Partial Recursive Functions: Functions defined by recursion and allowing for undefined outputs; they capture the idea that some algorithmic processes may not halt for all inputs.

GERHARD JÄGER, TIMOTEJ ROSEBROCK, SATO KENTARO, TRUNCATION AND SEMI-DECIDABILITY NOTIONS IN APPLICATIVE THEORIES, The Journal of Symbolic Logic, Vol. 83, No. 3 (SEPTEMBER 2018), pp. 967-990 ...