Abstract
In this paper, we study modular aspects of hierarchical combinations of term rewriting systems. A combination R 0 ∪ R 1 is hierarchical if the defined symbols of the two subsystems R 0 and R 1 are disjoint, some of the defined symbols of R 0 are constructors in R 1 and the defined symbols of R 1 do not occur in R 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 R 0 ∪ R 1 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.
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