Reduction Strategies for Left-Linear Term Rewriting Systems [chapter]

Yoshihito Toyama
2005 Lecture Notes in Computer Science  
Huet and Lévy (1979) showed that needed reduction is a normalizing strategy for orthogonal (i.e., left-linear and non-overlapping) term rewriting systems. In order to obtain a decidable needed reduction strategy, they proposed the notion of strongly sequential approximation. Extending their seminal work, several better decidable approximations of left-linear term rewriting systems, for example, NV approximation, shallow approximation, growing approximation, etc., have been investigated in the
more » ... terature. In all of these works, orthogonality is required to guarantee approximated decidable needed reductions are actually normalizing strategies. This paper extends these decidable normalizing strategies to left-linear overlapping term rewriting systems. The key idea is the balanced weak Church-Rosser property. We prove that approximated external reduction is a computable normalizing strategy for the class of left-linear term rewriting systems in which every critical pair can be joined with root balanced reductions. This class includes all weakly orthogonal left-normal systems, for example, combinatory logic CL with the overlapping rules pred · (succ · x) → x and succ · (pred · x) → x, for which leftmost-outermost reduction is a computable normalizing strategy. A part of this paper was published as preliminary version in [24] .
doi:10.1007/11601548_13 fatcat:zhejvqtolrdcrcs72xobfnhjs4