Termination of Computational Restrictions of Rewriting and Termination of Programs?

Termination of rewriting [Der87,Zan03] is often proposed as a suitable theory for proving termination of programs which are executed by rewriting. Programming languages and systems whose operational principle is based on reduction (e.g., functional, algebraic, and equational programming languages as well as theorem provers based on rewriting techniques) need, however, to break down the non-determinism which is inherent to reduction relations to make computations feasible. This is usually done by means of some reduction strategy, i.e., a concrete rule to specify the (non-empty set of) reduction steps which can be issued on any term which is not a normal form. Thus, termination of a program R (where R is a TRS) can be more precisely defined as the termination of the strategy S which is used to execute R . Here, by termination of a strategy S for a TRS R , we mean the termination of the reduction relation →S⊆→ associated to S. Traditionally, the most important question about a rewriting strategy S is whether it is normalizing, i.e., no infinite S-sequence t →S t ′→S · · · starts from a term t having a normal form. Obviously, every rewriting strategy S is forced to run forever when faced to terms t having no normal form. Then, the following property is obvious.