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We present a generalization of standard Turing machines based on allowingunusual tapes. We present a set of reasonable constraints on tape geometry andclassify all tapes conforming to these constraints. Surprisingly, thisgeneralization does not lead to yet another equivalent formulation of thenotion of computable function. Rather, it gives an alternative definition ofthe recursively enumerable Turing degrees that does not rely on oracles. Thedefinitions give rise to a number of questions about computable paths insideCayley graphs of finitely generated groups, and several of these questions areanswered.

Turing Machines on Graphs and Inescapable Groups