Fixed-point logics extend some basic logical formalism (like first-order logic, propositional modal logic, or conjunctive queries) by fixed points of definable relational operators. The most popular fixed point logics (LFP, mu-calculus, and Datalog) are based on least and greatest fixed points of monotone operators. We will compare them with logics based on inflationary and deflationary fixed points (IFP, MIC, etc) that are often more expressive, but algorithmically less manageable.

Typical greatest fixed points are, for instance, bisimilarity and common knowledge; interesting deflationary fixed points arise via iterated relativisations (problem of the lazy engineer) or in dynamic epistemic logics with repeated public announcement. We discuss the question whether inflationary and deflationary fixed points can be eliminated in favour of least and greatest fixed points, and compare the model-theoretic and algorithmic properties of different fixed-point logics.