Church turing thesis relevant proofs non computability

The classical version speaks about classical a. It is the classical thesis that is addressed in the Dershowitz-Gurevich paper. Not all classical algorithms are covered by Turing's analysis. Ruler-and-compas algorithms are classical, Gauss elimination procedure is classical. Classical algorithms may use randomization,. I axiomatized classical algorithms article at my website and proved that every classical algorithm can be step-for-step simulated by bona fide algorithms. The Dershowitz-Gurevich paper added one additional axiom to tame classical algorithms and to derive the thesis.

The axiomatization of classical algorithm was extended by Andreas Blass and myself to that of synchronous-parallel algorithms [article at my webpage]. Some specialist would know for sure I've been wanting to ask , but I think current physical theory whether it really describes Nature or not allows generation of unlimited amounts of Kolmogorov-random data through quantum processes, which can't be simulated with a Turing machine. As is well known, the absence of algorithmic solutions is no obstacle when the requirements do not make a solution unique.

A notable example is generating strings of linear Kolmogorov complexity, e. Algorithms fail, but a set of dice does a perfect job! I've been wondering for a while based on Levin's example about a thought experiment: Fix a universal Turing machine U. Flip a coin 2 million times and call the resulting random bit string S. Let P be the proposition that there is no program for U less than 1 million bits long, that writes S onto the tape i.

P says that S's Kolmogorov complexity is more than 1 million bits. There is of course some very small probability that P is false you might have flipped 2 million heads completely by chance , but P is almost certainly true. And you could always use 2 billion flips instead of 2 million. So you've got a simple experimental apparatus that can create unlimited amounts of unprovable but almost certainly true mathematical statements. This can't be done with classical algorithms. So where does our creaky old physical universe get such knowledge? There's a very nontechnical paper by Geroch and Hartle, " Computability and Physical Theories ," which discusses the question of whether there are dimensionless physical quantities that are measurable i.

They argue that while currently accepted physical theories produce only computable quantities, there are quantities that we can measure, like the fine structure constant, which are not specified by current theories, and that these could potentially be measurable but not computable. They also give an example of a formulation of quantum gravity that could potentially give rise to noncomputable physical quantities. As an aside, ever since I read this paper, I've never been able to understand the third-to-last paragraph.

In other words, in a world with noncomputable physical quantities, we can refer to experiment as an oracle in our computations. But this seems to beg the question, since by definition, our quantity is computable by an "experiment" computer. Anyway, I'm far from an expert in any of these matters, so I could just be completely missing their point, but I'm putting this out there in case anybody can enlighten me on this. Robin Gandy once wrote a paper listing axioms about physics that implied the Church-Turing thesis. I'm sure you could find it by searching for all articles written by him.

Also, Frank Tipler once wrote a paper claiming that relativity refutes the Church-Turing thesis. I saw this as an unpublished manuscript and I do not know if it ever appeared. The argument went this way: Send a rocket into space. The rocket A carries a computer carrying out an infinite search which will take infinite time to complete. The trajectory leads into a black hole. The point is that it takes infinite time in the rocket's frame to fall into the black hole. When the search succeeds, it radios back the report of success.

But in an external observer's frame, the entire fall into the black hole will take finite time. So if you don't get the report, the infinite search failed. Of course we have to neglect the difficulty of building a computer that lasts forever in this philosophical analysis.

PHYS Lecture 4: Minds and Machines

I believe that for some reason a second rocket was used to relay the result. I don't know if the physics is correct or not but the logic of the argument is OK. On the issue of "generation of unlimited amounts of Kolmogorov-random data through quantum processes," remember that a dice throw, or data generated via any classical chaotic process, has the same Kolmogorov complexity as the initial conditions since deterministic laws of classical dynamics have small K-complexity.

2. Misunderstandings of the Thesis

Quantum mechanics appears to be essential if you want to generate K-random sequences from non-random initial data, as I discussed eg in "Complexity, Vol. The most surprising point is that the result depends on non-Newtonian physics, in particular the impossibility of instantaneous action at a distance. I would answer that the question is uninteresting, because even if single physical devices were restricted to calculating computable functions, interacting or communicating physical devices are not.

This can be seen by considering two devices which are able to communicate. Both devices are Turing-like and have tapes. One device is programmed to space to the right until it reaches the end of its input. It then communicates with the second device and outputs what it has communicated. The second device alternates between two states: ready to communicate 0 or ready to communicate 1.

The behaviour of the two devices together is time-dependent. When applied to physics, the thesis has several possible meanings:. There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept. One can formally define functions that are not computable.

The physical Church–Turing thesis and non-deterministic computation over the real numbers

A well known example of such a function is the busy beaver function. This function takes an input n and returns the largest number of steps that a Turing machine with n states can execute before halting, when run with no input. Using particular models of Turing machines, researchers have computed the value of this function for small values of n : 0 through 4. Simulations of Turing machines with 5 and 6 states have been performed, but without conclusive results.

For higher values, only lower bounds have been given. Finding an upper bound on the busy beaver function is equivalent to solving the halting problem , a problem known to be unsolvable by Turing machines. Since the busy beaver function cannot be computed by Turing machines, the Church-Turing thesis asserts that this function cannot be effectively computed by any method. Mark Burgin, Eugene Eberbach, Peter Kugel, and other researchers argue that super-recursive algorithms such as inductive Turing machines disprove the Church—Turing thesis.

Their argument relies on a definition of algorithm broader than the ordinary one, so that non-computable functions obtained from some inductive Turing machines are called computable. This interpretation of the Church-Turing thesis differs from the interpretation commonly accepted in computability theory, discussed above.

The argument that super-recursive algorithms are indeed algorithms in the sense of the Church-Turing thesis has not found broad acceptance within the computability research community. Sign In Don't have an account? Please help to improve this page yourself if you can.. Contents [ show ]. He calls this "Church's Thesis", a peculiar choice of moniker. Kleene would, in Kleene in Davis , present a way to construct functions not recursive. These axioms are found by analysis or by experience i. We remark that, when extending the notion of a computable function to real numbers, we do not extend the execution modalities of algorithms. We are still using the same execution modalities—Turing machines, rewrite systems, etc. Simply, these computation mechanisms are now used with rational numbers approximating the real numbers. The thesis that real computable functions defined in such a way are sufficient for describing the laws of nature, or that an analogue machine cannot compute a function that would exceed this notion of computability, implies that all that an analogue machine can compute can also be computed by a digital machine, or that a physical process cannot access in a finite time the infinite amount of information contained in a real number.

This can be reformulated in more physical terms as the fact that a physical process cannot be used to identify, in a finite time, the difference between two close enough magnitudes. This definition of computability over real numbers can be generalized easily to the notion of a computable partial function over real numbers.

In the second case, the attempt to obtain any approximation of f x by providing more and more accurate approximations of x leads to a computation that does not terminate: the function f is not defined at x. We can prove in this way that the partial characteristic function of the set is computable.

We can, in this way, express formally what is often stated informally: if x is strictly positive real, then there is an effective way of proving it. We can prove that the domain of a partial computable function is always an open set.

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Thus, if we call semi-decidable a set whose partial characteristic function is computable, then all semi-decidable sets of real numbers are open. We have extended above the notion of computability of functions operating on natural numbers to relations between natural numbers. We can try to extend, in a similar way, the notion of computability of functions operating on real numbers to relations between real numbers.

To guide us in the choice of an extension of the notion of computability of functions operating on real numbers to relations between real numbers, we shall require, as a minimal condition, that the proposed definition extends the notion of a computable function, that is, that a function operating on real numbers is computable as a function if and only if it is computable as a relation. Representing the set R x by its characteristic function leads to representing a relation R by its characteristic function as well. As the computable functions over real numbers are always continuous, the relations that have computable characteristic functions are only the empty and the full relation.

Thus, this idea leads to a dead end. Representing the set R x by its partial characteristic function leads to representing the relation R by its partial characteristic function, as well. Thus, if this solution allows more relations to be represented than the previous ones, it is far from being sufficient, as the identity, which is computable as a function, would not be computable as a relation.

  • theses dissertations online.
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As the characteristic function of equality over the real numbers is not computable, the argument developed in the case of natural numbers that effectively enumerable relations are semi-decidable cannot be generalized to the case of real numbers: there are a greater number of effectively enumerable relations than semi-decidable ones. It can, for instance, be the set of natural numbers, the real line, an interval of the set of natural numbers, an interval of the real line, etc.

More generally, all computable functions g mapping real numbers to real numbers can be represented as relations, as it suffices to define the function f x , i as g x , ignoring the index i. Conversely, a functional relation that can be represented is computable as a function, because this function g associates with x the value f x , i 0 , where i 0 is an arbitrary index. It is probably the only one that can be used as a description language for the laws of physics.