Why do guys half smile?

Because guys generally think they look like dorks when they smile... the stereo-typical cool guy in action movies who gets the babe rarely ever gives a big open mouth smile. The half smile also indicates that he may like you. He is not laughing at you... or he would.

Ask him right out. "Why do you always give me that smile? "
He probably won't give you a straight answer - but it'll be fun watching him squirm!

Seriously - a lot of us men are pretty useless at working out what you're thinking. We're probably more petrified than you at making the first move. If you like him (and I'm assuming you do), ask him out. At the very least he'll be very flattered and politely decline. More likely he'll be so chuffed that he'll pay for dinner.

If being so direct is not your style, try to give him some more obvious clues. Most men aren't terribly good at the subtle ones. Anything that leaves us in any doubt and we have no confidence to proceed. Frankly it's amazing some of us manage to ever reproduce!

Try asking how he usually goes about asking a lady out. You may get some clues to his level of confidence and you could always ask him to practice on you!

Good luck.