Every Sequence Is Reducible to a Random One

Charles Bennett asked whether every infinite binary sequence can be obtained from an "incompressible" one by a Turing machine, He proved that this is the case for arithmetical sequences. The question has some philosophical interest because it permits us to view even very pathological sequences as the result of the combination of two relatively wellunderstood processes: the completely chaotic outcome of coin-tossing, and a transducer algorithm. A related problem was stated much earlier in Proposition 3 of (Levin, 1976). An interesting generalization of Levin's problem is still open, but its statement would require more definitions.