Well-foundedness for Termination of Term Rewriting Systems

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.

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.

Journal of Symbolic Computation, 1994

We investigate proving termination of term rewriting systems by interpretation of terms in a well-founded monotone algebra. The well-known polynomial interpretations can be considered as a particular case in this framework. A classi cation of types of termination, including simple termination, is proposed based on properties in the semantic level. A transformation on term rewriting systems eliminating distributive rules is introduced. Using this distribution elimination a new termination proof of the system SUBST of Hardin and Laville (1986) is given. This system describes explicit substitution incalculus. Another tool for proving termination is based on introduction and removal of type restrictions. A property of many-sorted term rewriting systems is called persistent if it is not a ected by removing the corresponding typing restriction. Persistence turns out to be a generalization of direct sum modularity, but is more powerful for both proving con uence and termination. Termination is proved to be persistent for the class of term rewriting systems for which not both duplicating rules and collapsing rules occur, generalizing a similar result of Rusinowitch for modularity. This result has nice applications, in particular in undecidability proofs.

Yi and Sakai [7] showed that the termination problem is decidable for a daae of semiconstructor term rewriting systems, which is a $supercl\epsilon\ s$ of the class of right ground term rewriting systems. The decidability w&8 $8hown$ by the fact that every non-terminating TRS in the cl&ae has aloop. In this paper we modify the $pr\infty f$ of [7] to show that innermost termination is decidable for the class of semi-constnictor TRSs.

IEICE Transactions on Information and Systems, 2010

In this paper, we show that the termination and the innermost termination properties are decidable for the class of term rewriting systems (TRSs for short) all of whose dependency pairs are right-linear and right-shallow. We also show that the innermost termination is decidable for the class of TRSs all of whose dependency pairs are shallow. The key observation common to these two classes is as follows: for every TRS in the class, we can construct, by using the dependency-pairs information, a finite set of terms such that if the TRS is non-terminating then there is a looping sequence beginning with a term in the finite set. This fact is obtained by modifying the analysis of argument propagation in shallow dependency pairs proposed by Wang and Sakai in 2006. However we gained a great benefit that the resulted procedures do not require any decision procedure of reachability problem used in Wang's procedure for shallow case, because known decidable classes of reachability problem are not larger than classes discussing in this paper.

Theoretical Computer Science, 1995

It is shown that a termination proof for a term rewriting system using a lexicographic path ordering yields a multiply recursive bound on the length of derivations, measured in the depth of the starting term. This result is essentially optimal since for every multiply recursive function f a rewrite system (which reduces under the lexicographic path ordering) can be found such that its derivation length cannot be bounded by f.

2004

We carry out a detailed analysis of Thatte's transformation of term rewriting systems. We refute an earlier claim that this transformation preserves confluence for weakly persistent systems. We prove the preservation of weak normalization, and of confluence in weakly normalizing systems and in nonoverlapping systems with linear subtemplates. We conclude by proving that weak persistence is an undecidable property of term rewriting systems.

Journal of Symbolic Computation, 1994

For the whole class of linear term rewriting systems and for each integer k, we define k-bounded rewriting as a restriction of the usual notion of rewriting. We show that the k-bounded uniform termination, the k-bounded termination, the inverse k-bounded uniform, and the inverse k-bounded problems are decidable. The k-bounded class (BO(k)) is, by definition, the set of linear systems for which every derivation can be replaced by a k-bounded derivation. In general, for BO(k) systems, the uniform (respectively inverse uniform) k-bounded termination problem is not equivalent to the uniform (resp. inverse uniform) termination problem, and the k-bounded (respectively inverse k-bounded) termination problem is not equivalent to the termination (respectively inverse termination) problem. This leads us to define more restricted classes for which these problems are equivalent: the classes BOLP(k) of k-bounded systems that have the length preservation property. By definition, a system is BOLP(k) if every derivation of length n can be replaced by a k-bounded derivation of length n. We define the class BOLP of bounded systems that have the length preservation property as the union of all the BOLP(k) classes. The class BOLP contains (strictly) several already known classes of systems: the inverse left-basic semi-Thue systems, the linear growing term rewriting systems, the inverse Linear-Finite-Path-Ordering systems, the strongly bottom-up systems.

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 ...