Both deterministic and nondeterministic variants of a two-way finite-state accepter are equivalent in power to a one-way deterministic finite-state accepter.
The original proof was given by Rabin and Scott (1959),
who also proposed the idea of two-way accepters.
Shepherdson (1959) subsequently published a simpler proof and showed that an n-state deterministic two-way accepter
can be converted to a deterministic one-way accepter with O((n + 1)n + 1) states.
Recently, Hulden (2015) proposed a simpler conversion regular expression-based strategy for converting a
two-way deterministic finite-state accepter to a deterministic one-way accepter.
Sadoka and Sipser (1978) posed
questions on the cost of simulating nondeterministic finite-state accepters via two-way accepters —
which remain open problems (Pighizzini, 2012):
Cost of simulating a nondeterministic one-way finite-state accepter via a deterministic two-way accepter
Cost of simulating a nondeterministic two-way finite-state accepter via a deterministic two-way accepter