At 22:43 27/08/04 -0700, George Levy wrote:
>Bruno
>
>I am trying to visualize Lob formula as a block diagram to be implemented
>either in neural net, as computer program or as a digital cicuit. Digital
>circuits have the advantage of being very simple (binary) so let's try to
>express Lob's formula as a truth table that could be implemented with NAND
>gates.
>
>Let's write Lob's formula as B2(B1p -> p) -> B1p
>where B1, B2, and p are binary variables.
I am not sure I understand. It is better to see B as a (non truth
functional) connector.
>Note that B1 applies to p and B2 applies to the implication (B1p -> p).
>(Should I have done this differently?)
>Let
>~ = NOT
>+ = OR
>. = AND
>
>We can convert the implication B1p -> p to ~(B1.p) + p
>
>The Boolean equivalent to Lob is
>
> ~B2(~(B1p)+ p) + B1p
>
>The truth table is
>
>B2 B1 p B1p ~B1p+p ~B2(~(B1p)+ p)) ~B2(~(B1p)+ p) + B1p
>
>0 0 0 0 1 1
> 1
>0 0 1 0 1 1
> 1
>0 1 0 0 1 1
> 1
>0 1 1 1 1 1
> 1
>1 0 0 0 1 0
> 0
>1 0 1 0 1 0
> 0
>1 1 0 0 1 0
> 0
>1 1 1 1 1 0
> 1
>
>I am now confused. The fifth column ~B1p+p surprisingly is all 1's. The
>last column ~B2(~(B1p)+ p) + B1p which is Lob's statement and which I
>expected to be all 1's is not. I have rechecked this table and I don't see
>anything wrong. Is there something wrong?
>
>It may be that Boolean algebra is not adequate to express Lob. The
>question is how can Lob's formula be expressed simply by a digital circuit
>a block diagram or a neural net?
By making a system enough rich to represent B, like Godel did show
for provability in arithmetic. It is more comparable to a (lisp) interpretor
described in Lisp. In terme on electrical or neuronal nets it means it will
have feedback loops. Modal logic is not truth functional. You need delay and
close circuits (like flip-flop).
Boyer, if I remember well, as explicitly build Lobian theorem prover.
Smullyan presents toy systems in chapter 26 and 27 of FU. Of course
any reasonable formalisation of arithmetic is enough, so any universal
machine is Lobian extendible.
Bruno
http://iridia.ulb.ac.be/~marchal/
Received on Sat Aug 28 2004 - 13:38:11 PDT