Infinite Time Recognizability from Random Oracles and the Recognizable Jump Operator

Abstract

By a theorem of Sacks, if a real x is recursive relative to all elements of a set of positive Lebesgue measure, x is recursive. This statement, and the analogous statement for non-meagerness instead of positive Lebesgue measure, have been shown to carry over to many models of transfinite computations. Here, we start exploring another analogue concerning recognizability rather than computability. We introduce a notion of relativized recognizability and show that, for Infinite Time Turing Machines (ITTMs), if a real x is recognizable relative to all elements of a non-meager Borel set Y, then x is recognizable. We also show that a relativized version of this statement holds for Infinite Time Register Machines (ITRMs). This extends our earlier work where we obtained the (unrelativized) result for ITRMs. We then introduce a jump operator for recognizability, examine its set-theoretical content and show that the recognizable jumps for ITRMs and ITTMs are primitive-recursively equivalent, even though these two models are otherwise of vastly different strength. Finally, we introduce degrees of recognizability by considering the transitive closure of relativized recognizability and connect it with the recognizable jump operator to obtain a solution to Post's problem for degrees of recognizability.