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.