A general condition for collapse results

In BST99] the collapse result theorem was proved for locally generic queries over ordered domain with Pseudo-nite Homogene-ity Property. In a very interesting paper of Baldwin and Benedikt BB98] the collapse result theorem was proved for locally generic queries over ordered domains without the independence property. It means that over such a domain, order-generic extended queries fail to express more than restricted queries. It was observed by Baldwin and Benedikt in BB98] that any theory without the independence property is P-reducible. I prove that, for the P-reducible theories, a version of Pseudo-nite Homogeneity Property holds.