## Supplement: Interdefinability of $$\mathsf{HS}$$ Modalities

• In the semantics without point-intervals, the six modalities $$\langle A \rangle$$, $$\langle B \rangle$$, $$\langle E \rangle$$, $$\langle \overline{A} \rangle$$, $$\langle \overline{B} \rangle$$, $$\langle \overline{E} \rangle$$ suffice to express all others, as shown by the following equivalences:

$\begin{array}{l@{\hspace{2cm}}l} \langle L \rangle \varphi \equiv \langle A \rangle \langle A \rangle\varphi, & \langle \overline{L} \rangle \varphi \equiv \langle \overline{A} \rangle \langle \overline{A} \rangle \varphi, \\ \langle D \rangle \varphi \equiv \langle B \rangle \langle E \rangle \varphi, & \langle \overline{D} \rangle \varphi \equiv \langle \overline{B} \rangle \langle \overline{E} \rangle\varphi, \\ \langle O \rangle \varphi \equiv \langle E \rangle \langle \overline{B} \rangle \varphi, & \langle \overline{O} \rangle \varphi \equiv \langle B \rangle \langle \overline{E} \rangle \varphi. \end{array}$
• In the semantics with point intervals, the four modalities $$\langle B \rangle$$, $$\langle E \rangle$$, $$\langle \overline{B} \rangle$$, $$\langle \overline{E} \rangle$$ suffice to express all others, as shown by the following equivalences: \begin{aligned} & \langle A \rangle\varphi \equiv ([E]\bot \wedge (\varphi \vee \langle \overline{B} \rangle\varphi)) \vee \langle E \rangle([E]\bot \wedge (\varphi \vee \langle \overline{B} \rangle\varphi)), \\ & \langle \overline{A} \rangle\varphi \equiv ([B]\bot \wedge (\varphi \vee \langle \overline{E} \rangle\varphi)) \vee \langle B \rangle([B]\bot \wedge (\varphi \vee \langle \overline{E} \rangle\varphi)), \\ & \langle L \rangle\varphi \equiv \langle A \rangle(\langle E \rangle\top \wedge \langle A \rangle\varphi), \\ & \langle \overline{L} \rangle\varphi \equiv \langle \overline{A} \rangle(\langle B \rangle\top \wedge \langle \overline{A} \rangle\varphi), \\ & \langle D \rangle\varphi \equiv \langle B \rangle\langle E \rangle\varphi, \\ & \langle \overline{D} \rangle\varphi \equiv \langle \overline{B} \rangle\langle \overline{E} \rangle\varphi, \\ & \langle O \rangle\varphi \equiv \langle E \rangle(\langle E \rangle\top \wedge \langle \overline{B} \rangle\varphi), \\ & \langle \overline{O} \rangle\varphi \equiv \langle B \rangle(\langle B \rangle\top \wedge \langle \overline{E} \rangle\varphi). \end{aligned}

Also, the modal constant $$\pi$$ is definable in terms of $$\langle B \rangle$$ and $$\langle E \rangle$$, as $$[B]\bot$$ or $$[E]\bot$$, respectively.