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Least Upper Bounds on the Size of ChurchRosser Diagrams in Term Rewriting and λCalculus
"... Abstract. We study the ChurchRosser property—which is also known as confluence—in term rewriting and λcalculus. Given a system R and a peak t ∗ ← s → ∗ t ′ in R, we are interested in the length of the reductions in the smallest corresponding valley t → ∗ s ′ ∗ ← t ′ as a function vsR(m, n) of ..."
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Abstract. We study the ChurchRosser property—which is also known as confluence—in term rewriting and λcalculus. Given a system R and a peak t ∗ ← s → ∗ t ′ in R, we are interested in the length of the reductions in the smallest corresponding valley t → ∗ s ′ ∗ ← t ′ as a function vsR(m, n) of the size m of s and the maximum length n of the reductions in the peak. For confluent term rewriting systems (TRSs), we prove the (expected) result that vsR(m, n) is a computable function. Conversely, for every total computable function ϕ(n) there is a TRS with a single term s such that vsR(s, n) ≥ ϕ(n) for all n. In contrast, for orthogonal term rewriting systems R we prove that there is a constant k such that vsR(m, n) is bounded from above by a function exponential in k and independent of the size of s. For λcalculus, we show that vsR(m, n) is bounded from above by a function contained in the fourth level of the Grzegorczyk hierarchy. 1