> On 22 Apr 2018, at 06:21, Russell Standish <li...@hpcoders.com.au> wrote: > > On Sat, Apr 21, 2018 at 08:08:50PM -0400, John Clark wrote: >> On Sat, Apr 21, 2018 at 6:15 PM, Russell Standish <li...@hpcoders.com.au> >> wrote: >> >> > >>> *Yes, of course a Loebian machine is a type of Turing machine.* >> >> >> How can I determine if that particular Turing Machine is doing something >> fundamentally different from what every other Turing Machine is doing? >> > > I would say that it is a machine that proves Loeb's theorem. Not all > Turing machines are capable of that, even universal machines absent > the right software. But I may have misunderstood this :).
Not at all. That is correct. The main difference can be sum up by saying that a machine, having a believability predicate, noted “[]” 1) is universal if p -> []p is true for them on all p sigma_1. (And that can be proved equivalent with “identifiable” with a number u such that phi_u(x, y) = phi_x(y). For example “p -> []p” is true for RA = [], and indeed RA can compute all phi_x(y) for any enumeration of the phi_i. 2) is a Löbian machine if it is a universal machine which can prove its own universality and the consequences, like its own incompleteness, in particular Gödel and Löb theorems, and much more. That means that is is a believer-machine “[]", for which not only p -> []p is true, but the machine is rich enough (in term of beliefs) to be able to prove it. The main ingredient to become Löbian, is in believing enough induction axioms. p sigma_1 means that p has the shape ExP(x, y), with P decidable (sigma_0). If you can convince yourself that you can search for a number having some decidable property P, and find it if it exists, with no bounding time of research, than you are Löbian. Note that all Löbian entity can know that they are Löbian, bit none can known that they are consistent, and none can define their own soundness. Universal machine have logical limitation, and Löbian machine are the one knowing that they are universal, and the limitations this entails. The price to pay for being universal, as I have explained a lot some years ago, is that you need the ability to search numbers which might not exist, and this never stop, without knowing it for sure. Machines are like kids, we can forbid them to ever escape some collection of total computable function (security), but then they cannot be universal (liberty). Löbian machine are aware of that tension between security and liberty, from the start. > >> >>> > ** >>> >>> >>> *The question I want to ask is has Hod Lipson built a Loebian machine in >>> physical matter?* >> >> >> There is no way I can ever know if >> Hod Lipson >> 's robots are self aware, I don't even know if >> Hod Lipson >> is self aware, all I know for sure is that both behave intelligently. >> > > His argument is that his robot is self-aware, for some operational > definition of self-aware. Of course this claim is bound to be > controversial. Regardless, I'm curious as to the relationship between > that and Loebianity. >From the video, it is hard to say. If they can find induction rules, which are >often the base of learning, and planning in AI, then there will be Löbian if >they can reason and talk in classical logic. Bruno > > > -- > > ---------------------------------------------------------------------------- > Dr Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders > Visiting Senior Research Fellow hpco...@hpcoders.com.au > Economics, Kingston University http://www.hpcoders.com.au > ---------------------------------------------------------------------------- > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
