Geodesic
Solvability of the halting problem for certain classes of Turing machines
Matematičeskie zametki, Tome 13 (1973) no. 6, pp. 899-909.

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One method of proving that some Turing machine is not universal is to prove that the halting problem is solvable for it. Therefore, to obtain a lower bound on the complexity of a universal machine, it is convenient to have a criterion of solvability of the halting problem. In the present paper, we establish some of these criteria; they are formulated in terms of properties of machine graphs and computations. 
@article{MZM_1973_13_6_a12,
     author = {L. M. Pavlotskaya},
     title = {Solvability of the halting problem for certain classes of {Turing} machines},
     journal = {Matemati\v{c}eskie zametki},
     pages = {899--909},
     publisher = {mathdoc},
     volume = {13},
     number = {6},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_13_6_a12/}
}
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L. M. Pavlotskaya. Solvability of the halting problem for certain classes of Turing machines. Matematičeskie zametki, Tome 13 (1973) no. 6, pp. 899-909. http://geodesic.mathdoc.fr/item/MZM_1973_13_6_a12/