A Purely Logical Approach to Program Termination

We present our work in progress concerning the logical foundations of the analysis of termination for imperative recursive programs. The analysis is based on forward symbolic execution [12] and functional semantics. The distinctive feature of our approach is the formulation of the termination condition as an induction principle developed from the structure of the program with respect to iterative structures (recursive calls and while loops). Moreover the termination condition insures the existence and the uniqueness of the function implemented by the program. Note that the existence is not automatic, because a recursive program corresponds, logically, to an implicit definition. It is interesting that this inductive termination condition can be also used for proving the uniqueness of the function as well as the total correctness of the program. We show in this paper how to prove the existence of the implemented function in the case of single recursion programs (programs with at most one recursive call on each branch). The method can be applied however to all imperative recursive programs, where recursive calls are outside the loops. For other programs, termination analysis appears to involve co-recursive functions and it is subject to further investigation. The methods presented here are under implementation in the Theorema system [2].