gwern comments on Open Thread: July 2010 - Less Wrong
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Information theory challenge: A few posters have mentioned here that the average entropy of a character in English is about one bit. This carries an interesting implication: you should be able to create an interface using only two of the keyboards keys, such that composing an English message requires just as many keystrokes, on average, as it takes on a regular keyboard.
To do so, you'd have to exploit all the regularities of English to offer suggestions that save the user from having to specify individual letters. Most of the entropy is in the intial charaters of a word or message, so you would probably spend more strokes on specifying those, but then make it up with some "autocomplete" feature for large portions of the message.
If that's too hard, it should be a lot easier to do a 3-input method, which only requires your message set to have an entropy of less than ~1.5 bits per character.
Just thought I'd point that out, as it might be something worth thinking about.
Already done; see Dasher and especially its Google Tech Talk.
It doesn't reach the 0.7-1 bit per character limit, of course, but then, according to the Hutter challenge no compression program (online or offline) has.
Wow, and Dasher was invented by David MacKay, author of the famous free textbook on information theory!
According to Google Books, the textbook mentions Dasher, too.