Total Termination of Term Rewriting is Undecidable

Mathematical Foundations of …, 1985

A new partial ordering scheme for proving uniform termination of term rewriting systems is presented. The basic idea is that two terms are compared by comparing the paths through them. It is shown that the ordering is a well-founded simplification ordering and also a ...

Information and Computation, 1990

Lecture Notes in Computer Science, 1981

i. Introduction 2. Combinatory Logic Rewriting Systems and preliminary definitions. 3. The Non-Ascending Property as a sufficient condition for strong termination 4. Comparing the Non-Ascending Property, Recursive Path Orderings and Simplification Orderings.

Information and Computation, 2002

For two hierarchies of properties of term rewriting systems related to con uence and termination, respectively, we prove relative undecidability : for implications X ) Y in the hierarchies the property X is undecidable for term rewriting systems satisfying Y .

Foundations of Software Science and Computational …, 2005

Term rewriting systems provide a versatile model of computation. An important property which allows to abstract from potential nondeterminism of parallel execution of the modelled program is confluence. In this paper we prove that confluence of a fairly large class of systems, namely right ground term rewriting systems, is decidable. We introduce a labelling of variables with colours and constrain substitutions according to these colours. We show how right ground rewriting systems can be reduced to simple systems with coloured variables. Such systems can be analysed using reduction-automata techniques which leads to an interesting decision procedure for confluence.

1993

This paper deals with termination proofs for Higher-Order Rewrite Systems (HRSs), introduced in [Nip9l, Nip93]. This formalism combines the computational aspects of term rewriting and simply typed lambda calculus. Our result is a proof technique for the termination of a HRS, similar to the proof technique "Termination by interpretation in a well-founded monotone algebra" described in [Zan93]. The resulting technique is as follows: Choose a higher-order algebra with operations for each function symbol in the HRS, equipped with some well-founded partial ordering. The operations must be strictly monotonic in this ordering. This choice generates a model for the HRS. If the choice can be made in such a way that for each rule and for each valuation of the free variables in that rule the value of the left hand side is greater than the value of the right hand side, then the HRS is terminating. At the end of the paper two applications of this technique are given, which show that this technique is natural and can easily be applied. A nice characterisation of termination is given in [Zan93]. The function symbols of a TRS 1Z have to be interpreted as strictly monotonic operations in some well-founded algebra. This interpretation is extended to closed terms as a usual algebraic homomorphism. Now the associated rewrite relation is terminating if every left hand side is greater (under the chosen interpretation) than the belonging right hand side, for each possible interpretation of the variables in that rule. The strength of this characterisation is that one can concentrate on the "intuitive reason" for termination. This intuition can be translated in suited operations on well-founded orderings, thus using semantical arguments. The real termination proof consists of testing a simple condition on the rules only instead of on all possible rewrite steps or all possible redexes. This semantical approach is more convenient than a syntactical technique. The aim of this paper is to generalise this semantical characterisation of termination for TRSs to one for HRSs. We use an extension of the definition of an HRS in [Nip93] because we do not need the restrictions of the formalism (for instance, that the rules should be of base type and the left hand sides should be patterns). Our result is also applicable to the HRSs of the definition in [Nip9l, Nip93]. The main result is that such a generalisation is possible. The interpretation of terms can be extended to the interpretation of higher-order terms. The orderings and the notion of strictness can also be generalised. The techniques to achieve this are similar to those used in [Gan8O, dV87]. Moreover, the result that termination proofs can be given with a well-founded monotone algebra in [Zan93] carries over to HRSs with simple conditions on the well-founded ordering. With this technique some natural HRSs are proved to be terminating (see Section 7.

Information Processing Letters, 1985

The self-embedding property of term rewriting systems is closely related to the uniform termination property, since a nonself-embedding term rewriting system is uniform terminating. The self-embedding property is shown to be undecidable and partially decidable. It follows that the nonself-embedding property is not partially decidable. This is true even for globally finite term rewriting systems. The same construction gives an easy alternate proof that uniform termination is undecidable in general and also for globally finite term rewriting systems. Also, the looping property is shown to be undecidable in the same way.

Applicable Algebra in Engineering, Communication and Computing, 2005

, a new size-change principle was proposed to verify termination of functional programs automatically. We extend this principle in order to prove termination and innermost termination of arbitrary term rewrite systems (TRSs). Moreover, we compare this approach with existing techniques for termination analysis of TRSs (such as recursive path orders or dependency pairs). It turns out that the size-change principle on its own fails for many examples that can be handled by standard techniques for rewriting, but there are also TRSs where it succeeds whereas existing rewriting techniques fail. Moreover, we also compare the complexity of the respective methods. To this end, we develop the first complexity analysis for the dependency pair approach. While the size-change principle is PSPACE-complete, we prove that the dependency pair approach (in combination with classical path orders) is only NP-complete. To benefit from their respective advantages, we show how to combine the size-change principle with classical orders and with dependency pairs. In this way, we obtain a new approach for automated termination proofs of TRSs which is more powerful than previous approaches. We also show that the combination with dependency pairs does not increase the complexity of the size-change principle, i.e., the combined approach is still PSPACE-complete.

Theoretical Computer Science, 1997

In this paper we investigate the concept of simple termination. A term rewriting system is called simply terminating if its termination can be proved by means of a simplification order. The basic ingredient of a simplification order is the subterm property, but in the literature two different definitions are given: one based on (strict) partial orders and another one based on preorders (or quasi-orders). We argue that there is no reason to choose the second one, while the first one has certain advantages.

Information and Computation, 1990

It is undecidable whether or not a finite Noetherian term rewriting system is ground-confluent. This undecidability result holds even when systems involving only unary function symbols and one constant are being considered, or when left-linear or right-linear ...