The Halting Problem

One of the classic unsolvable problems in computability is the halting problem. In universal computation systems, there are ways to cause computations to repeat themselves. However, this leads to a possible problem—if a function is poorly written, the function may get caught in a repetitive portion and not be able to leave. This computation would be a non-halter, and therefore, left to itself, would never complete. Most familiar computations are halting computations, as demonstrated in the following computer program. All programming examples are given in JavaScript for readability. 

This program defines a function called double which obviously doubles its input. It creates a temporary variable called y to hold the result of the computation and then returns y as the final value for the function. So, after defining it, the function can be used by saying double 4) which would give 8, or double z) which would take the value currently denoted by 2 and return whatever is double of z. 

The next example will demonstrate the operation of a loop. This program computes the factorial of a number which is the result of multiplying a number by all of the numbers below it down to 1. For instance, factorial 5) is5 4 3 2 l. factorial(3) is 3 2 1. So, the number of computations performed, while always finite for a finite number, varies with the value given. A typical way to program a factorial function follows  

Using Turing Oracles in Cognitive Models of Problem-Solving 

The next statement returns the value in val as the result of the entire computation. Thus, since val currently holds 6, this function returns 6 as the result of factorial 3), which is the correct result. Since it does eventually return a value, it is considered a halting program. It will take longer to return a value if the input is bigger (since it has to run the loop computation process more times), and it will return invalid values if the input is less than one (or not an integer), but it will always return a value. Therefore, since it will always complete in a finite number of steps, it is a halter. 

---

The halting problem thus asks whether it is possible to tell in advance if a general program will halt its execution or, put another way, whether an arbitrary program s... 

Considerations of the real sense of the Gibbs Paradox are used to illustrate the idea of the term step-aside which is our main methodological tool for looking for the infinite cycle in a Turing computing process and which enables us to avoid the commonly used attempts to solve the Halting Problem. 

Minski s proof for the undecidability of the Halting Problem (Entscheidungsproblem type). 

The Auto-Reference step that is to solve the Halting solve the Halting Problem proves, only, its own disusability creates just a certain image of what is to be possibly discovered - the infinite cycle in the form of the infinite constant time sequences [when the time expansion (58) for p>l is considered). 

The second type of the unsolvable or undecidable problems are those which are given mistakenly by having an Auto-Reference embedded. They are the paradoxes, which invokes the infinite cycles when they are solved and just for this they are reducible to the Halting Problem their... 

Notice that the emptiness problem for LBC (which is equivalent to the halting problem for multicounter machines) is known undecidable. According to Theorem 4, for (single) RBCS, the emptiness problem (i.e., Lang(M, L) = 0 for a given RBCS M and regular bond type L) is undecidable as well. Hence,... 

Calculating Software Complexity Using the Halting Problem... 

In this example, since multiplier is never decreased, then, for any input greater than 1, this function will never stop computing Therefore, in terms of the halting problem, it doesn t halt. 

As mentioned previously, if humans are able to solve incomputable functions, then the physicalism hypothesis is false. The halting problem makes a good test case for this idea because it is one of the most widely studied class of incomputable problems on both a theoretical and a practical level. 

One may then conclude from the experience of the process of programming that significant evidence exists that humans are able to at least solve a similar problem to the halting problem. However, there are some important caveats. 

As an example, one could solve the halting problem on fixed-size memories using a counter (Gurari, 1989). Since the number of possible machine states is 2 , then if machine states are counted,... 

Therefore, if the above program halts, then there is an odd perfect number. If it does not halt, then there is not one. However, no human currently knows the answer to this question. Therefore, whatever it is that humans are doing, it is not directly knowing the answer to the halting problem. 

Bringsjord s group has considerable experience with the halting problem, but... 

Q—a decision problem (such as the halting problem) p—a program i—the input to program p... 

Second, specific reasons must exist in order to believe that the proposed process is incomputable. Since solving the halting problem is known to be incomputable and adding axioms is incomputable by definition (otherwise they would be theorems), then specific evidence indicates that the proposed process is incomputable. 

Bartlett, J. (2014). Calculating software complexity using the halting problem. In J. Bartlett, D. Halsmer, M. R. Hall (Eds.), Engineering and the ultimate (pp. 123 130). Broken Arrow, OK Blyth Institute Press. 

Chaitin, G. (2007). The halting problem omega Irreducible complexity in pure mathematics. Milan Journal of Mathematics, 75(1), 291-304. Available from http //www.cs.auckland.ac.nz/ chaitin/mjm.html... 

---