We survey one branch of algebraic logic, namely modal semirings. They provide compact algebraic definitions of actions, with choice + and sequential composition . , together with multi-modal operators box and diamond that allow reasoning about successors and predecessors of states/worlds. Particular instances are homogeneous binary relations or sets of finite and infinite non-empty traces under fusing concatenation.

As main examples of applications we present obstacle analysis for geographic wayfinders, Hoare Logic, O’Hearn’s Incorrectness Logic, General Correctness Logic, as well as the temporal logic CTL^* and its sublogics CTL and LTL. We also give glimpses at Epistemic Logics of belief and knowledge, pointer structures and Separation Logic, and preference database queries.