Definable fixed points in modal and temporal logics — a survey

Abstract

The paper presents a survey of author's results on definable fixed points in modal, temporal, and intuitionistic propositional logics. The well-known Fixed Point Theorem considers the modalized case, but here we investigate the positive case. We give a classification of fixed point theorems, describe some classes of models with definable least fixed points of positive operators, special positive operators, and give some examples of undefinable least fixed points. Some other interesting phenomena are discovered – definability by formulas that do not preserve positivity of parameters and definability by finite sets of formulas. We also consider negative operators, graded modalities, construct undefinable inflationary fixed points, and put some problems.