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The LISP S-expression ((A bc) 39 x-y-z) denotes the following nested tuples: A, bc , 39, x-y-z Programs and data in LISP are always S-expressions. Everything is an S-expression! The S in S-expression stands for Symbolic. S-expressions that are not lists are called atoms. A, bc, 39 and x-y-z are the atoms in the above S-expression. The empty list () = is also an atom, usually denoted by nil. 50 CHAITIN, FROM PHILOSOPHY TO PROGRAM SIZE LISP is a functional programming language, not an imperative programming language.

Over all requirements s, o Note that if there are duplicate requirements in the list, then several p will yield the same output o. , we will have several p with the same value for C(p). This way of constructing C may be thought of as a first-fit storage allocation algorithm for one infinitely-divisible unit of storage. 14 Three-Level Proofs: Program Size, Probabilities, Geometry The proof of this crucial version of the Kraft inequality involves simultaneously thinking of C as a computer, as an assignment of probabilities to outputs, and as an assignment of outputs to segments of the unit interval, segments which are halves, or halves of halves, or halves of halves of halves.

1 Formal Definition of the Complexity of a Formal Theory Before starting with LISP, let me finish some things I started in the last lecture, and also answer some questions that were raised about that lecture. First of all, I never stated the formal version of the incompleteness results about Ω that I explained informally in Day II. How can we measure the complexity of a formal mathematical theory in bits of information? ) program p 45 46 CHAITIN, FROM PHILOSOPHY TO PROGRAM SIZE such that U (p) = {the infinite set of all the theorems in the theory}.