### Re: recursion theory

Title: Re: recursion theory At 11:41 -0700 25/06/2002, Wei Dai wrote: On Mon, Jun 17, 2002 at 12:39:49PM +0200, Bruno Marchal wrote: Recursion theory is the same as Computability Theory, although the name recursion theory is used when the emphasis is on the abstract theory. Recursion theory has born with Emil Post 1944 paper, which is still a good introduction. Post paper can be find in Davis The Undecidable. I found a 38 page paper on the web (http://www.math.ucla.edu/~asl/bsl/0203/0203-002.ps) Ah! A paper by Soare. Very good. about why computability theory is historically called recursion theory, and why we should stop calling it that. I knew that recursive functions are the same as Turing-computable functions, which is a key result of computability/recursion theory. What I didn't realize until recently was how much work has been done on unsolvable problems, for example classifying them according to the degree of unsolvability, and that also falls under the name recursion theory. Yes. Classification of unsolvable problems has been the main purpose in recursion theory since the fundamental papers by Emil Post. I'm hoping this theory will offer some insights about what uncomputable universes might be like, and whether we should assign a non-zero probability of being inside a uncomputable universe (i.e. whether there could be uncomputable physical laws). The problem is that there is no clear standart definition of computability with the reals. Well, there exists *too much* standart definitions I should say. Richard Pour El has wriiten a book showing that some differential equation can have non computable solutions but I have never understand its motivation for its definition of computable reals. At another extreme you have the entire relativisation of computability *on a ring* by Blum Smale and Shub which is far more interesting and also admit natural extension for quantum computing ... Now, and we have discussed this before, I have no understanding of the expression being inside a universe. I really believe that the uda shows such an expression being senseless. The *apparent* partially computable universe is itself a result of averaging on all relative computation going through states closed to my *actual* state. This shows comp implies that our local universe must have non computable and computable features. *The* classical treatise is the book by Hartley Rogers: ROGERS H., Theory of Recursive Functions and Effective Computability, McGraw-Hill, 1967. (2ed Ed. MIT Press, 1987). I got this book yesterday. It seems to cover everything I want but hasn't been updated since 1967. (The second edition only fixes a few errors.) I wonder what interesting new results I might miss. You just miss a bunch of highly technical results which will find applications in two or three centuries! Modern recursion theory has lose its grip with the original motivation and looks like, imo, a machine working for itself. I may be wrong of course. I have try to use some results and methodologies inspired by Drake work on the degree below O' but without success. I think that recursion has given important tools for more mundane theoretical computer science, for exemple in complexity theory or in theoretical inductive inference (Case and Smith, Oherson Stob Weinstein), though. I assume Odifreddi's book is more up to date, but why is it called Classical Recursion Theory? Which parts of recursion theory is considered not classical? Classical is a not well chosen term because in logic not classical is used for non classical logic. Here classical means the elementary theory on which the rest (of recursion theory) is based. Not classical means either generalisation of recursion theory, like alpha-recursion theory (where natural numbers are replaced by church-kleene constructive ordinals), or axiomatic approach to recursion theory, where the axioms fit different sort of recursion (including alpha recursion, or beta recursion with beta some other high ordinal, etc.). It is beyond the elementary need I make in my thesis. Actually I believe that high ordinal recursion theory can be used to show that some result in my thesis can be applied in the non comp context. I have written some text on alpha-comp (alpha constructive ordinal) and comp-in-a-ring. Could be usefull for non-computationalist approach, or for weaker sort of comp. I recall for the interested one that I have send two introductory post on recursion theory on line: Diagonalisation 1: http://www.escribe.com/science/theory/m3079.html Diagonalisation 1 (answers) http://www.escribe.com/science/theory/m3344.html Bruno

### Re: recursion theory

On Mon, Jun 17, 2002 at 12:39:49PM +0200, Bruno Marchal wrote: Recursion theory is the same as Computability Theory, although the name recursion theory is used when the emphasis is on the abstract theory. Recursion theory has born with Emil Post 1944 paper, which is still a good introduction. Post paper can be find in Davis The Undecidable. I found a 38 page paper on the web (http://www.math.ucla.edu/~asl/bsl/0203/0203-002.ps) about why computability theory is historically called recursion theory, and why we should stop calling it that. I knew that recursive functions are the same as Turing-computable functions, which is a key result of computability/recursion theory. What I didn't realize until recently was how much work has been done on unsolvable problems, for example classifying them according to the degree of unsolvability, and that also falls under the name recursion theory. I'm hoping this theory will offer some insights about what uncomputable universes might be like, and whether we should assign a non-zero probability of being inside a uncomputable universe (i.e. whether there could be uncomputable physical laws). *The* classical treatise is the book by Hartley Rogers: ROGERS H., Theory of Recursive Functions and Effective Computability, McGraw-Hill, 1967. (2ed Ed. MIT Press, 1987). I got this book yesterday. It seems to cover everything I want but hasn't been updated since 1967. (The second edition only fixes a few errors.) I wonder what interesting new results I might miss. I assume Odifreddi's book is more up to date, but why is it called Classical Recursion Theory? Which parts of recursion theory is considered not classical?

### recursion theory

I'm trying to learn about recursive ordinals and related topics such as the arithmetical hierarchy, which are studied under the name recursion theory. Is anyone here familiar with recursion theory? Can anyone recommend a book on it?