# The Modal mu-calculus Alternation Hierarchy is Strict

@inproceedings{Bradfield1996TheMM, title={The Modal mu-calculus Alternation Hierarchy is Strict}, author={Julian Bradfield}, booktitle={CONCUR}, year={1996} }

One of the open questions about the modal mu-calculus is whether the alternation hierarchy collapses; that is, whether all modal fixpoint properties can be expressed with only a few alternations of least and greatest fixpoints. In this paper, we resolve this question by showing that the hierarchy does not collapse. @ 1998 Elsevier Science B.V. All rights reserved

## 92 Citations

### Fixpoint alternation: Arithmetic, transition systems, and the binary tree

- Computer Science, MathematicsRAIRO Theor. Informatics Appl.
- 1999

We provide an elementary proof of the fixpoint alternation hierarchy in arithmetic, which in turn allows us to simplify the proof of the modal mu-calculus alternation hierarchy. We further show that…

### Alternation Is Strict For Higher-Order Modal Fixpoint Logic

- Computer ScienceGandALF
- 2016

It is shown that the number and parity of priorities available to an APKA form a proper hierarchy of expressive power as in the modal mu-calculus, which induces a strict alternation hierarchy on HFL.

### Deciding the First Levels of the Modal mu Alternation Hierarchy by Formula Construction

- MathematicsCSL
- 2015

The blow-up incurred by turning Psi into the modal formula is shown to be necessary: there are modal formulas that can be expressed sub-exponentially more efficiently with the use of fixpoints.

### The modal μ-calculus hierarchy over restricted classes of transition systems

- Mathematics, Computer ScienceThe Journal of Symbolic Logic
- 2009

First, it is proved that over transitive systems the hierarchy collapses to the alternation-free fragment, and it is shown that the hierarchy is strict over reflexive frames by proving the finite model theorem for reflexive systems.

### Disjunctive form and the modal μ alternation hierarchy

- SociologyFICS
- 2015

The classes of formulas studied here illustrate a previously undocumented type of avoidable syntactic complexity which may contribute to the understanding of why deciding the alternation hierarchy is still an open problem.

### Canonical completeness of infinitary mu

- MathematicsJ. Log. Algebraic Methods Program.
- 2008

### On closure ordinals for the modal mu-calculus

- MathematicsCSL
- 2013

It is proved that the closure ordinals arising from the alternation-free fragment of modal mu-calculus (the syntactic class capturing Sigma_2 \cap Pi_2) are bounded by omega^2.

### The μ-calculus alternation hierarchy collapses over structures with restricted connectivity

- MathematicsTheor. Comput. Sci.
- 2014

The alternation hierarchy of the mu-calculus collapses to the alternation-free fragment over some classes of structures, including infinite nested words and finite graphs with feedback vertex sets of a bounded size.

### Deciding low levels of tree-automata hierarchy

- Computer ScienceElectron. Notes Theor. Comput. Sci.
- 2002

### Canonical completeness of infinitary μ

- Mathematics

This paper presents a new model construction for a natural cut-free infinitary version Kω (μ) of the propositional modal μ-calculus. Based on that the completeness of Kω (μ) and the related system…

## References

SHOWING 1-10 OF 25 REFERENCES

### A Hierarchy Theorem for the µ-Calculus

- MathematicsICALP
- 1996

We consider the positive mu-calculus with successors PμS, namely a variant of Kozen's modal mu-calculus Lμ [9] where negation is suppressed and where the basic modalities are a sequence of successor…

### On the Expressivity of the Modal Mu-Calculus

- MathematicsSTACS
- 1996

A strong upper bound is established on the complexity of the sets of states, in certain classes of infinite systems, that satisfy formulae of the modal mu-calculus.

### Verifying Temporal Properties of Systems

- Computer ScienceProgress in Theoretical Computer Science
- 1992

The generalized tableau technique is exhibited on Petri nets, and various standard notions from net theory are shown to play a part in the use of the technique on nets-in particular, the invariant calculus has a major role.

### On Model-Checking for Fragments of µ-Calculus

- Computer ScienceCAV
- 1993

It is shown that the logic L2 is as expressive as ECTL* given in [13], and the model checking problem for the μ-calculus is equivalent to the non-emptiness problem of parity tree automata.

### On Fixed-Point Clones (Extended Abstract)

- MathematicsICALP
- 1986

Initiality of the tree algebra is established via a "Mezei-and Wright-like" result on interpretation of fixed point terms, and a reduction of these terms to Rabin automata on infinite trees is shown which yields some decidability results.

### Verification of Temporal Properties of Concurrent Systems

- Computer Science
- 1993

This thesis is concerned with the verification of concurrent systems modelled by process algebras. It provides methods and techniques for reasoning about temporal properties as described by…

### A finite model theorem for the propositional μ-calculus

- MathematicsStud Logica
- 1988

A finite model theorem and infinitary completeness result for the propositional μ-calculus is proved and a link between finite model theorems for propositional program logics and the theory of well-quasi-orders is established.

### μ-definable sets of integers

- MathematicsJournal of Symbolic Logic
- 1993

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the…

### An Improved Algorithm for the Evaluation of Fixpoint Expressions

- Computer ScienceTheor. Comput. Sci.
- 1994