, The equivalence is given by the conjunction of Theorem V.59 and V, vol.58

, Let z ? 2 ? . The following are equivalent: 1. z is generic over L ? , 2. z is ITTM-decidable generic, 3. z is co-ITTM generic

, Let z be a real generic over L ? . Let M be a machine that halts on a co-meager set. By Corollary V.59.1 we have that the set {x ? 2 ? : ? x > ?} is meager. Note also that if z is generic over L ? we have L ? (x) ? 1 L ? (x) together with L ? (x) ? 2 L ? (x). Thus the set {x ? 2 ? : ? x > ?} is actually also meager, The implications (3) ? (2) and (2) ? (1) are trivial. Thus, it remains only to prove

, By admissibility of ?, there must exists ? < ? such that the set {? : ? ? ?? < ? M (x) ? [?]} is already a dense set of strings. It follows that {x ? 2 ? : M (x) ? [?]} is co-meager in a dense open set and thus comeager. Furthermore its complement is a union of nowhere dense closed sets with Borel code in L ?, particular the set {? : ? ? ?? M (x) ? [?]} must be a dense set of strings

, Thus z also is co-ITTM generic

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