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of what exactly that means, and see if it succeeds in capturing our intuitive picture of complexity.

“Amount of surprise” is a good first approximation for the quantity that I have in mind here, so let’s start by thinking through a simple analogy. I converse with both my roommate and my Siamese cat on a fairly regular basis. In both cases, the conversation consists in my making particular sounds and my interlocutor responding by making different sounds. Likewise, in both cases there is a certain amount of information exchanged between my interlocutor and me. In the case of my roommate, the nature of this information might vary wildly from conversation to conversation: sometimes we will talk about philosophy, sometimes about a television show, and sometimes what to have for dinner. Moreover, he’s a rather unusual fellow—I’m never quite sure what he’s going to say, or how he’ll respond to a particular topic of conversation. Our exchanges are frequently surprising in a very intuitive sense: I never know what’s going to come out of his mouth, or what information he’ll convey. My Siamese cat, on the other hand, is far less surprising. While I can’t predict precisely what’s going to come out of her mouth (or when), I have a pretty general sense: most of the time, it’s a sound that’s in the vicinity of “meow,” and there are very specific situations in which I can expect particular noises. She’s quite grandiloquent for a cat (that’s a Siamese breed trait), and the sight of the can opener (or, in the evening, just someone going near the can opener) will often elicit torrent of very high-pitched vocalizations. I’m not surprised to hear these noises, and can predict when I'll hear them with a very high degree of accuracy.

The difference between conversing with these two creatures should be fairly clear. While my cat is not like a recording—that is, while I’m not sure precisely what she’s going to say (in the way that, for instance, I’m precisely sure what Han Solo will say in his negotiations with Jabba