Arguably, humans require more energy per operation. So, presumably such an argument hinges upon what types of operations are performed in conducting automated proof search?
The context here is that certain problems cannot be solved by an algorithm, which doesn't necessarily translate into "cannot be performed by a machine".
It only means that there cannot be any Turing machine capable of solving them, no matter what "operations" it represents.
Either the (unreferenced) study was actually arguing that "automated proof search" can't be done at all, or that human neural computation is categorically non-algorothmic.
Grid search of all combinations of bits that correspond to [symbolic] classical or quantum models.
Or better: evolutionary algorithms and/or neural nets.
I don't quite understand the role of neural nets in that context, though. Those just classical computations in the end and should be bound by the same limits that every other algorithms are, shouldn't they?
Neuromorphic engineering has expanded since the 1980s. https://en.wikipedia.org/wiki/Neuromorphic_engineering
Quantum computing is the best known method for simulating chemical reactions and thereby possibly also neurochemical reactions. But, Is quantum computing necessary to functionally emulate human cognition?
It may be that a different computation medium can accomplish the same tasks without emulating all of the complexity of the brain.
If the brain is only classical and some people are using their brains to perform quantum computations, there may be something there.
Quantum cognition: https://en.wikipedia.org/wiki/Quantum_cognition
From "Quantum Memristors in Frequency-Entangled Optical Fields" (2020) https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7079656/ :
> Apart from the advantages of using these devices for computation [12] (such as energy efficiency [13], compared to transistor-based computers), memristors can be also used in machine learning schemes [14,15]. The relevance of the memristor lies in its ubiquitous presence in models which describe natural processes, especially those involving biological systems. For example, memristors inherently describe voltage-dependent ion-channel conductances in the axon membrane in neurons, present in the Hodgkin–Huxley model [16,17].
> Due to the inherent linearity of quantum mechanics, it is not straightforward to describe a dissipative non-linear memory element, such as the memristor, in the quantum realm, since nonlinearities usually lead to the violation of fundamental quantum principles, such as no-cloning theorem. Nonetheless, the challenge was already constructively addressed in Ref. [18]. This consists of a harmonic oscillator coupled to a dissipative environment, where the coupling is changed based on the results of a weak measurement scheme with classical feedback. As a result of the development of quantum platforms in recent years, and their improvement in controllability and scalability, different constructions of a quantum memristor in such platforms have been presented. There is a proposal for implementing it in superconducting circuits [7], exploiting memory effects that naturally arise in Josephson junctions. The second proposal is based on integrated photonics [19]: a Mach–Zehnder interferometer can behave as a beam splitter with a tunable reflectivity by introducing a phase in one of the beams, which can be manipulated to study the system as a quantum memristor subject to different quantum state inputs.
Quantum harmonic oscillators have also found application in modeling financial markets. Quantum harmonic oscillator: https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator
But there are real-world instances that contradict this hypothesis already. Humans don't exhibit Turing-computable behaviour because they did indeed solve problems that aren't solvable by Turing Machines.
I can give a very concrete example as well. Consider the following program (in Python for simplicity's sake):
def nat(x):
yield x
yield from nat(x+1)
for a in nat(2):
for b in nat(2):
for c in nat(2):
if a*a*a + b*b*b == c*c*c:
print(f"{a}^3 * {b}^3 = {c}^3")
exit()
Yet it has has been proven centuries ago that the above program won't halt.
The mathematicians couldn't have used a Turing Machine-compliant computation, so something else must be involved.
Now this of course doesn't mean that humans are generally capable of solving every problem that's not solvable by an algorithm, but it means that they can solve problems that algorithms can't solve.
Yes, there is. There's a machine which does nothing but print that exact proof.
You seem to have conflated solving an instance of a problem with solving the entire problem. Turing's theorem states that no one Turing machine will get the answer right for every instance of the halting problem, but for any instance that one specific machine gets wrong, there's another machine that also gets that instance right (in addition to everything else the original machine gets right).
One such result was that certain problems are undecidable, which means that any automated approach would be prone to become "stuck" and never terminate.
Another results showed that the complexity of general automated proof generation is exponential in nature.
Both these results mean, that it's impossible to tell whether a computation just takes a very long time (say years) or whether the problem is indeed undecidable (in which case the program would never halt.)
A human on the hand, is very well capable of solving the Halting Problem and can discern whether a problem is undecidable [1].
This is of course a corner case, but it goes to show that there are indeed limits to automated theorem proving.
A human would therefore - in principle - always need to proof that a problem is decidable in the first place before passing it to an ATP.
[1] https://en.wikipedia.org/wiki/List_of_undecidable_problems
Which humans can not do.
And yes, humans are very well capable of solving that, because humans can in fact proof whether any program halts.
This doesn't mean every human can or that it has been done for every program, but it means that humans are - in general - able capable of deciding this, while algorithms (programs) can't.
For now at least, we don't know of any counter-example to the Church-Turing thesis. Humans are certainly not able to tell whether a program in general will halt for any input it recieves. If you believe otherwise, please tell me whether the following program halts:
Foo():
if P = NP:
return true
else:
Foo()That's a logical fallacy - just because I can't show whether P=NP doesn't mean nobody can!
Turing-Church is a conjecture, not a theorem. There have been arguments made against it [1].
Also, the link you replied provides an argument of efficiency, not an argument of possibility. It is arguing that we might use things that are not Turing machines because they can solve the problem more efficiently, not because they can solve problems Turing machines could never hope to solve.
The "given any program" part is crucial.
And we aren’t doing some kind of magical logic when we humans create proofs. Sure naively general automated proof generation could be exponential in nature, but that hasn’t stopped us from mathematics research. There’s no reason it should stop computers either.
Humans can solve problems that aren't solvable by algorithms and that's why humans can - in general - decide whether a given program will halt. That doesn't mean every human can or that it has been done for every program (which would be impossible, since there are infinitely many instances anyway).
But the difference between humans and Turing Machines (computers) is, that humans can and did proof undecidable problems, while algorithms provably can't.
This is a fundamental difference and there's no way around it. So whatever might be done in ATP, it can't be algorithmic in nature or based on a Turing Machine in order to be applied universally.
Unless of course, you know of a way to disprove Gödel and Turing...
Basically, why can't I write an algorithm that can be given input for "is the halting problem decidable?" and return "no"?