Abstract
We study confluence and the Church-Rosser property in term rewriting and λ-calculus with explicit bounds on term sizes and reduction lengths. Given a system R, we are interested in the lengths of the reductions in the smallest valleys t → * s′ * ← t′ expressed as a function:
—for confluence a function vsR(m, n) where the valleys are for peaks t ← s → * t′ with s of size at most m and the reductions of maximum length n, and
—for the Church-Rosser property a function cvsR(m,n) where the valleys are for conversions t ↔ * t′ with t and t′ of size at most m and the conversion of maximum length n.
For confluent Term Rewriting Systems (TRSs), we prove that vsR is a total computable function, and for linear such systems that cvsR is a total computable function. Conversely, we show that every total computable function is the lower bound on the functions vsR(m, n) and cvsR(m,n) for some TRS R: In particular, we show that for every total computable function φ: N → N there is a TRS R with a single term s such that vsR(|s|, n) ≥ φ(n) and cvsR(n, n) ≥ φ(n) for all n. For orthogonal TRSs R we prove that there is a constant k such that: (a) vsR(m, n) is bounded from above by a function exponential in k and (b) cvsR(m, n) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy. Similarly, for λ-calculus, we show that vsR(m,n) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy.
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Least upper bounds on the size of confluence and church-rosser diagrams in term rewriting and λ-calculus
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