Halting Problem Free Essay


The computability theory defines halting problem as the challenge of using a random computer input and program to determine whether the software will run forever or will stop at one point. As such, it is a decision that concerns computer program properties based on a fixed Turing-complete computation model. Turing is a test created in 1950 by Allan Turing. It checks the ability of machines to exhibit intelligent behavior as compared to that of humans. According to the Turing-complete model, the programs in this category include all of those written or writable in a programing language. That is broad enough to become equivalent to a Turing machine. Thus, a problem arises when deciding whether or not a program will come to a halt when a particular input is used to run it. The abstract in this framework does not have resource limitations, especially concerning the amount of time or memory necessary for the program to successfully execute [1]. As such, it can take longer just before halting due to utilizing more storage space. Therefore, the core question is whether a program can ever stop when a given input has been used.

In this paper, the focus will be on the reasons why the analysis of the finite-state systems is applicable to the halting problem. As such, it will pay attention to algorithmic calculations and the possible results they will likely post [2]. Besides, it will also explore the reasons why the impossibility outcomes related to the halting problem do not become instantiated as finite-state, resulting in practitioner perceptions about the assessment of programs that comprise of finite state space.

Halting Problem

The challenge involves a specification of an algorithm as to whether or not a specified program p will ultimately come to a stop in case a given input i. For example, there could be a program such as halt (p,i), where, in case it is terminated, a return entry True will be posted and if not then it will indicate False. Thus, a pseudo code of the halting problem could be presented as:

1 halt (p, i)

2 {

3      if (program p halts on input i)

4        return true; // p halts

5      else

6        return false; // p doesn’t halt

7 }

Impossibility Outcomes

The impossibility results could entail several proofs of the halting problem. As such, there could be disproof of the existence of the halting problem based on the difference in programs and inputs[2]. Thus, it is not always the case that an outcome will be available. Therefore, it becomes usable in modeling programs, which could produce an outcome or fail to halt [2]. The second case involves a variant that may be more unexpected for people with little knowledge of the halting problem. In fact, the earlier disproof indicates that, in its original creation, it rarely uses realistic computers which have full superior binding of the likely state space [4]. Most probably, this is due to the fact that the rebuttal is founded on an infinite number of inputs, as well as programs or machines that are different.

Direct Proof

While looking at direct proof, a computability assumption is made as a way of ensuring the recognition of present contradictions or inconsistencies. Later on, the data is used to determine whether the assumption made was true or false. However, in some instances it happens so that inconsistencies could have existed even before the computability assumption was made[1]. It may as well have originated from an unnoted, unrecognized assumption that was unstated. Besides, proofs of incompatibility have come to be considered as significant.

Reversed Halting Problem

In case one wishes to discover the sources of inconsistencies, a reverse process has been considered to be a reliable starting point [1]. That means that the process commences with the inconsistency and continues to build towards the halting problem. For example, assuming a culmination of program executes property, the reversal process can begin by defining the terms [2]. As such, it could begin by recognizing the program containing the property, which in this case will be given as D. In that case, there could be several choices, such as the culmination of the implementation or the the opposite. In case it does not culminate, it continues. That covers the two sides of proof in this particular case. While the first one is the original state, the second one acts as the reversal of the previous process, highlighting the areas that could have inconsistencies.

Reasons Why Impossibility Outcomes Fail to Apply Finite-State Systems

The computation of model of finite state space could have different possibilities, including the parametric state bound and the absolute state bound ones. The former involves theoretical prototypes, entailing finite-state computations with no imposition of tangible upper limits for the state space. For example, LBA (linear bounded automation) is a Turing machine that has a finite tape, whereby its length becomes similar to the linear function of input’s length [1]. As such, an LBA becomes of finite state without an upper limit on the length of input tape, as well as on the alphabet’s size. On the other hand, absolute state bound programs comprise of realistic computers, whereby the state space becomes limited by a concrete value. In doing all this, it becomes obvious that modeling resources become properly utilized for the assessment of the state space finiteness [3]. Besides, it should entail an additional memory that has ability to perform a project.

Importance and Concerns of Halting

The assessment of the halting problem has become an important aspect in the world of computing due to its undecidable characteristics, despite being a seemingly simple problem. As such, it has forced experts to invest their time into analyzing and understanding the challenge further. Nevertheless, that leads to multiple consequences that threaten the success of a program. For example, a general algorithm does not get to decide the truthfulness of untruthfulness of a particular statement concerning natural numbers. As such, it does not tell whether or not it is true [1]. That is caused by proposals that indicate the halting of a program based on a particular input, which may become converted to natural numbers. According to Rice’s theorem, the halting problem cannot be solved. The theory observes that decision procedures do not exist for non-trivial properties [4]. On the other hand, Gregory Chaitin’s definition of halting probability states that it is a real number thought to represent the possibility of casually created program halts. As such, the numbers are considered to entail a similar Turing degree positioned as the halting problem [5]. Besides, it has come to be considered normal and inspiring that a number may be defined with no possibility of computation. Therefore, it becomes clear that one could be in a position to clearly indicate the non-existence of an algorithm, providing the digits produced for the halting probability. However, few starting cases have become computable. 

Key Drawbacks in Halting Problem

The core challenge of the halting problem arises from a prerequisite that all the decision processes are expected to work with any program or input. However, programs act differently and could either halt on one input or many of them. Similarly, another program may function properly under such an arrangement. As such, due to these inconsistencies, it has been observed that no developed algorithms could claim to fully respond to all the questions [3]. Again, there are interpreters that have contributed to simulation of the effecting of the provided source code. As such, it indicates that the program continues to function under such circumstances.

At some point, the interpreter will be expected to stop the simulation process, which will signify the halting of the original program. Nonetheless, in case the program continues to operate properly and does not halt, the interpreter will carry on with the simulation as well [4]. Therefore, this has been considered to be a drawback, because it fails to address the shortcomings of the halting process as noted. In essence, it does not effectively respond to the ‘doesn’t halt’ situation for the programs that do not stop.


The concept of formalization of algorithm was developed by Turing during his introduction of Turing machines. Nevertheless, the outcomes have not always been explicit. That means that it can be utilized in other computational models equal is calculating power to Turing machines. Such computations include Lambda calculus, register machines, tag systems, post systems, and Markov algorithms. The process of formalization is now considered significant due to its ability to permit forthright charting of algorithms related to certain data types [2]. For example, in instances where formalism allows algorithms to describe functions over strings, their mapping should take place while allowing them to define natural numbers at the same time. The charting of the strings have become a forthright issue; however, alphabetical strings which may include n characters can get mapped through their interpretation as n-ary numerical system numbers.


The assessment of the halting problem has indicated that no single algorithm could address it in its entirety. Again, it becomes evident that the ability to solve the challenge is dependent on the input, as well as on the program. As such, it becomes difficult to have an algorithm that solves all the problems the same way. Therefore, it is evident that there exists no clear way through which a halting problem could become solved. Besides, on the issue of impossibility, it was established that disproof created under Turing construction did not show any obvious signs of their non-applicability to the finite-state system.

Moreover, the existing halting problem disproof did not disallow the survival of a total stoppage system. Moreover, the formalization has become an easy way to try and address several challenges in the halting process. However, it has been faced with its own difficulties, which make it quite challenging to fully address the problem. Such hitches arise from the alphabetical strings, which may limit the success rate of the program. Another challenge involves the presence of an interpreter whose responsibility is the stimulation of the input. However, in the reversal role, it is expected that the interpreter may stop at one point, causing the entire process to halt. As such, it does not provide a proper mechanism through which the halting problem may be solved. Therefore, despite being perceived as easy to understand, the halting problem has rightfully earned attention in the computing world due to the time it has taken to uncover a solution. Besides, there is still no reliable information indicating the challenge will be resolved soon.

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