Discovering complex and incomplete periodic patterns in the logs of events is a complicated and time consuming task. This work shows that it is possible to discover complex and incomplete periodic patterns through finding simple patterns first and through logical derivations of complex and incomplete patterns later on. The paper defines a syntax and semantics of a class of periodic patterns that frequently occur in the logs of events. A system of derivation rules proposed in the paper can be used to transform a set of periodic patterns into a logically equivalent set of patterns. The rules are used in the algorithms that derive complex and incomplete periodic patterns. A prototype implementation of the algorithms that discover complex and incomplete periodic patterns in the logs of events is presented.