Termination is an important property of term rewriting systems. For a finite terminating rewrite system, a normal form of a given term can be found by a simple depth-first search. If the system is also confluent, the normal forms are unique, which makes the word problem for the corresponding equational theory decidable. Unfortunately, as shown in the first section of this chapter, termination is an undecidable property of term rewriting systems. This is true even if one allows for only unary function symbols in the rules, or for only one rewrite rule (but then for function symbols of arity greater than 1). In the restricted case of ground rewrite systems, i.e. rewrite systems whose rules must not contain variables, termination becomes decidable, though. In the second section of this chapter, we introduce the notion of a reduction order. These orders are an important tool for proving termination of rewrite systems. The main problem for a given rewrite system is to find an appropriate reduction order that shows its termination. Thus, it is desirable to have a wide range of different possible reduction orders available. In the third and fourth sections of the chapter, we introduce two different ways of defining reduction orders.
The decision problem
First, we show undecidability of the termination problem for term rewriting systems, and then we consider the decidable subcase of right-ground term rewriting systems (which can be treated by a slight generalization of the well-known proof for ground systems).