The Construction and Computation of for-loop Programs for SCMPDS

This article defines two for-loop statements for SCMPDS. One is called for-up, which corresponds to ”for (i=x; i<0; i+=n) S” in C language. Another is called for-down, which corresponds to ”for (i=x; i>0; i-=n) S”. Here, we do not present their unconditional halting (called parahalting) property, because we have not found that there exists a useful for-loop statement with unconditional halting, and the proof of unconditional halting is much simpler than that of conditional halting. It is hard to formalize all halting conditions, but some cases can be formalized. We choose loop invariants as halting conditions to prove halting problem of for-up/down statements. When some variables (except the loop control variable) keep undestroyed on a set for the loop invariant, and the loop body is halting for this condition, the corresponding for-up/down is halting and computable under this condition. The computation of for-loop statements can be realized by evaluating its body. At the end of the article, we verify for-down statements by two examples for summing.