Behaviours are Dualto Events

Philip WadlerUniversity of Edinburgh

Hudak Symposium, 29 April 2016

�, A ` � ⇤-L�, ⇤A ` �

⇤� ` A, ⌃� ⇤-R⇤� ` ⇤A, ⌃�

⇤�, A ` ⌃� ⌃-L⇤�, ⌃A ` ⌃�� ` A, � ⌃-R

� ` ⌃A, �

⇤A = Time ! A

⌃A = Time⇥ A

¬A = A! ?

¬⇤¬A

= (Time ! (A! ?))! ?= ((Time⇥ A)! ?)! ?= ¬¬⌃A

= ⌃A

¬⌃¬A

= (Time⇥ (A! ?))! ?= Time ! ((A! ?)! ?)

= ⇤¬¬A

= ⇤A