# The Modal mu-calculus Alternation Hierarchy is Strict

```@inproceedings{Bradfield1996TheMM,
title={The Modal mu-calculus Alternation Hierarchy is Strict},
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

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

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

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

• Mathematics
CSL
• 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 Science
The 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

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

• Mathematics
J. Log. Algebraic Methods Program.
• 2008

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

• Mathematics
CSL
• 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

• Mathematics
Theor. 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.

### 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

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

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

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 Science
CAV
• 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)

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

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

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

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