Program-Size Complexity of Initial Segments and Domination Reducibility

A theorem of Solovay states that there is a noncomputable Δ 2 0 real x = (x n ) n such that H(x n ) ⩽ H(n) + O(1). We improve this result by showing there is a noncomputable c.e. real x with the same property. This answers a question concerning the relationship between the domination relation and program-size complexity of initial segments of reals (informally, a real x dominates a real y if from a good approximation of x from below one can compute a good approximation of y from below). Solovay proved that if x and y are two c.e. reals and y dominates x, then H(x n ) ⩽ H(y n ) + O(1). The result above shows that the converse is false, namely there are c.e. reals x and y such that H(x n ) ⩽ H(y n ) + O(1) and y does not dominate x.