The opacity of backbones

This paper uses structural complexity theory to study whether there is a chasm between knowing an object exists and getting one's hands on the object or its properties. In particular, we study the nontransparency of backbones. We show that, under the widely believed assumption that integer factoring is hard, there exist sets of boolean formulas that have obvious, nontrivial backbones yet finding the values of those backbones is intractable. We also show that, under the same assumption, there exist sets of boolean formulas that obviously have large backbones yet producing such a backbone is intractable. Furthermore, we show that if integer factoring is not merely worst-case hard but is frequently hard, as is widely believed, then the frequency of hardness in our two results is not too much less than that frequency. These results hold even if one's assumptions are, respectively, P≠NP∩coNP or that some NP∩coNP problem is frequently hard.