Two heads are better than two tapes

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We show that a Turing machine with two single-head one-dimensional tapes cannot recognize the set x2x'| x ∈ 0,1* and x' is a prefix of x in real time, although it can do so with three tapes, two two-dimensional tapes, or one two-head tape, or in linear time with just one tape. In particular, this settles the longstanding conjecture that a two-head Turing machine can recognize more languages in real time if its heads are on the same one-dimensional tape than if they are on separate one-dimensional tapes.