Completeness of Combinations of Conditional Constructor Systems

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 refuted the modularity of strong normalisation. His counterexample inspired many researchers to look for conditions which are sufficient to recover the modularity of strong normalisation (e.g. Rusinowitch (1987), Middeldorp (1989), Kurihara and Ohuchi (1990), and Toyama et al. (1989).) Recently Gramlich (1992a) proved an interesting theorem which generalises the results of Rusinowitch (1987), Middeldorp (1989), and Kurihara and Ohuchi (1990), in case of finitely branching term rewriting systems. Ohlebusch (1993b) extended Gramlich’s result to arbitrary term rewriting systems. Another recent contribution to the topic of modularity is the work of Caron (1992) who investigates the decidability problem of reachability from a modularity perspective. The disjointness requirement limits the practical applicability of the results mentioned above. The results of Rusinowitch (1987), Middeldorp (1989), and Kurihara and Ohuchi (1990) were generalised to combinations of term rewriting systems that possibly share constructors—function symbols which do not occur at the leftmost position in left-hand