Commonsense Reasoning, 2nd Edition

Appendix C

Events with Duration

Abstract In most versions of the event calculus, event occurrences are instantaneous. We present a revised version of the axiomatization EC that allows event occurrences with duration. A three-argument Happens predicate Happens3(e,t₁,t₂) is introduced, representing that event e occurs between t₁ and t₂.

Keywords Event calculus, Events with duration, Three-argument Happens

Definition EC1′

$\begin{array}{c}\hfill \mathit{Clipped}\left(t_1,f,t_4\right)\stackrel{def}{\equiv }\hfill \\ \hfill \exists e,t_2,t_3\left(\mathit{Happens}\mathit{3}\left(e,t_2,t_3\right)\wedge t_1\le t_3\wedge t_2<t_4\wedge \mathit{Terminates}\left(e,f,t_2\right)\right)\hfill \end{array}$

Definition EC2′

$\begin{array}{c}\hfill \mathit{Declipped}\left(t_1,f,t_4\right)\stackrel{def}{\equiv }\hfill \\ \hfill \exists e,t_2,t_3\left(\mathit{Happens}\mathit{3}\left(e,t_2,t_3\right)\wedge t_1\le t_3\wedge t_2<t_4\wedge \mathit{Initiates}\left(e,f,t_2\right)\right)\hfill \end{array}$

Definition EC3′

$\begin{array}{c}\hfill \mathit{StoppedIn}\left(t_1,f,t_4\right)\stackrel{def}{\equiv }\hfill \\ \hfill \exists e,t_2,t_3\left(\mathit{Happens}\mathit{3}\left(e,t_2,t_3\right)\right)\hfill \end{array}$