Simplification Orders for Term Graph Rewriting

Term graph rewriting differs from term rewriting in that common subexpressions can be shared, improving the efficiency of rewriting in space and time. Moreover, computations by term graph rewriting terminate more often than computations by term rewriting. In this paper, simplification orders on term graphs are introduced as a means for proving termination of term graph rewriting. Simplification orders are based on an extension of the homeomorphic embedding relation from trees to term graphs. By generalizing Kruskal's Tree Theorem to term graphs, it is shown that simplification orders are well-founded. Then a recursive path order on term graphs is defined by analogy with the well-known order on terms, and is shown to be a simplification order. Examples of termination proofs with the recursive path order are given for rewrite systems that are non-terminating under term rewriting.

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