Rudiments of &mgr;calculus
Edited by A. Arnold
 D. Niwinski
This book presents what in our opinion constitutes the basis of the theory of the mucalculus, 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 fixedpoint notation, and on the connections between mucalculus, 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 selfcontained, except for the proof of the McNaughton's Determinization Theorem (see, e.g., [97]. 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.
Studies in Logic and the Foundations of Mathematics
Hardbound, 298 Pages
Published: February 2001
Imprint: Northholland
ISBN: 9780444506207
Reviews

....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
Valentin F. Goranko, Zentralblatt f. Mathematik
Contents
 Complete lattices and fixedpoint 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.Fixedpoint 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 mucalculi: Syntax and semantics. mucalculi. Functional mucalculi. Fixedpoint terms. Syntax. The mucalculus of fixedpoint terms.Semantics. Quotient mucalculi. Families of interpretations. Variants of fixedpoint terms. Powerset interpretations. Alternationdepth hierarchy. Clones in a mucalculus. A hierarchy of clones. The syntactic hierarchy. The EmersonLei hierarchy. Vectorial mucalculi. Vectorial extension of a mucalculus. Interpretations of vectorial fixed point terms. The vectorial hierarchy. Vectorial fixedpoint terms in normal form.Bibliographic notes and sources. The Boolean mucalculus. Monotone Boolean functions. Powerset interpretations and the Boolean mucalculus. The selection property. Finite vectorial fixedpoint terms.Infinite vectors of fixedpoint terms. Infinite vectors of infinite fixedpoint terms. Bibliographic notes and sources. Parity games. Games and strategies. Positional strategies. The mucalculus of games. Boolean terms for games. Game terms. Games for the mucalculus. Games for Boolean terms. Games for powerset interpretations. Weak parity games. Bibliographical notes and sources. The mucalculus on words. Rational languages. Preliminary definitions. Rational languages. Arden's lemma. The mucalculus of extended languages. Nondeterministic automata. Automata on finite words. Automata on infinite words. Parity automata. Recognizable languages. McNaughton determination theorem. Terms with intersection. The mucalculus of constrained languages. Duality. Interpretation of terms by constrained languages. Recognizable constrained languages. Bibliographical notes and sources. The mucalculus over powerset algebras. Powerset algebras. Semialgebras. Powerset algebras. Logical operations. Modal mucalculus. Homomorphisms, muhomomorphisms and bisimulations. Bibliographic notes and sources. The mucalculus versus automata. Automata over semialgebras. Informal description. Basic definitions. Hierarchy of indices. Dual automata. Relation to classical automata. Automata in the mucalculus perspective. The mucalculus of automata. The interpretations as homomorphism. Equivalences. From fixed point terms to automata. From automata to fxedpoint terms. Conclusion. Bibliographic notes and sources. Hierarchy problems. Introduction. The hierarchy of alternating parity tree automata. Games on the binary tree. Alternating automata on trees. A diagonal argument. The hierarchy of Mostowski indices for tree languages. Universal languages. Weak alternating automata. Bibliographical notes and sources. Distributivity and normal form results. The propositional mucalculus. Guarded terms. Introductory example. Guarded terms. Intersectionfree terms. Definition. A syntactic notion of intersection. The powerset construction. The vmu case. The simulation theorem. McNaughton's Theorem revisited. The simulation theorem. The Rabin complementation lemma. Bibliographic notes and sources. Decision problems. Disjunctive mappings.Decidability of emptiness for disjunctive muterms. Compact lattices. Disjunctiveness of fixedpoints. Emptiness of nondeterministic tree automata. The regularity theorem. The satisfiability over powerset algebras. Bounded decomposition. Tree models. Finite models and decidability. Bibliographic notes and sources. Algorithms. Evaluation of vectoral fixedpoint terms. A naive algorithm. Improved algorithms. Winning positions and winning strategies. Computations of winning positions. Computations of winning strategies. Bibliographical notes and sources.