The operator gap

This paper continues investigations pertaining to recursive bounds on computing resources (such as time or memory) and the amount by which these bounds must be increased if new computations are to occur within the new bound. The paper proves that no recursive operator can increase every recursive bound enough to reach new computations. In other words, given any general recursive operator F[ ], there is an arbitarily large recursive t( ) such that between bound t( ) and bound F[t( )]( ) there is a gap in which no new computation runs. This demonstrates that the gap phenomenon first discovered by Borodin for composition is a deeply intrinsic property of computational complexity measures. Moreover, the Operator Gap Theorem proved here is shown to be the strongest possible gap theorem for general recursive operators. The proof involves a priority argument but is sufficiently self-contained that it can easily be read by a wide audience. The paper also discusses interesting connections between the Operator Gap Theorem and McCreight & Meyer's important result that every complexity class can be named by a function from a measured set.