# RE: Belief Statements

`Hi Hal,`

`At 22:30 17/01/05 -0500, Hal Ruhl wrote:`

I reject Schmidhuber Comp because a sequence of world states [world kernels] which may indeed be Turing machine [or some extension there of] emulable is nevertheless managed by the system's dynamic which is external to the machine.

Any sub component of a world kernel [such as myself] is subject to the same result thus my rejection of Personal Comp.

I understand the sentences but I am completely lost to see relations with previous sentences of you. Have you try to sum up your theory (your theories) in a web page (like you did some years ago) ?

I really think we should focus on a well established theory or language to make progress. As you
know my results are based on classical first order arithmetic. But recently I came back to an old craving of me: lambda calculus and combinatory logic. I have discovered that combinators make it possible to simplify considerably my work (and this at many different levels). I will hardly resist to say more on this in a near future. The advantage of combinatory logic is that it is more quickly funny (as opposed to first order logic which beginning is tedious and hard (and I usually refer to textbook or to Podnieks pages).
Informal Combinatory logic can be presented as an "everything theory". It has a static and a dynamic:

STATIC: K is a molecule
S is a molecule
if x and y are molecules then (x y) is a molecule. From this you can easily enumerate all
possible molecules: K, S, (K K), (K S), (S K), (S S), ((K K) K), ((K S) K) ...

DYNAMIC: For all molecules x and y, the molecules ((K x) y) produce x
For all molecules x, y z, the molecules (((S x) y) z) produce y

```Exercices: what gives ((K K) K) ? what gives (K (K K)) ?
Is there a molecule M such that (M x) gives x?```

That theory has been discovered by Shoenfinkel in 1920 and rediscovered by Curry and Church (the same as in "Church thesis") in the 1930, and has had since many applications in practical and theoretical computer science. It is very fine grained and useful to compare many theories. I am sure that if you were willing to study it a little , it would inspire you for finding more understandable presentation of your ideas. Note that it does not subsumes comp, but, given that this theory is Turing complete, it is perhaps one of the better road toward computer science and its philosophies. It could interest everyone in this list and I am pretty sure many know it or have heard about it. I am currently translating my thesis in combinators language if only because I have discovered it provides huge helps to make concrete some otherwise too much abstract notion for good willing students and philosophers.
I am leaving Brussels for Paris (First European Congress in Philo of Sciences) and I will be back in a few days.

`Regards,`

`Bruno`

`http://iridia.ulb.ac.be/~marchal/`

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