Compact event notation

When following Toaq's semantics algorithm to turn Toaq sentences into logical formulas, many simple clauses translate into an existential quantification of an event, combined with its aspect and verb participant information. For example, Luı heaqdo jí súq máq becomes

$\exists e.\tau (e)<\mathrm {t} \wedge {\text{heaqdo}}_{w}({\text{suq}},{\text{maq}})(e)\wedge {\text{AGENT}}(e)(w)={\text{jı}}$

There is an event e of heaqdo-ing súq máq in world w, whose runtime precedes the implicit tense t, and whose agent is jí.

Kuna supports a compact notation for this in its denotation outputs:

$\exists \mathop {\text{heaqdo}} \limits _{<\mathrm {t} }{}_{w}^{e}\left({\text{jı}};{\text{suq}},{\text{maq}}\right)\color {gray}.P(e)$

It works as follows:

1. When an existential quantifier $\exists$ is followed by a Toaq verb, it asserts the existence of an event of that verb.
2. The event variable $e$ being bound is given by the following superscript.
3. The world variable $w$ in which the event is situated is given by the following subscript.
4. The aspect information is given underneath the verb. If it starts with a relational operator $<t$ it abbreviates $\tau (e)<t$.
5. The participants are listed in parentheses. If there is an agent, it's separated from the non-agent participants by a semicolon.
6. Optionally, a final dot announces the rest of the formula $P(e)$ in which $e$ is bound.

There are some variants of the notation, depending on the presence of an agent, of non-agent participants, and of a subsequent statement $P(e)$:

Compact event notation

Compact notation	Expanded notation
$\exists \mathop {\text{tıjuı}} \limits _{<{\text{t}}}{}_{w}^{e}\left({\text{y}},{\text{z}}\right)$	$\exists e.\tau (e)<{\text{t}}\wedge {\text{tıjuı}}_{w}({\text{y}},{\text{z}})(e)$
$\exists \mathop {\text{tıjuı}} \limits _{<{\text{t}}}{}_{w}^{e}\left({\text{y}},{\text{z}}\right).P(e)$	$\exists e.\tau (e)<{\text{t}}\wedge {\text{tıjuı}}_{w}({\text{y}},{\text{z}})(e)\wedge P(e)$
$\exists \mathop {\text{heaqdo}} \limits _{<{\text{t}}}{}_{w}^{e}\left({\text{x}};{\text{y}},{\text{z}}\right)$	$\exists e.\tau (e)<{\text{t}}\wedge {\text{heaqdo}}_{w}({\text{y}},{\text{z}})(e)\wedge {\text{AGENT}}(e)(w)={\text{x}}$
$\exists \mathop {\text{heaqdo}} \limits _{<{\text{t}}}{}_{w}^{e}\left({\text{x}};{\text{y}},{\text{z}}\right).P(e)$	$\exists e.\tau (e)<{\text{t}}\wedge {\text{heaqdo}}_{w}({\text{y}},{\text{z}})(e)\wedge {\text{AGENT}}(e)(w)={\text{x}}\wedge P(e)$
$\exists \mathop {\text{ruqshua}} \limits _{<{\text{t}}}{}_{w}^{e}$	$\exists e.\tau (e)<{\text{t}}\wedge {\text{ruqshua}}_{w}(e)$
$\exists \mathop {\text{ruqshua}} \limits _{<{\text{t}}}{}_{w}^{e}.P(e)$	$\exists e.\tau (e)<{\text{t}}\wedge {\text{ruqshua}}_{w}(e)\wedge P(e)$
$\exists \mathop {\text{marao}} \limits _{<{\text{t}}}{}_{w}^{e}({\text{jı}};)$	$\exists e.\tau (e)<{\text{t}}\wedge {\text{marao}}_{w}(e)\wedge {\text{AGENT}}(e)(w)={\text{jı}}$
$\exists \mathop {\text{marao}} \limits _{<{\text{t}}}{}_{w}^{e}({\text{jı}};).P(e)$	$\exists e.\tau (e)<{\text{t}}\wedge {\text{marao}}_{w}(e)\wedge {\text{AGENT}}(e)(w)={\text{jı}}\wedge P(e)$

Example

The full denotation of Pu tao jí hóq da (i.e. Ruaq jí ꝡä pu tao jí hóq ka) is:

$\exists e.\ \tau (e)\subseteq {}{\text{t}}_{0}\land {}{\text{AGENT}}(e)({\text{w}})={\text{jı}}\land {}{\text{ruaq}}_{\text{w}}(\lambda w.\ \exists e'.\ \tau (e')\subseteq {}{\text{t}}\land {}{\text{AGENT}}(e')(w)={\text{jı}}\land {}{\text{tao}}_{w}({\text{a}})(e'))(e)\ |\ {\text{t}}<{\text{t}}_{0}\ |\ {\text{abstract}}({\text{a}})$

Using compact notation, it becomes a bit less daunting:

$\exists \mathop {\text{ruaq}} \limits _{\subseteq {}{\text{t}}_{0}}{}_{\text{w}}^{e}\left({\text{jı}};\lambda w.\ \exists \mathop {\text{tao}} \limits _{\subseteq {}{\text{t}}}{}_{w}^{e'}\left({\text{jı}};{\text{a}}\right)\right)\ |\ {\text{t}}<{\text{t}}_{0}\ |\ {\text{abstract}}({\text{a}})$