In this paper, we begin with an approach to non-termination inference of logic programs. Our framework relies on an extension of the Lifting Theorem, where some specific argument positions can be instantiated while others are generalized. Atomic left looping queries are generated bottom-up from selected subsets of the binary unfoldings of the program of interest. Then non-termination inference is tailored to attempt proofs of optimality of left termination conditions computed by a termination inference tool. For each class of atomic queries not covered by a termination condition, the aim is to ensure the existence of one query from this class which leads to an infinite search tree. An experimental evaluation is reported. When termination and non-termination analysis produce complementary results for a logic procedure, they induce a characterization of the operational behavior of the logic procedure with respect to the left most selection rule and the language used to describe sets of atomic queries.