Abstract rewriting machine

The Abstract Rewriting Machine (ARM) is a virtual machine which implements term rewriting for minimal term rewriting systems. Minimal term rewriting systems are left-linear term rewriting systems in which each rule takes on one of six forms:

Continuation: [math]\displaystyle{ f(\vec{x},\vec{y},\vec{z})\rightarrow g(\vec{x},h(\vec{y}),\vec{z}) }[/math]
Return: [math]\displaystyle{ f(x)\rightarrow x }[/math]
Match: [math]\displaystyle{ f(\vec{x},g(\vec{y}),\vec{z})\rightarrow h(\vec{x},\vec{y},\vec{z}) }[/math]
Add: [math]\displaystyle{ f(\vec{x},\vec{z})\rightarrow g(\vec{x},y,\vec{z}) - {\rm for }~y\in\vec{x}\cup\vec{z} }[/math]
Delete: [math]\displaystyle{ f(\vec{x},\vec{y},\vec{z})\rightarrow g(\vec{x},\vec{z}) }[/math]
Ident: [math]\displaystyle{ f(\vec{x})\rightarrow g(\vec{x}) }[/math]

Each of these six forms is mapped (in ARM) to one or a few processor instructions on most contemporary micro processors. Accordingly, minimal term rewriting is achieved at tens to hundreds of clock cycles per reduction step—millions of reduction steps per second.

ARM implements general term rewriting, in that every single-sorted unconditional left-linear term rewriting system can be transformed (compiled) into a minimal term rewriting system that gives rise to the same normal form relation.

An overview with references to this compilation process for innermost rewriting, as well as a detailed overview of ARM, can be found in "Within ARM's reach: compilation of left-linear rewrite systems via minimal rewrite systems". A description for lazy (non-innermost) rewriting can be found in "Lazy rewriting on eager machinery".

A documented implementation of ARM (with the term rewriting language Epic) is available here. Note that site and software are no longer being actively maintained.

References

Giesl, J. R.; Middeldorp, A. (July 2004). "Transformation techniques for context-sensitive rewrite systems". Journal of Functional Programming 14 (4): 379–427. doi:10.1017/S0956796803004945. http://cl-informatik.uibk.ac.at/users/ami/research/publications/journals/04jfp.pdf.
Lucas, Salvador (2002). "Lazy Rewriting and Context-Sensitive Rewriting". Electronic Notes in Theoretical Computer Science 64: 234–254. doi:10.1016/S1571-0661(04)80353-0. http://www.dsic.upv.es/users/elp/berlanga/slucas/entcsV64/entcsV64.pdf. Retrieved 2015-08-29.
Nguyen, Quang-Huy (2001). "Compact Normalisation Trace via Lazy Rewriting". Electronic Notes in Theoretical Computer Science 57: 87–108. doi:10.1016/S1571-0661(04)00269-5. http://hal.inria.fr/docs/00/10/78/74/PDF/A01-R-139.pdf.
Schernhammer, F.; Gramlich, B. (April 2008). "Termination of Lazy Rewriting Revisited". Electronic Notes in Theoretical Computer Science 204: 35–51. doi:10.1016/j.entcs.2008.03.052. http://www.logic.at/staff/gramlich/papers/wrs07-techrep.pdf.
Kirchner, C.; Kirchner, H. (2014). "Equational Logic and Rewriting". Handbook of the History of Logic 9: 255–282. doi:10.1016/B978-0-444-51624-4.50006-X. ISBN 9780444516244. https://hal.inria.fr/hal-01183817/file/eqLogicRw.pdf.
Antoy, S.; Johannsen, J.; Libby, S. (2015). "Needed Computations Shortcutting Needed Steps". Proceedings 8th International Workshop on Computing with Terms and Graphs 183: 18–32. doi:10.4204/EPTCS.183.2.

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