Computation of recursive functionals using minimal initial segments

The following problem in the computation of partial recursive functionals is considered: Minimizing the length of initial segments of input functions containing all function values requested by a machine computing a partial recursive functional. A recursive functional F is constructed such that any algorithm for F has unbounded redundancy, i.e. it requests function values on inputs unboundedly larger than those on which the output of F depends. However, for any recursive functional F such that the length of the segment on which F depends is itself a recursive functional, a non-redundant machine for F can be effectively constructed. Also considered are machines on 0-1 sequences for which it is shown that a machine realizing a given level of significance in a universal test of randomness must have unbounded redundancy.