Chapter

# Closure Ordinals of the Two-Way Modal $$\mu$$ -Calculus

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## Abstract

The closure ordinal of a $$\mu$$-calculus formula $$\varphi (x)$$ is the least ordinal $$\alpha$$, if it exists, such that, in any model, the least fixed point of $$\varphi (x)$$ can be computed in at most $$\alpha$$ many steps, by iteration of the meaning function associated with $$\varphi (x)$$, starting from the empty set. In this paper we focus on closure ordinals of the two-way modal $$\mu$$-calculus. Our main technical contribution is the construction of a two-way formula $$\varphi _n$$ with closure ordinal $$\omega ^n$$ for an arbitrary $$n\in \omega$$. Building on this construction, as our main result we define a two-way formula $$\varphi _\alpha$$ with closure ordinal $$\alpha$$ for an arbitrary $$\alpha <\omega ^\omega$$.

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