In this paper, we investigate a problem dual to the unification problem, namely theCommon Term (CT) problem for string rewriting systems. Our main motivation was in computing fixed points in systems, such as loop invariants in programming languages. We show that the fixed point problem is reducible to the common term problem. We also prove that the common term problem is undecidable for dwindli...

Skew confluence was introduced as a characterization of nonconfluent term rewriting systems that had unique infinite normal forms or Böhm like trees. This notion however is not expressive enough to deal with all possible sources of non-confluence in the context of infinite terms or terms extended with letrec. We present a new notion called ωskew confluence which constitutes a sufficient and nec...

A property of term rewriting systems is modular if it is preserved under disjoint union. In the past few years the modularity of properties of term rewriting systems has been extensively studied. The first results in this direction were obtained by Toyama. In (Toyama, 1987a) he showed that confluence is a modular property (see Klop et al. (1994) for a simplified proof) and in (Toyama, 1987b) he...

Usually termination of term rewriting systems (TRS's) is proved by means of a mono-tonic well-founded order. If this order is total on ground terms, the TRS is called totally terminating. In this paper we prove that total termination is an undecidable property of nite term rewriting systems. The proof is given by means of Post's Correspondence Problem.

We address the problem of efficient rewriting and narrowing strategies for general term rewriting systems. Several strategies have been proposed over the last two decades, the most efficient of all being the natural rewriting and narrowing strategies of Escobar. All the strategies so far, including natural rewriting and narrowing, assume that the given term rewriting system is left-linear and c...

An induction method called term rewriting induction is proposed for proving properties of term rewriting systems. It is shown that the Knuth-Bendix completion-based induc-tive proof procedures construct term rewriting induction proofs. It has been widely held heretofore that these procedures construct proofs by consistency, and cannot be justiied as induction methods. Our formulation shows othe...

Simply-typed term rewriting systems (STRSs) are an extension of term rewriting systems. STRSs can be naturally handle higher order functions, which are widely used in existing functional programming languages. In this paper we design recursive and lexicographic path orders, which can efficiently prove the termination of STRSs. Moreover we discuss an application to the dependency pair and the ar...

The formal framework of Logically Constrained Term Rewriting Systems (LCTRSs), introduced in [Kop and Nishida 2013], combines term rewriting with constraints and calculations over an arbitrary theory. This for instance allows users to specify rules with integers, arrays and strings, and can be used to analyze both imperative and functional programs (without higher-order variables) in a natural ...

Our goal is to give a list of rewriting properties, and then automatically find a term rewriting system (TRS) satisfying these properties. In earlier work we did this for finite abstract reduction systems; in this paper we extend the approach to ground term rewriting systems over constants and one unary symbol. In particular, we fully automatically find a TRS that is locally confluent but not c...

We present a formalism called Addressed Term Rewriting Systems, which can be used to define the operational semantics of programming languages, especially those involving sharing, recursive computations and cyclic data structures. Addressed Term Rewriting Systems are therefore well suited for describing object-based languages, as for instance the family of languages called λObja, involving both...