Termination of Linear Bounded Term Rewriting Systems

Electronic Notes in Theoretical Computer Science, 2012

This paper is on several basic properties of term rewrite systems: reachability, joinability, uniqueness of normal forms, unique normalization, confluence, and existence of normal forms, for subclasses of rewrite systems defined by syntactic restrictions on variables. All these properties are known to be undecidable for the general class and decidable for ground (variable-free) systems. Recently, there has been impressive progress on efficient algorithms or decidability results for many of these properties. The aim of this paper is to present new results and organize existing ones to clarify further the boundary between decidability and undecidability for these properties. Another goal is to spur research towards a complete classification of these properties for subclasses defined by syntactic restrictions on variables. The proofs of the presented results may be intrinsically interesting as well due to their economy, which is partly based on improved reductions between some of the properties.

Lecture Notes in Computer Science, 1989

The direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols. It is shown that two term rewriting systems both are left-linear and complete if and only if the direct sum of these systems is so.

IEICE Transactions on Information and Systems, 2005

This paper explores how to extend the dependency pair technique for proving termination of higher-order rewrite systems. In the first order case, the termination of term rewriting systems are proved by showing the non-existence of an infinite R-chain of the dependency pairs. However, the termination and the non-existence of an infinite R-chain do not coincide in the higher-order case. We introduce a new notion of dependency forest that characterize infinite reductions and infinite R-chains, and show that the termination property of higher-order rewrite systems R can be checked by showing the non-existence of an infinite R-chain, if R is strongly linear or non-nested.

Journal of Symbolic Computation - JSC, 1995

Usually termination of term rewriting systems (TRS's) is proved bymeans of a monotonic well-founded order. If this order is total on groundterms, the TRS is called totally terminating. In this paper we prove thattotal termination is an undecidable property of finite term rewriting systems.The proof is given by means of Post's Correspondence Problem.1 IntroductionTermination of term rewriting systems (TRS's) is an important property. Oftentermination proofs are given by defining an order ...

Lecture Notes in Computer Science, 2005

We show that confluence of shallow and right-linear term rewriting systems is decidable. This class of rewriting system is expressive enough to include nontrivial nonground rules such as commutativity, identity, and idempotence. Our proof uses the fact that this class of rewrite systems is known to be regularity-preserving, which implies that its reachability and joinability problems are decidable. The new decidability result is obtained by building upon our prior work for the class of ground term rewriting systems and shallow linear term rewriting systems. The proof relies on the concept of extracting more general rewrite derivations from a given rewrite derivation.

2016

The ground term reachability problem consists in determining whether a given variable-free term t can be transformed into a given variable-free term t' by the application of rules from a term rewriting system R. The joinability problem, on the other hand, consists in determining whether there exists a variable-free term t'' which is reachable both from t and from t'. Both problems have proven to be of fundamental importance for several subfields of computer science. Nevertheless, these problems are undecidable even when restricted to linear term rewriting systems. In this work, we approach reachability and joinability in linear term rewriting systems from the perspective of parameterized complexity theory, and show that these problems are fixed parameter tractable with respect to the depth of derivations. More precisely, we consider a notion of parallel rewriting, in which an unbounded number of rules can be applied simultaneously to a term as long as these rules do ...

2013

Well-foundedness is related to an important property of rewriting systems, namely termination. A well-known technique to prove well-foundedness on term orderings is Kruskal's theorem, which implies that a monotonic term ordering over a finite signature satisfying the subterm property is well-founded. However, it does not seem to work for a number of terminating term rewriting systems. In this paper, it is shown that a term ordering possessing subterm property and decomposability ( in place of monotonicity) does yield a simpler proof of well-foundedness for terminating term rewriting systems than the techniques depending on Kruskal's theorem.

IPSJ Online Transactions, 2009

The present paper discusses a head-needed strategy and its decidable classes of higher-order rewrite systems (HRSs), which is an extension of the headneeded strategy of term rewriting systems (TRSs). We discuss strong sequential and NV-sequential classes having the following three properties, which are mandatory for practical use: (1) the strategy reducing a head-needed redex is head normalizing (2) whether a redex is head-needed is decidable, and (3) whether an HRS belongs to the class is decidable. The main difficulty in realizing (1) is caused by the β-reductions induced from the higher-order reductions. Since β-reduction changes the structure of higher-order terms, the definition of descendants for HRSs becomes complicated. In order to overcome this difficulty, we introduce a function, PV, to follow occurrences moved by βreductions. We present a concrete definition of descendants for HRSs by using PV and then show property (1) for orthogonal systems. We also show properties (2) and (3) using tree automata techniques, a ground tree transducer (GTT), and recognizability of redexes.

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.