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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. . Modal logic in Russia Let us first recall some standard definitions. Modal propositional formulas are constructed from propositional variables p, q, r . . . and the constant ⊥ (falsity) using the binary connectives ∧, ∨, and the unary connectives ¬ and . We introduce the following abbreviations:A modal Kripke frame W, R consists of a non-empty set W with a binary relation R. A Kripke model based on a frame W, R is a triple W, R, v , where v is a valuation function assigning a subset of W (the value) to every propositional variable. This function is extended to formulas in the usual way as follows: the value of the constant ⊥ is always the empty set; the connectives ¬, ∧, ∨ respectively correspond to the complement, the intersection, and the union; and ♦ correspond to the following operations on sets:A = {x | ∀y (xRy ⇒ y ∈ A)}, ♦A = {x | ∃y (xRy ∧ y ∈ A)}.

The concepts of a E-formula and E-scheme are used in the predicate calculus [1]. In this paper we transfer them to the modal propositional logic, which holds a position between the classical propositional and predicate logics. In our case, the use of schemes considerably, but not completely, simplifies the situation as compared to that in the predicate logic setting. We will prove that a scheme over a transitive Kripke model arrives at its least fixed point in finitely many steps, and that this point is definable by suitable formulas. Other theorems on fixed points can be found in [2].

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