What does it really mean by saying "ordered before"? Because even if
action_a happens-before action_b, action_a can be executed after
action_b in some implementation, right?
A Happens before relationship creates a memory barrier which prevents the execution of action-b before action-a. Thus some underlying JVM optimizations cannot be applied. So, NO action-a cannot be executed after or along with action-b.
If action_a happens-before action_b, does it mean action_a MUST NOT
see action_b? Or action_a may see or may not see action_b?
It means action-b must see all the changes brought about by action-a.
If action_a does NOT happen-before action_b, and action_b does NOT
happen-before action_a, does it mean action_a may see or may not see
action_b?
Happens-before is a transitive relationship. So, if action-a happens before action-b which happens before action-c ... so on upto action-y, and action-y happens before action-z, then action-a happens before action-z.
A happens before relationship ensures that the actions which follow the current action, will see the changes made by the current action. If the changes are not seen, then a happens before doesn't exist.
There could not be any cyclic happens-before, right?
Right, If action-a happens before action-b,action-c, action-d, then none among b,c,d can happen before action-a.
Edit :
The JLS says It should be noted that the presence of a happens-before relationship between two actions does not necessarily imply that they have to take place in that order in an implementation. If the reordering produces results consistent with a legal execution, it is not illegal.. So, if action-a has a happens before relationship with action-b, then action-b can execute first provided the final is equivalent to the sate if action a had executed before action b. This is implementation specific. The JIT might decide to run action-b earlier than action a provided this change in order does not affect the final result.
Well, action-a is independent of action-b. atleast theoretically :)
Happens before specifies, sequential actions. If the actions are parallel, then a happens before doesn't exist.
Note : All this confusion is because of the removal of happens before by the the JIT if there is no dependency between two actions. Please read about Escape analysis.