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. Bekiĉ 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. Auto
This book presents what in our opinion constitutes the basis of the theory of the mu-calculus, considered as an algebraic system rather than a logic. We have wished to present the subject in a unified way, and in a form as general as possible. Therefore, our emphasis is on the generality of the fixed-point notation, and on the connections between mu-calculus, games, and automata, which we also explain in an algebraic way.
This book should be accessible for graduate or advanced undergraduate students both in mathematics and computer science. We have designed this book especially for researchers and students interested in logic in computer science, comuter aided verification, and general aspects of automata theory. We have aimed at gathering in a single place the fundamental results of the theory, that are currently very scattered in the literature, and often hardly accessible for interested readers.
The presentation is self-contained, except for the proof of the Mc-Naughton's Determinization Theorem (see, e.g., . However, we suppose that the reader is already familiar with some basic automata theory and universal algebra. The references, credits, and suggestions for further reading are given at the end of each chapter.
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- © North Holland 2001
- 7th February 2001
- North Holland
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"....the book is a solid, conceptually and mathematically profound, modern treatment of the subject, worth reading for an audience ranging from graduate students to working mathematicians and computer scientist" --Zentralblatt f. Mathematik
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