Alternating automata, the weak monadic theory of the tree, and its complexity
International Colloquium on Automata, Languages and Programming on Automata, languages and programming
International Colloquium on Automata, Languages and Programming on Automata, languages and programming
The greatest fixed-points and rational omega-tree languages
Theoretical Computer Science
Theory of recursive functions and effective computability
Theory of recursive functions and effective computability
Alternating automata on infinite trees
Theoretical Computer Science
Fixed point characterization of Bu¨chi automata on infinite trees
Journal of Information Processing and Cybernetics
On modal mu-calculus and Bu¨chi tree automata
Information Processing Letters
The modal mu-calculus alternation hierarchy is strict
Theoretical Computer Science
An automata-theoretic approach to branching-time model checking
Journal of the ACM (JACM)
Symbolic Model Checking
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Automata for the Modal mu-Calculus and related Results
MFCS '95 Proceedings of the 20th International Symposium on Mathematical Foundations of Computer Science
A Linear-Time Model-Checking Algorithm for the Alternation-Free Modal Mu-Calculus
CAV '91 Proceedings of the 3rd International Workshop on Computer Aided Verification
A Decision Procedure for the Propositional µ-Calculus
Proceedings of the Carnegie Mellon Workshop on Logic of Programs
Freedom, Weakness, and Determinism: From Linear-Time to Branching-Time
LICS '98 Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science
The weakness of self-complementation
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
A Characterization Theorem for the Alternation-Free Fragment of the Modal µ-Calculus
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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The µ-calculus is an expressive specification language in which modal logic is extended with fixpoint operators, subsuming many dynamic, temporal, and description logics. Formulas of µ-calculus are classified according to their alternation depth, which is the maximal length of a chain of nested alternating least and greatest fixpoint operators. Alternation depth is the major factor in the complexity of µ-calculus model-checking algorithms. A refined classification of µ-calculus formulas distinguishes between formulas in which the outermost fixpoint operator in the nested chain is a least fixpoint operator (Σi formulas, where i is the alternation depth) and formulas where it is a greatest fixpoint operator (Πi formulas). The alternation-free µ-calculus (AFMC) consists of µ-calculus formulas with no alternation between least and greatest fixpoint operators. Thus, AFMC is a natural closure of Σ1 ∪ Π1, which is contained in both Σ2 and Π2. In this work we show that Σ2 ∩ Π2 ≡ AFMC. In other words, if we can express a property ξ both as a least fixpoint nested inside a greatest fixpoint and as a greatest fixpoint nested inside a least fixpoint, then we can express ξ also with no alternation between greatest and least fixpoints. Our result refers to µ-calculus over arbitrary Kripke structures. A similar result, for directed µ-calculus formulas interpreted over trees with a fixed finite branching degree, follows from results by Arnold and Niwinski. Their proofs there cannot be easily extended to Kripke structures, and our extension involves symmetric nondeterministic Büchi tree automata, and new constructions for them.