Open Access

Bernhard Möller

• Algebraic structures, such as modal idempotent semirings or Kleene algebras, offer a large variety of applications, while requiring only a small set of operators and axioms. Such algebras abstractly capture so-called Kripke structures, i.e., access relations over a set of worlds or states. In addition they provide the associated multi-modal operators box and diamond that allow reasoning, e.g., about possible actions of agents in a system or about state transitions in general. Particular instances of modal semirings are provided by the algebra of homogeneous binary relations and by abstract relation algebras. This setting allows many general considerations and results, ranging from epistemic logics with knowledge and belief to propositional dynamic Hoare logic and resource-based settings such as separation logic. Moreover, many further applications are covered, like abstract reasoning about bisimulations for model refinement, formal concept analysis, simple and concise correctness proofsAlgebraic structures, such as modal idempotent semirings or Kleene algebras, offer a large variety of applications, while requiring only a small set of operators and axioms. Such algebras abstractly capture so-called Kripke structures, i.e., access relations over a set of worlds or states. In addition they provide the associated multi-modal operators box and diamond that allow reasoning, e.g., about possible actions of agents in a system or about state transitions in general. Particular instances of modal semirings are provided by the algebra of homogeneous binary relations and by abstract relation algebras. This setting allows many general considerations and results, ranging from epistemic logics with knowledge and belief to propositional dynamic Hoare logic and resource-based settings such as separation logic. Moreover, many further applications are covered, like abstract reasoning about bisimulations for model refinement, formal concept analysis, simple and concise correctness proofs for the optimisation of database preference queries , Petri nets or generally applicable models of module hierarchies in a feature oriented software development process. A large collection of such examples is treated in the forthcoming book Modal semirings and applications by the two authors, of which this report presents the first three chapters with the basic algebraic definitions and essential theorems about them. It serves as a reference for the current state of the theory.…