A recent direction within belief revision theory is to develop a theory of belief change for the Horn knowledge representation framework. We consider questions related to the complexity aspects of previous work, leading to questions about Horn envelopes (or Horn LUB's), introduced earlier in the context of knowledge compilation. A characterization is obtained of the remainders of a Horn belief set with respect to a consequence to be contracted, as the Horn envelopes of the belief set and an elementary conjunction corresponding to a truth assignment satisfying a certain body building formula. This gives an efficient algorithm to generate all remainders, each represented by a truth assignment. On the negative side, examples are given of Horn belief sets and consequences where Horn formulas representing the result of most contraction operators, based either on remainders or on weak remainders, must have exponential size.