Most sentences are unique

How long does a sentence have to be to gaurantee its uniqueness? This question can be answered by looking at a huge corpora of indexed sentences, e.g. the text of the internet. All that has been written on the internet is not nessecarily a million monkeys typing on a keyboard (although a million monkeys typing on a keyboard could produce all the current content),¹ with exception of a website that generates random text.² Still, I was wondering, out of the current 25 billion or so webpages³, how long does a sentence have to be to make it unique? Surprisingly short, as it turns out.

A quick experiment

Here’s a quick experiment to see how few words you need to actually produce a totally and completely unique sentence into the universe.

Lets take the sentence (which I’m making up as I type):

A painting of a donkey in the desert.

This sentence comes from a painting I made that I’m looking at. For searching “A” there are 25.3 billion results. The exact phrase “A painting” narrows it down to 22.4 million. Already we’ve decreased by a factor of 10³. The phrase “A painting of” decreases it to 5.3 million (another 10^~1 decrease) and adding “a donkey” decreases it to 131,000 (another 10^~2 decrease). The smallest number of results is “a painting of a donkey in the” which yields only a single result (another 10^~5 decrease).⁴ There are no results for “A painting of a donkey in the desert”, at least until this post is published.

This makes sense, as there are increasing combinatorial factors that limit the probability of a given sentence for each new word. Each word has many possibilities as there are hundreds of thousands of English words. Although, most of the words are not possible because there are requirements on grammar and context. We can very roughly approximate this by solving for the 7th root (since we only had seven words that gave us a few results) as the total number of sites (25 billion), which gives ~30. Although this number is inflated for the nouns and deflated for the prepositions.

But the size of this number, 30, gives pause to ever think that the internet will contain every possible sentence. Though the internet may contain most 6-7 word sentences, it would need 30x as many pages to have 8 word sentence. It would need ~900x more pages to have 9 word sentences, and so on exponentially.

So far now, it is really amusing and satisfying to relish in the thought that most 8 word sentences we create, are totally unique to the internet and have never existed before. And if we extrapolate a little bit, and say that the sum of total language ever spoken may be only a million times bigger than the internet, than that would mean that most 11 word sentences have never been uttered before!*

Take these calculations with a big big big grain of salt, but since these numbers lie in the world of logarithms, they won’t be off my much more than an order of magnitude.