Decision versus search problems in super-polynomial time

The following propositions are considered: (1) E=NE (i.e. it is decidable in exponential time whether there is a solution for an exponential-type search problem). (2) Every exponential-type search problem is solvable in exponential time. (3) The first solution to every exponential-type search problem can be found in exponential time. (4) E=E^NP. It is easy to see that (4) implies (3) implies (2) implies (1). It has been conjectured that the first and last of these assumptions are equivalent in every relativized world. It is proved here that there exist relativized words in which the last two implications are not reversible. This is evidence that the search problem is not reducible to decision problems in exponential time. It is also proved that the third and fourth assumptions are equivalent. The combinatorial core of the separation results is a lower bound on the parallel complexity of a generalized version of the X-search problem