The Formal Termination Problem for Programs with Finite ALGOL 68-Modes

In f 1 l] the importance of the formal termination problem for a high level programming language has been pointed out. If this problem is algorithmically solvable, then the translator is at least principally able to check whether certain properties closely related to termination like macro property of programs or nonrecursivity or block behaviour of procedures are fulfilled and then to produce better object code. Whereas formal termination of ALGOL 60or 68programs is undecidable it has been conjectured in [lo] that the macro property is decidable for programs with finite modes in the sense of ALGOL 68 [ 141. In [ 1,2] the solvability of the formal termination and macro problem for finite mode programs is proved: Such programs T may be considered in a certain sense as n-rational program schemes S and vice versa. Formal non-termination of ?r means emptiness of the level n tree language generated by S and the authors show by initial algebra methods [S] that the emptiness problem is solvable. Now we should like to present a quite different solvability proof purely in phrases well Izown to (system-) programmers and compiler constructors. Firstly we observe that we may restrict ourselves to programs of ALGOL 60-type where procedures do not yield procedures as their values. Secondly, in order to study the formal termination problem for ALGOL 60-programs it suffices to investigate the actual termination proble basic statements are dummy statements or procedure statements only and where we have nondeterministic alternative statements instead of conditional statements. We present a contextfree-like production system P for A6OD: