We prove that a group has word problem that is a growing context-sensitive language precisely if its word problem can be solved using a non-deterministic Cannon's algorithm (the deterministic algorithms being defined by Goodman and Shapiro in ). We generalize results of  to find many examples of groups not admitting non-deterministic Cannon's algorithms. This adds to the examples of Kambites and Otto in  of groups separating context-sensitive and growing context-sensitive word problems, and provides a new language-theoretic separation result.
Holt, D. F., Rees, S., & Shapiro, M. (2008). Groups that do and do not have context-sensitive word problem. International Journal of Algebra and Computation, 18(7), 1179-1191. https://doi.org/10.1142/S0218196708004834