## 2009-04-09

### Conclusion of lambda

Japanese version

Recently I saw this advertisement, ``For the finance specialists: Let's start from the simple thing.'' I assume the simple thing means, 1+1=2. I try to explain in this blog that how to teach 1 + 1 to a machine. I took more than eight months and yet not quite complete. (By the way, this is a tobacco advertisement.)

For human beings, this seems simple. But once you want to teach what 1+1 means to a machine, you must know more about it. For example, we discussed what is the numbers, and we represent it as Church numbers since a machine does not know what the meaning of '1' or '2''s sign. Someone may think this is paranoia since this is so natural.

I believe ``natural'' does not mean simple. It is just familiar to us. It is not simple at all for me. Some of you might feel it is natural to spend time with your family or your lover. But it is just you are familiar with that, it is not simple thing. It is important for me to see back into the natural things.

I would like to conclude this Hitchhiker's guide to λ calculus at the moment.

One day, I searched lambda calculus in Wikipedia. It said, ``it can be verified that PLUS 2 3 and 5 are equivalent lambda expressions.'' on the PLUS function. However, I did not understand how it works. I needed a large help of my friends. I do not want to forget about this. This is the motivation of writing this blog.

We talked about what is λ calculus, why people care about that, and several concrete examples. I hope this blog could help the people like me.

But this is not everything about the λ calculus. λ calculus is deep, I am hardly just open the door of this area. I still do not understand combinators. If I could understand it, I would like to continue this blog. I learned that writing is learning or teaching is learning. Also I learned that I could write an article only if I really like it.

I try to keep this article more understandable in informal way. I did neither mention about formal λexpression construction method, nor conversion procedure (α-conversion, β-conversion, and η-conversion). If you wan to know further, Wikipedia would be a good starting point.

Acknowledgments

Thanks to Uchida to give me the cue to write this blog. Thanks to my friends Hoedicke and Rehders to help to understand the examples. Thanks to Maeda who gave me an implementation of Y combinator. This blog is dedicated to my friend Tateoka. I wanted to show him this blog, but unfortunately it is not possible anymore.