On the Uniquely Converging Property of Nonlinear Term Rewriting Systems

A uniquely converging (UC) property for a possibly nonlinear term rewriting system (TRS) is investigated. UC, which is an intermediate property between conventional Church-Rosser (CR) and uniquely normalizing (UN), is newly proposed in connection with the consistency of continuous semantics. Continuous semantics is de ned by constructing free-continuous algebra which is required in algebraic speci cation on a lazy space. In fact, freecontinuous algebra can specify a lazy space, whereas neither initial algebra nor nal algebra can. This paper also clari es a su cient condition for UC. The statement is, an !-nonoverlapping TRS is UC (irrespective of linearlity). This makes the contrast with the well-known facts that a nonoverlapping TRS is possibly non-UN when nonlinear, although CR when left linear. The di erence between !-nonoverlapping and usual nonoverlapping is that uni cation with in nite terms is applied instead of usual uni cation with occur-check.