Dana Scott's Stochastic Lambda-Calculus Lecture Summarized by Assembly AI
Summary
- Dana has connected the whole field with the great traditions of mathematical logic. His vision of the subject has been a source of wisdom and guidance for all of us over the years. So I think it's very fitting that he should give the opening lecture of this boot camp.
- I was hoping to see Raymond Smolliam, an old, old friend, but his health is not too good at the moment. There is a theory that you can just call on ordinary probability theory to introduce random elements. For computer science applications there's been a big development. But I'm going to be very informal today about doing it.
- Here's a model using enumeration operators. You have the integers and we have a small amount of Gödel numbering on the integers. Why does this model work as a model for the lambda calculus?
- If you just take Pure lambda terms that have lambda and application as the only components then they will be Computable. But even Better, you can do a recursion by getting a fixed point of a continuous Operator. This is exactly the analog of the universal Turing machine.
- A random variable is just a parameterized family of sets of integers. It's easy to prove that the random variables form a model for the lambda calculus on real variables. Now for further thinking about this, and I hope there will be time for some sessions where we can have a reading group or something.
- Helmut Helmuth was one of the organizers of the Vienna Summer of Logic back in 2014. He was incredibly active in doing things in Vienna. His wife told me that the Formal Methods and Computer Design Conference in October will have a section devoted to his work.