.
Although in Dijkstra's theory the order of guard evaluations is undefined, in this book the guards are
assumed to be evaluated (and branches selected for execution) in their textual order.
In most programming languages the Dijkstra loop has to be modelled. In the older versions of Oberon
(including Component Pascal) it is sufficient to use the LOOP construct, with the body consisting of a
multibranch IF operator that contains the single loop EXIT operator in the ELSE branch: