Solvability of the halting problem for certain classes of Turing machines

1973 ◽  
Vol 13 (6) ◽  
pp. 537-541 ◽  
Author(s):  
L. M. Pavlotskaya
Keyword(s):  
Turing Machines ◽  
Halting Problem

The unsolvability of the uniform halting problem for two state Turing machines

1969 ◽  
Vol 34 (2) ◽  
pp. 161-165 ◽  
Author(s):  
Gabor T. Herman
Keyword(s):  
Finite Number ◽  
Turing Machine ◽  
Decision Procedure ◽  
Turing Machines ◽  
Halting Problem

The uniform halting problem (UH) can be stated as follows:Give a decision procedure which for any given Turing machine (TM) will decide whether or not it has an immortal instantaneous description (ID).An ID is called immortal if it has no terminal successor. As it is generally the case in the literature (see e.g. Minsky [4, p. 118]) we assume that in an ID the tape must be blank except for some finite number of squares. If we remove this restriction the UH becomes the immortality problem (IP).

High and low Kleene degrees of coanalytic sets

1983 ◽  
Vol 48 (2) ◽  
pp. 356-368 ◽  
Author(s):  
Stephen G. Simpson ◽  
Galen Weitkamp
Keyword(s):  
Characteristic Function ◽  
Cross Section ◽  
Finite Time ◽  
Infinite Sequence ◽  
Turing Machines ◽  
Jump Operator ◽  
Halting Problem ◽  
Computing Machine ◽  
Definition Of ◽  
Countably Infinite

We say that a set A of reals is recursive in a real y together with a set B of reals if one can imagine a computing machine with an ability to perform a countably infinite sequence of program steps in finite time and with oracles for B and y so that decides membership in A for any real x input to by way of an oracle for x. We write A ≤ yB. A precise definition of this notion of recursion was first considered in Kleene [9]. In the notation of that paper, A ≤yB if there is an integer e so that χA(x) = {e}(x y, χB, 2E). Here χA is the characteristic function of A. Thus Kleene would say that A is recursive in (y, B, 2E), where 2E is the existential integer quantifier.Gandy [5] observes that the halting problem for infinitary machines such as , as in the case of Turing machines, gives rise to a jump operator for higher type recursion. Thus given a set B of reals, the superjump B′ of B is defined to be the set of all triples 〈e, x, y〉 such that the eth machine with oracles for y and B eventually halts when given input x. A set A is said to be semirecursive in y together with B if for some integer e, A is the cross section {x: 〈e, x, y 〉 ∈ B′}. In Kleene [9] it is demonstrated that a set A is semirecursive in y alone if and only if it is

scholarly journals Translatability Problem of Geodesics on Algorithmic Manifolds

2016 ◽  
Author(s):  
Takuya Yabu
Keyword(s):  
Turing Machine ◽  
Optimal Algorithm ◽  
Turing Machines ◽  
Halting Problem ◽  
End Point ◽  
Np Problem

In the previous paper, I define algorithmic manifolds simulating deterministic Turing machines and by determining the start point and end point of the algorithm in a P problem on the algorithmic manifold, there is the optimal algorithm as the length minimizing geodesic between the start point and the end point, and the length minimizing geodesic can be derived by determining the start point and the end point also in a NP problem.　In this paper, I show that the possibility of translating algorithms from geodesics on algorithmic manifolds is equivalent to the halting problem of Turing machine. I will also discuss the problems of translating from geodesics using existing algorithms.

Many-one degrees associated with problems of tag

1973 ◽  
Vol 38 (1) ◽  
pp. 1-17 ◽  
Author(s):  
C. E. Hughes
Keyword(s):  
Formal Proof ◽  
Extensive Study ◽  
Decision Problems ◽  
Turing Machines ◽  
Natural Numbers ◽  
Halting Problem ◽  
The Family ◽  
Form Part ◽  
The Many ◽  
Degrees Of Unsolvability

Tag systems were defined by Post [9], [10] and have been studied by a number of researchers including Minsky [7], Maslov [6] and Aanderaa and Belsnes [1]. In their recent paper Aanderaa and Belsnes demonstrated that every r.e. many-one degree (exclusive of the degree of the empty set) is represented by the general halting problem for tag systems, that is, by the family of halting problems ranging over all tag systems. Their result depends upon an informal proof of this property for Turing machines but may be seen to be correct in light of a formal proof due to Overbeek [8]. Our aim is to extend their results to the general word problem for these systems. Specifically, we shall present an effective method which, when applied to an arbitrary r.e. set S, where S is neither empty nor the set of all natural numbers, produces a tag system R′ whose word and halting problems are both of the same many-one degree as the decision problem for S. The proof is realized by first constructing, from the description of an arbitrary Turing machine M, which machine has at least one mortal and one immortal configuration, a 5-register machine R, whose word and halting problems are both of the same many-one degree as the halting problem for M. From R we then construct the desired tag system R′. This construction combined with Overbeek's [8] shows that every r.e. many-one degree (exclusive of the degrees of the empty set and the set of all natural numbers) is represented by the general word and halting problems for tag systems. Moreover our results are seen to be best possible with regard to degrees of unsolvability in that it is not the case that every nonrecursive r.e. one-one degree is represented by either of the general decision problems for tag systems which are considered here. These results were first shown in the author's thesis [3] and were announced in [4], They form part of an extensive study into the many-one equivalence of general decision problems. An overview of the initial findings of this research project may be found in [5].

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