$\aleph_1$ and the modal $\mu$-calculus

TL;DR

This paper characterizes the $eth_1$-continuity fragment of the modal $oldsymbol{ extmu}$-calculus, proves the decidability of $eth_1$-continuity, and explores the properties of closure ordinals, including the significance of $oldsymbol{ extomega_1}$.

Contribution

It introduces the $C_{eth_1}(x)$ fragment, proves all formulas in it are $eth_1$-continuous, and establishes the decidability of the $eth_1$-continuity problem.

Findings

All formulas in $C_{eth_1}(x)$ are $eth_1$-continuous.

The problem of determining $eth_1$-continuity is decidable.

Closure ordinals include $ extomega_1$ and are closed under ordinal sum.

Abstract

For a regular cardinal $\kappa$, a formula of the modal $\mu$-calculus is $\kappa$-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of $\kappa$-directed sets. We define the fragment $C_{\aleph_1}(x)$ of the modal $\mu$-calculus and prove that all the formulas in this fragment are $\aleph_1$-continuous. For each formula $\phi(x)$ of the modal $\mu$-calculus, we construct a formula $\psi(x) \in C_{\aleph_1 }(x)$ such that $\phi(x)$ is $\kappa$-continuous, for some $\kappa$, if and only if $\phi(x)$ is equivalent to $\psi(x)$. Consequently, we prove that (i) the problem whether a formula is $\kappa$-continuous for some $\kappa$ is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment $C_{\aleph_0}(x)$ studied by Fontaine and the fragment $C_{\aleph_1}(x)$. We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal $\mu$-calculus. An ordinal $\alpha$ is the closure ordinal of a formula $\phi(x)$ if its interpretation on every model converges to its least fixed-point in at most $\alpha$ steps and if there is a model where the convergence occurs exactly in $\alpha$ steps. We prove that $\omega_1$, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, $\omega$, $\omega_1$ by using the binary operator symbol + gives rise to a closure ordinal.

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