It is shown that it is undecidable in general whether a graph rewriting system (in the "double pushout approach") is terminating. The proof is by a reduction of the Post Correspondence Problem. It is also argued that there is no straightforward reduction of the halting problem for Turing machines or of the termination problem for string rewriting systems to the present problem.

. For a hierarchy of properties of term rewriting systems, related to termination, we prove relative undecidability even in the case of single rewrite rules: for implications X ) Y in the hierarchy the property X is undecidable for rewrite rules satisfying Y . 1 Introduction A fundamental problem in the theory of term rewriting is the detection of termination: for a fixed system of rewrite rules, determine whether there are infinite rewrite sequences. Besides termination a number of related properties are of interest, linearly ordered by implication: polynomial termination ) !-termination ) total termination ) simple termination ) non-self-embeddingness ) termination ) non-loopingness ) acyclicity We call this the termination hierarchy. Apart from polynomial termination, all properties in the termination hierarchy are known to be undecidable ([11, 15, 13, 18, 8, 9]). In [9] we showed the stronger result of relative undecidability : for all implications X ) Y in the termination hier...

The first undecidability results in rewriting were proved by reduction to undecidable problems of Turing machines [7]. More recently Post's Correspondence Problem [19] has been used (e.g. [14]). The most natural candidate for proofs regarding term rewriting, however, is Recursion Theory, a direction we promote in this paper. We present some undecidability results for "primitive" term rewriting systems, which encode primitive-recursive definitions. We also show how to faithfully encode partial recursive functions. Then we reprove and improve some undecidability results for orthogonal and non-orthogonal rewriting by applying standard results in recursion theory. 1 Introduction A number of models of computation compete for the role of most "basic". These include: semi-Thue systems, Markov algorithms, lambda calculus, combinatory logic, Turing machines, and recursive functions. First-order term-rewriting allows for a very natural programming paradigm based on subterm replacemen...