Least upper bounds on the size of confluence and church-rosser diagrams in term rewriting and λ-calculus

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 vs R ( 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 cvs R ( 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 vs R is a total computable function, and for linear such systems that cvs R is a total computable function. Conversely, we show that every total computable function is the lower bound on the functions vs R ( m , n ) and cvs R ( 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 vs R (| s |, n ) ≥ φ( n ) and cvs R ( n , n ) ≥ φ( n ) for all n . For orthogonal TRSs R we prove that there is a constant k such that: (a) vs R ( m , n ) is bounded from above by a function exponential in k and (b) cvs R ( m , n ) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy. Similarly, for λ-calculus, we show that vs R ( m , n ) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy.