Simple section interchange and properties of non-computable functions

We here extend our earlier work on the theory of partially defined computer instructions and guards to cover partially defined computer expressions and programs. The notion of the relevant region of an expression is generalized to conditional relevant regions, and we specify upper bounds on these for expressions built up from simpler ones. We then proceed to specify upper bounds on the input and output regions of programs containing goto statements, even though the state transformations of such programs are not necessarily computable. This is then combined with our earlier commutativity results to obtain a general condition under which two simple sections of such a program commute with each other, and therefore may be interchanged, or possibly done in parallel.