Andy Liu 1p gave an interesting lecture at this year's US Go Congress on the topic of shoulder hits and attachments. He has been studying AlphaGo and also FineArt.
He views the shoulder hits and attachments seen in the AlphaGo not as probes but as ways to simplify the game. He calls the 3-3 invasion a shoulder hit (I prefer the more general angle play) and cites the same argument that it a) usually will get a response and b) reduces the opponent's responses compared to the knight's approach. (I have doubts about his insinuations that one can end in sente. If your opponent has more stones than you in an area, I don't think it's you that necessarily decides who ends in sente.)
For attachments, he made the same argument. The opponent will usually choose one of two hanes, and as long as you can handle either, it's okay.
There are a couple of things he did not say, but I thought about later. The first is that if the opponent plays nobi instead, that the sequence transposes to a shoulder hit sequence, which is also "simplifying." The second is that although we think of attachments as starting fights, the player who attaches does not have to crosscut after the opponent's hane. A counter-hane can be played instead, and this usually has fewer follow-ups. In fact, he didn't show any crosscut variations in his examples, only counter-hane.
He has been playing early 3-3 invasions recently himself and claims some good results. Of course, Andy is strong and may get good results no matter what he does.
None of his lecture had to do with influence vs. territory. That's a separate topic, but in another lecture he suggested that kyu players assume their groups are alive because their opponents would have a hard time killing them anyway and that even if they are killed the more important thing is whether the stones are useful than whether they are dead or alive. He is not suggesting a total disrespect of influence, though and I saw one review in which he suggested keeping a safe distance, so he is not rejecting all traditional theory.
I think if you like the center, maybe you can take Bill Spight's suggestion and play 5-5, but instead of invading at 3-3, play the shoulder hit at 5-5 against your opponent's 4-4 if and when you think the timing is right. Then you "simplify" the position in Andy's terms. (That's just my idea, it's not something I've tried but I don't think it's too bad if the whole board position is okay.)
Andy emphasized that reducing the opponent's responses and seeking sente are a big part of his philosophy and he is seeing this kind of thing in current strong AI games.
FWIW, when I was 10k I had a decent record against players who invaded early at 3-3 but had poor results playing directly at 3-3 myself in an empty corner. I am looking forward to more people trying that against me.
Maybe in another thread I'll try to share some of what Andy said in his AI talk.
Is 4-4 a mistake in the opening?
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Calvin Clark
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