Annotated Bibliography Example on Hypercomputation

The concept of hypercomputation is new in the field of computation, dealing with computing methods and tools that play along with the Church-Turing thesis. The Church-Turing thesis shows that there is an the ease in computation of any function that is termed as computable and that which can be computed normally through a pen and paper can also be solved using Turing machines as long as there is a finite set of algorithms. With hypercomputation as the new technology in science, various challenges including those regarding halting can be solved.

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Michaelson, Greg, and Paul Cockshott. "Constraints on Hypercomputation." Logical Approaches to Computational Barriers Lecture Notes in Computer Science, 2006, pp. 378-387., doi:10.1007/11780342_40.

A research by Michaelson & Paul has stated that hypercomputation has become a developing challenge in the computer science field (Michaelson & Paul). It is through hypercomputation that the Turing model is surpassed and the von Neumann architecture is exceeded. Algebraic constructions are known to lead to a viable pseudo-recursive equation theory. However, each given number does not seem recursive as the equations written in the variables of the given numbers form a recursive set. The concept of hypercomputation is a requirement as long as an algorithmic answer is expected to solve the membership problem of the recursive set.

Arkoudas, Konstantine. "Computation, Hypercomputation, and Physical Science." Journal of Applied Logic, vol. 6, no. 4, 2008, pp. 461-475., doi:10.1016/j.jal.2008.09.007.

According to Arkoudas, one of the tools that can solve the challenge of hypercomputation is that which can address the halting problem (Arkoudas). The main advantage of using such a tool is that hypercomputers can simplify tasks that a Turing machine cannot. The Church-Turing sense cannot be effectively used to solve the equations solvable by hypercomputers. Turing machines tend to produce incomputable data even as most of the hypercomputing literature tends to focus on the random, incomputable functions instead of useful computations. The models of hypercomputers range from useful and unrealizable to the less useful and generators of random functions. There are hypercomputers that have incomputable functions. However, these machines cannot effectively solve the useful incomputable functions including the halting problem.

Ziegler, Martin. "Real Hypercomputation and Continuity." Theory of Computing Systems, vol. 41, no. 1, July 2007, pp. 177-206., doi:10.1007/s00224-006-1343-6.

Yet it is impossible to come up with a machine that solves the halting problem under the laws of quantum mechanics, Ziegler seeks to come up with such a tool through some bizarre laws of physics including measurable physical constant with an oracular calve including the Chaitin's constant. It is evident that a symbol can be computable in the limit as long as there is a finite, non-halting program on a Turing machine. This program should have the intent of outputting each symbol embedded in the sequence including the computable real. Even though Turing machines cannot be understood as problem-solvers of their first outputs, generalized Turing tools can. The symbol sequence is that which has a finite program that runs smoothly on a generalized Turing machine in the sense that any output symbol reaches a converging point by itself and through a halting program, hence, a solution to a halting problem (Ziegler). Generalized Turing machines can only come up to a correct solution of the challenge through an evaluation of the existing sequence. This proposal uses a tool that can read the advice function to the accomplishment of the task. This tool can be placed in a previous classical machine and under the common assumptions, one can predicate another club.

Lindell, Steven, "Revisiting Finite-Visit Computations." 2004, Available at http://www.haverford.edu/cmsc/slindell/Presentations/Revisiting finite-visit computations.pdf

A research by Lindell has seen various energy requirements into physically built machines to shed new light on computing and computability. However, depending on the conditions, conventional mathematical models tend to be realizable as a result of the availability of funds, which is required for storing the information and execute instructions. On the other hand, the same author by in 2006 notes that employing the Turning's model creates an autonomous situation as it is grounded on the various psychological advances of the natural calculator by man. This idea tends to showcase its ability to change the state of mind, especially in a stepwise fashion as a result of the realization of symbolic configurations. The aim of employing this may be to lead a mathematical model of the best of an effective computability, along with the best capacity and limitations. In that case, scientists should be able to extend their analysis to the idealized psychological characteristics of an effective machine computer. One of these characteristics is that which the machine computer has the ability to manipulate matter through the employment of energy. This may, in turn, a new intention where feasibility studies. In the development of the tool, the machine can be able to understand the ability of a person. The usual notions of space and time complexity can be challenged through mass and energy constraints as soon as hypercomputation is employed.

Sharma, Ashish. "Algorithms Simulating Natural Phenomena and Hypercomputation." 2016 IEEE Students Conference on Electrical, Electronics and Computer Science (SCEECS), 2016, doi:10.1109/sceecs.2016.7509337.

According to Sharma, there are several attempts that aim at transcending the basic limitations to computability as implied by a halting challenge for Turing machines. These attempts may be dependent on the use of the various existing modes of hypercomputation that attract the goals of infinite or continuous computation. In that case, these schemes could encompass the deployment of actualized infinities of physical resources or representations of real algorithms (Sharma). In that case, the various cases of hypercomputation could not just be naturally realizable hence; no new forms of calculability can be constituted.

The halting problem can be addressed by hypercomputers that access to a resource that allows the challenge to be solved. The decision made by the hypercomputer can be adjusted to decide on whether there could be termination of a standard Tuning machine. The machine could be designed in such a way that it could analyze itself in a way that has to halt in conditions that involve halting and running. There should, however, he the vanishing in a fury of contradiction by the mythical machine.

Syropoulos, Apostolos. Hypercomputation: Computing Beyond the Church-Turing Barrier. New York: Springer, 2006. Print.

Syropoulos notes that Trial and error machines tend to draw the capability of the ordinary Turing machines hence, should be able to create a solution for the halting challenge.

In the tool, a certain program can also be embedded. When there is a program, an input and an output, a simulation can always be possible to make. However, of the simulation halts, the whole system can be seen feasible. In a clear mode, the last output should always be addressing the halting challenge as soon as one gets to accept the results just as they arrived at a limit. This does not tend to prove that limiting a computation can perform no ordinary things, nor should they be recursive, but all about computation.

Hypercomputation is, therefore, a clear way of convincing oneself that one is completely possible, just like in obvious confusions. For instance, as long as one is not convinced that a certain number is not a real number and hence notice that it later would be non-computable, only to notice that a Turing machine could be the best producer of the answer. Even though it is physically implausible to make the idea of such a measurement, the mistake can be torn up between certain ways including talking about traditional computers and making assumptions that the components can take a real value.

References

Arkoudas, Konstantine. "Computation, Hypercomputation, and Physical Science." Journal of Applied Logic, vol. 6, no. 4, 2008, pp. 461-475., doi:10.1016/j.jal.2008.09.007.

Lindell, Steven, "Revisiting Finite-Visit Computations." 2004, Available at http://www.haverford.edu/cmsc/slindell/Presentations/Revisiting finite-visit computations.pdf.

Michaelson, Greg, and Paul Cockshott. "Constraints on Hypercomputation." Logical Approaches to Computational Barriers Lecture Notes in Computer Science, 2006, pp. 378-387., doi:10.1007/11780342_40.

Sharma, Ashish. "Algorithms Simulating Natural Phenomena and Hypercomputation." 2016 IEEE Students Conference on Electrical, Electronics and Computer Science (SCEECS), 2016, doi:10.1109/sceecs.2016.7509337.

Syropoulos, Apostolos. Hypercomputation: Computing Beyond the Church-Turing Barrier. New York: Springer, 2006. Print.

Ziegler, Martin. "Real Hypercomputation and Continuity." Theory of Computing Systems, vol. 41, no. 1, July 2007, pp. 177-206., doi:10.1007/s00224-006-1343-6.