It started with a small bell on a hook next to the door and I would take their paw and hit the bell before letting them outside. Eventually they caught on and I'd give them a lot of praise when they did it on their own. The next iteration was an analog setup and the latest uses a Raspberry Pi. I will say, I have three dogs and only one (the smartest) ever caught on with the bell and analog version. Two of them are able to use the newest version which is the easiest to trigger, since it has a touch surface that is 4x4 inches. The third dog has never caught on. But, he hates going outside, so it's likely he doesn't want to catch on.
I'm not exactly sure how they know when to expect me home, but they seem to have a decent sense of time or are sensitive enough to external triggers that condition them, like neighbors arriving home after work, rush hour traffic, etc.
They do exhibit certain behaviors when it's feeding time (hasn't been automated yet), which is the same every night and that is consistent through daylight savings changes and day lengths varying with seasons. So I assume they have some rudimentary sense of time.