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We introduce the notion of eventually uniformly weak truth table array computable (eu wtt-ac) sets. As our main result, we show that a computably enumerable (ce) set has this property iff it is weak truth table (wtt-) reducible to a maximal set. Moreover, in this equivalence we may replace maximal sets by quasi-maximal sets, hyperhypersimple sets or dense simple sets and we may replace wtt-reducibility by identity-bounded Turing reducibility (or any intermediate reducibility).