Re: recursion theory

From: Bruno Marchal <>
Date: Thu, 27 Jun 2002 15:59:49 +0200

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

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:
Diagonalisation 1 (answers)

Received on Thu Jun 27 2002 - 06:57:27 PDT

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