Learning implied constraints lazily

Explanations are a technique for reasoning about constraint propagation, which have been applied in many learning, backjumping and user interaction algorithms. To date, explanations have been recorded eagerly when the propagation happens, which leads to ine cient use of time and space, because many will never be used. In this paper we show that it is possible and highly e ective to calculate explanations retrospectively when they are needed. To this end, we implement lazy explanations in a state of the art learning framework. Experimental results con rm the e ectiveness of the technique: we achieve reduction in the number of explanations calculated up to a factor of 200 and robust reductions in overall solve time up to a factor of 2.