Positional simulation of two-way automata: proof of a conjecture of R. Kannan and generalizations

R. Kannan conjectured that every non-deterministic two-way finite automaton can be positionally simulated by a deterministic two-way finite automaton. The conjecture is proved here by reduction to a similar problem about finite monoids. The method and the result are then generalized to alternating two-way finite automata, certain alternating one-tape linear-time Turing machines, and one-pebble automata.