New Undecidability Results for Properties of Term Rewrite Systems

Information Processing Letters, 2003

Ground reachability, ground joinability and confluence are shown undecidable for flat term rewriting systems, i.e. systems in which all left and right members of rule have depth at most one.

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 .

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.

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 ...

Applicable Algebra in Engineering, Communication and Computing, 2010

Term rewrite systems are useful in many areas of computer science. Two especially important areas are decision procedures for the word problem of some algebraic systems and rule-based programming. One of the most studied properties of rewrite systems is confluence, and one of the primary benefits of having a confluent rewrite system is that the system also has uniqueness of normal forms. However, uniqueness of normal forms is an interesting property in its own right and well studied. Also, confluence can be too strong a requirement for some applications. In this paper, we study the decidability of uniqueness of normal forms. Uniqueness of normal forms is decidable for ground rewrite systems, but is undecidable in general. This paper shows that the uniqueness of normal forms problem is decidable for the class of linear shallow term rewrite systems, and gives a decision procedure that is polynomial as long as the arities of the function symbols are bounded or the signature is fixed. Keywords Term rewrite systems • Uniqueness of normal forms • Basic properties of rewrite systems • Linear, shallow rewrite systems Research supported in part by NSF grants CCF 0306475, DUE 0755500 and DUE 0737404.

Information and Computation, 2005

The theorem of Huet and Lévy stating that for orthogonal rewrite systems (i) every reducible term contains a needed redex and (ii) repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form, forms the basis of all results on optimal normalizing strategies for orthogonal rewrite systems. However, needed redexes are not computable in general. In the paper we show how the use of approximations and elementary tree automata techniques allows one to obtain decidable conditions in a simple and elegant way. Surprisingly, by avoiding complicated concepts like index and sequentiality we are able to cover much larger classes of rewrite systems. We also study modularity aspects of the classes in our hierarchy. It turns out that none of the classes is preserved under signature extension. By imposing various conditions we recover the preservation under signature extension. By imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructor-sharing combinations.

Lecture Notes in Computer Science, 2004

Right-(ground or variable) rewrite systems (RGV systems for short) are term rewrite systems where all right hand sides of rules are restricted to be either ground or a variable. We define a minimal rewrite extension R of the rewrite relation induced by a RGV system R. This extension admits a rewrite closure presentation, which can be effectively constructed from R. The rewrite closure is used to obtain decidability of the reachability, joinability, termination, and confluence properties of the RGV system R. We also show that the word problem and the unification problem are decidable for confluent RGV systems. We analyze the time complexity of the obtained procedures; for shallow RGV systems, termination and confluence are exponential, which is the best possible result since all these problems are EXPTIME-hard for shallow RGV systems.

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.

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 ...

Theoretical Computer Science, 1995

In this paper, we study modular aspects of hierarchical combinations of term rewriting systems. A combination ~0 td ~l is hierarchical if the defined symbols of the two subsystems ~0 and ~l are disjoint, some of the defined symbols of ~0 are constructors in ~l and the defined symbols of ~l do not occur in ~0. It is shown that in hierarchical combinations, a reduction can increase the rank of a term. Therefore, techniques employed in proving the modularity results for direct sums and constructor sharing systems are not applicable for hierarchical combinations. We propose a set of sufficient conditions for the modularity of completeness of hierarchical combinations. The sufficient conditions are syntactic ones (about recursion) and can be easily tested for finite systems. First, the modularity of strong innermost normalization (SIN) for a class of hierarchical combinations is established. By imposing a restriction that ~0 tJ ~l is an overlay system, the modularity of local confluence is established for this class. Then the modularity of completeness is obtained using a recent result relating strong innermost normalization and termination properties of locally confluent overlay systems.