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In their paper, Böhm and Jacopini conjectured, but did not prove that it was necessary to introduce such additional variables for certain kinds of non-structured programs in order to transform them into structured programs.<Ref name="K">J. Computer and System Sciences, 9, 3 (December 1974), {{doi| 10.1016/S0022-0000(74)80043-7}}</ref>{{rp|236}} An example of program (that we now know) does require such additional variables is a loop with two conditional exits inside it. In order to address the conjecture of Böhm and Jacopini, Kosaraju defined a more restrictive notion of program reduction than the Turing equivalence used by Böhm and Jacopini. Essentially, Kosaraju's notion of reduction imposes, besides the obvious requirement that the two programs must compute the same value (or not finish) given the same inputs, that the two programs must use the same primitive actions and predicates, understood as expressions used in the conditionals. Because of these restrictions, Kosaraju's reduction does not allow the introduction of additional variables; assigning to these variables would create new primitive actions and testing their values would change the predicates used in the conditionals. Using this more restrictive notion of reduction, Kosaraju proved Böhm and Jacopini's conjecture, namely that a loop with two exists cannot be transformed into a structured program ''without introducing additional variables'', but went further and proved that programs containing multi-level breaks (from loops) form an hierarchy, such that one can always find a program with multi-level breaks of depth ''n'' that cannot be reduced to a program of multi-level depth less than ''n'', again without introducing additional variables.<Ref name="K"/><Ref>For more modern treatment of the same results see: Kozen, [http://www.cs.cornell.edu/~kozen/papers/bohmjacopini.pdf The Böhm–Jacopini Theorem is False, Propositionally]</ref>
McCabe notes in his paper that in view of Kosaraju's results, he intended to find a way to capture the essential properties of non-structured programs.<ref name="mccabe76"/>{{rp|315}} He proceeds by first identifying the control flow graphs corresponding to the smallest non-structured programs (these include branching into a loop, branching out of a loop, and their if-then-else counterparts) which he uses to formulate a theorem analogous to [[Kuratowski's theorem]], and thereafter introduces his notion of essential complexity in order to give a scale answer ("measure of the
structureness of a program" in his words) rather than a yes/no answer to the question of whether a program's control flow graph is structured or not.<ref name="mccabe76"/>{{rp|315}} Finally, the notion of reduction used by McCabe to shrink the CFG is not the same as Kosaraju's notion of reducing flowcharts. The reduction defined on the CFG do not know or care about the program's inputs, it is simply a [[graph transformation]].<ref>McCabe footnotes the two definitions of on pages 315 and 317.</ref>
For example, the following C program fragment has an essential complexity of 1, because the inner '''if''' statement and the '''for''' can be reduced, i.e. it is a structured program.
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The idea of CFG reducibility by successive collapses of sub-graphs (ultimately to a single node for well-behaved CFGs) is also used in modern compiler optimization. The areas of the CFG that cannot be reduced to "nice" programs, technically called [[natural loop]]s (which have a rather involved definition that we
==See also==
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