Great Pyrenees, as with other dogs, will do exactly what they are bred and trained for. Pyrenean Mountain Dogs were bred to guard the livelyhood and families of the mountain shepards.  This did not require a lot of running.  However it did require intelligence and a certain amount of independence.  Such as recognizing threats and taking whatever action was necessary to neutralize the threat, whether by force or by intercession. I have found my Pyr will most often just get between the protectees (goats usually, sometimes grandkids) and whatever is percieved to be a possible threat.  No threats or growling or even confrontation, just gets between them and lies down, situation neutralized.  Most intelligent dog I have ever owned(11).