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We present a new method for proving termination of term rewriting systems automatically. It is a generalization of the match bound method for string rewriting. To prove that a term rewriting system terminates on a given regular language of terms, we first construct an enriched system over a new signature that simulates the original derivations. The enriched system is an infinite system over an infinite signature, but it is locally terminating: every restriction of the enriched system to a finite signature is terminating. We then construct iteratively a finite tree automaton that accepts the enriched given regular language and is closed under rewriting modulo the enriched system. If this procedure stops, then the enriched system is compact: every enriched derivation involves only a finite signature. Therefore, the original system terminates. We present three methods to construct the enrichment: top heights, roof heights, and match heights. Top and roof heights work for left-linear systems, while match heights give a powerful method for linear systems. For linear systems, the method is strengthened further by a forward closure construction. Using these methods, we give examples for automated termination proofs that cannot be obtained by standard methods.

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We provide a critical assessment of the current set of benchmarks for relative SRS termination in the Termination Problems Database (TPDB): most of the benchmarks in Waldmann_19 and ICFP_10_relative are, in fact, strictly terminating (i. e., terminating when non-strict rules are considered strict), so these benchmarks should be removed, or relabelled. To fill this gap, we enumerate small relative string rewrite systems. At present, we have complete enumerations for a 2-letter alphabet up to size 11, and for a 3-letter alphabet up to size 8. For some selected benchmarks, old and new, we discuss how to prove termination, automated or not.

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