Friday, March 25, 2005

I'll lead a "real" life when my school gives me a bit more of the "real" time it steals from me ;-)

In the meantime, while I commute to and fro opposite corners of town, I'll continue to study and make weird, unprovable conjectures on the side.

In fact, after some time, I've reformulated my thoughts on the possibility of machines beyond Turing ...
let's call them Transcendental Machines with the following properties:

1.) A Transcendental Machine (TDM) (if it exists) recognizes/decides all languages recognizeable/decidable by Turing Machines, (regular languages, context-free languages, Turing-recognizable languages) and at least one or more non-Turing-decidable/recognizable..

2.) While TDMs can recognize and simulate Turing Machines (TM), The opposite is not necessarily true. As far as modern Theory of Computation is concerned (and as far as it should be concerned, realistically), the Turing Machine is the most versatile model of computation. But ironically, if TDMs do exist, then Turing Machines (and thus humans) may not be able to direcly detect them ... or even prove their existence as a certainty within Turing-recognizable rules).

3.)Due to the fact that I'm exploring these possibilities, I'm forced to accept the possibility that a Turing Machine can still indirectly describe higher-level machines, and that given the proper resources, perhaps a Turing Machine could even simulate a TDM.

4.) There may be even more higher-level supersets of machines than TDMs: These would be considered TDMs of Order 2, Order 3, etc - but there is a catch: while such a conjecture is easily plausible, the main problem with TDMs is that the set would hold rather large subclasses of various machines that recognize/decide wildly varying sets of Turing-unrecognizable/undecidable languages, each unioned with all Turing-recognizable/decidable languages. In such a case, ordering of sets would by necessity, also make sense on a per-subset basis, although some TDM classes might even be uncountably infinite.

In general, let us temporarily assume there exists a being/machine (who can do what we can't, essentially) who can determine the existence of integer roots from a multi-variable polynomial in the set of real numbers, given a finite time limit (preferably a short one) - then we have an example of a transcendental machine.

Regarding giving such an unwieldly name to these machines, I feel justified due to the definition(until someone gives a better name :-D after all, these machines may already have a name), since the term "transcendental" has also been given to a specific set of irrational numbers such as pi. The preliminary name was "Chaos Machine" but such a name conveys the scientific notion of a Chaotic system as described by Chaos Theory, which can be recognized by Turing Machines.