We are in mid-January 2021 when a report from the Max Planck Institute appears in the Journal of Artificial Intelligence Research which highlights how a Super AI equipped with self-learning capabilities (super algorithm) could escape the control of its programmers and, virtually/hypothetically, lead the Gender Homo Sapiens facing extinction. In this article we intend to highlight the theoretical non-existence (Alan Turing Incompleteness Theorem) of an algorithm that can decide whether an AI can act against Humankind.
Turing’s theorem: premise
This location is presumably unsuitable for even an elementary illustration of Gödel and Turing’s Theorems due to their intrinsic logical complexity; what, on the other hand, is interesting is how both theorems make use of self-referentiality.
Use: Self-referentiality = To Dialle (diallelos from Greek = circular reasoning) indicates a logical pseudo-demonstration where the conclusions derive from the initial hypotheses which, in turn, are obtained directly from the conclusions. In common lexicon: “snake that eats its own tail”
Liar paradox
The information reported on this site is always false.
It is well known, since its enunciation (Eubulides of Miletus 4th century BC), how this statement inevitably leads to the Paradox of the Liar. Here it is interesting to highlight its self-referential-circular content (potentially recursive) which leads to subsequent stages of the type:
if (the information is always false = A) then (the information that the information is always false is also false = B) if (the information that the information is always false is also false = B) then (it is true that the information is always false = A )
0 = false 1 = true
if A= 0 then B = 0
if B = 0 then A = 0
Symbolically A leads to B and B in turn leads to A.
The Liar’s Paradox has been proposed, over time, in various logically identical although formally different forms by a great variety of authors: Aristotle, Diogenes, Buridan, Cervantes, etc. The solution to the paradox proposed by Aristotle hypothesizes that it is generated by the confusion between a term (falsify) and its citation (false) (which adds no logical contribution) to the first term.
William of Ockham argues that the paradox is generated by the fusion between language and metalanguage where the sentence “information is always false” is written in metalanguage. Likewise for Alfred Tarski (1944) that the Paradox is generated by autonymy (a metalanguage that still indicates oneself).
Richard’s paradox
Richard’s Paradox (Jules Antoine Richard – 1905) is of much greater importance. Consider rigorously and completely defining all the properties of Rational Integers. For example:
prime number = divisible only by itself and by unity perfect square = product of a number by itself even number = divisible by two irrational number = cannot be represented as a fraction of rational integers etc.
For any axiomatization of Arithmetic the number of properties of Rational Integers is in finite quantity so that their ordering according to an arbitrary criterion and the consequent progressive numbering will lead to a finite numbered sequence. For example:
1-integer number = union between positive and negative natural numbers 2-perfect square = product of a number by itself 3-odd number = not divisible by two (set ℕ) 4- ……………………………… ………………………………………………………………………………… ………………………………………………… …………………………………………………………………… 17- prime number = divisible only by the unit and by sex
We immediately notice that some numbers in the sequence possess exactly the property with which they are associated. For example:
1 has exactly the property of being a positive natural 3 it is not divisible by two (remaining in the set ℕ) 17 has the property of being divisible only by itself and by unity
while other numbers are associated with properties that they do not possess. For example: 2 ≠ “is a perfect square”. Now the numbers that do NOT possess the property to which they are associated are defined as Richardians while, obviously, those that possess it are defined as NOT Richardians.
The property of being or not being Richardian is a property of Natural numbers therefore it too can be part of the list of properties of Naturals and in this list it will have a number R = being a Richardian number. Is R a Richardian number or is it not? This is Jules Richard’s paradoxical problem.
If R were Richardian it should not be associated with the property of being Richardian while if it were not Richardian it could not be associated with the property “being a Richardian number”. Naturally “to be or not to be Richardian” is a NON Arithmetic property of the Natural Numbers therefore it could not be admitted into the countable and numbered list of mathematical properties (of any axiomatization of Arithmetic); this reduces Jules Richard’s pseudo proof to a Paradox.
If R is Richardian then it does not have Richardian properties
if it does not have the properties of Richardians then it cannot enter the list of Richardians
Self-referentiality appears evident when R = Richardian is used to demonstrate that R ≠ Richardian while circularity is highlighted by the previous mini-graph.
Turing’s Theorem
Alan Turing’s Theorem affirms the existence of algorithms whose calculability (truth) cannot be established because they generate an infinite loop (Stopping Problem).
Let A be an algorithm operating on a given input; the Theorem concerns the possibility of establishing whether A(d) (calculability of A on the given d) stops in a finite time, i.e. after a finite number of steps). Let there then be a second AR algorithm implemented on a digital meta language, such that:
AR (A(d)) has as output V= true or F= false
that is, an AR algorithm that establishes through an output (True) that A(d) will end in a finite time, or establishes through the output (False) that A(d) will produce an infinite loop.
Shortly
if AR (A(d)) = true then A(d) terminates (it stops in a finite time) if AR (A(d)) = false then A(d) does not terminate (it does not stop in a finite time) .
Since both A and d are strings of symbols of a meta-language having as alphabet 0, 1 (in the case of a binomial digital alphabet) and therefore mutually exchangeable (not recognizable as program and input but only as digital signals) it is possible to operate on l ‘AR algorithm obtaining AR ((A,(A))
AR (A(A)) true if A(A) ends for the relation (1) AR (A(A)) false if A(A) does not stop in a finite time for the relation (2)
Now repeat the same operation performed for the creation of (A,A) creating NW(AR, AR) which gives the Output “true” if AR (A (A(d)) = does not terminate (as per (2)) therefore: NW (AR, AR) is a program that defines itself as “true” if AR defines itself as “false” which, in turn by relation (2) defines A(d) as an algorithm that does not terminate (infinite loop). Conclusion: There cannot be an algorithm that ends with a True or a False (Stop).
The demonstration technique is that of self-referentiality, powerfully used in the evaluation of AR(AA) and NW(AR, AR) where both the AR algorithm evaluates itself as a function of the A algorithm and the NW algorithm.
Conclusion on Turing’s Theorem
(Where, as usual, nothing is concluded but merely highlights some curiosities of human pseudo logic and the pseudo power of a SuperAI).
At the end of the 1930s Alonzo Church and Alan Turing published the result of their work (conducted separately and independently but ultimately unified with an identical result) aimed at solving the now long-standing 10th Hilbert problem and, in parallel, the Super AI problem. Their conjecture goes:
If a problem is humanly calculable, then there will be a Turing machine capable of solving it.
Alan Turing proved that there is no algorithm, much less any machine, that can solve anything that the human mind can solve. Even more analytically: if we wanted to create an algorithm that establishes a priori whether the data processing and self-learning programs of SuperIA could be a danger for the Homo Sapiens genus, we would obtain as a result that this hypothetical algorithm (implemented on a Turing Machine does not deterministic) it would go on forever without ever giving a True or False answer.
dr.ing. Alberto Sacchi, graduated in electrical engineering in 1962 from Polimi, former top manager of numerous SMEs of US and CH groups, passionate scientific communicator, author of dozens of technical publications in sector magazines and scientific-popular essays in mathematics and mathematics periodicals physics and science fiction stories and books. The experience acquired in both the engineering and management fields allows him to provide a unique contribution in the field of scientific dissemination.