RUDIMENTS OF μ-CALCULUS

Complete lattices and fixed-point theorems.Complete lattices. Least upper bounds and greatest lower bounds. Complete lattices. Some algebraic properties of lattices.Symmetry in lattices. Sublattices. Boolean algebras. Products of lattices. Functional lattices.Fixed-point theorems. Monotonic and continuous mappings. Fixed points of a function.Nested fixed points. Duality. Some properties of fixed points. Fixed points on product lattices. The replacement lemma. Bekic principle. Gauss elimination.Systems of equations. Conway identities. Bibliographical notes and sources. The mu-calculi: Syntax and semantics. mu-calculi. Functional mu-calculi. Fixed-point terms. Syntax. The mu-calculus of fixed-point terms.Semantics. Quotient mu-calculi. Families of interpretations. Variants of fixed-point terms. Powerset interpretations. Alternation-depth hierarchy. Clones in a mu-calculus. A hierarchy of clones. The syntactic hierarchy. The Emerson-Lei hierarchy. Vectorial mu-calculi. Vectorial extension of a mu-calculus. Interpretations of vectorial fixed point terms. The vectorial hierarchy. Vectorial fixed-point terms in normal form.Bibliographic notes and sources. The Boolean mu-calculus. Monotone Boolean functions. Powerset interpretations and the Boolean mu-calculus. The selection property. Finite vectorial fixed-point terms.Infinite vectors of fixed-point terms. Infinite vectors of infinite fixed-point terms. Bibliographic notes and sources. Parity games. Games and strategies. Positional strategies. The mu-calculus of games. Boolean terms for games. Game terms. Games for the mu-calculus. Games for Boolean terms. Games for powerset interpretations. Weak parity games. Bibliographical notes and sources. The mu-calculus on words. Rational languages. Preliminary definitions. Rational languages. Arden's lemma. The mu-calculus of extended languages. Nondeterministic automata. Automata on finite words. Automata on infinite words. Parity automata.