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Term graphs represent functional expressions such that common subexpressions can be shared, making expression evaluation more efficient than with strings or trees. Rewriting of term graphs proceeds by both applications of term rewrite rules and folding steps which enhance the degree of sharing. The present paper introduces critical pairs in term graph rewriting and establishes a Critical Pair Lemma as an analogue to the well-known result in term rewriting. This leads to a decision procedure for confluence in the presence of termination. As a by-product, the procedure can be used as a confluence test for term rewriting and as such it extends the classical test of Knuth and Bendix because it applies to all terminating and to certain non-terminating term rewriting systems.

Critical pairs in term graph rewriting