On the complexity of the ℰ2 Grzegorczyk class

Abstract

We define several abstract computing devices which allow us to characterise the E 2 class of the Grzegorczyk hierarchy. For each of these devices, we estimate the time needed to compute functions of the class E 2 . Grzegorczyk introduced in [11] a classification of the set of primitive recursive functions which have found a widespread use. Among the classes E n of this classification, the classes E 2 and E 3 are of particular interest. The class E 3 is the class of functions which are elementary following Kalman's classification and the class E 2 , which is simpler than the class E 3 , can be considered as the key class of the classification. For example, the class E 2 contains the majority of the arithmetic functions which are used in actual mathematical practice (excepting, maybe, some functions of exponential growth). The class E 2 has various machine descriptions, and these descriptions are easily extended to the other Grzegorczyk classes. Finally, the class E 2 has a finite basis with respect to superposition, and this result is also easily extended to the classes E n for n > 2. However, neither positive nor negative results on the finiteness of the basis are known for the classes E 0 and E 1 .In this paper, we would like to call attention to the complexity of computation of the functions of the Grzegorczyk class E 2 . As a rule, by the complexity of computation of a function is meant the time complexity or the capacity complexity of some type of computing device. The main results for the class E 2 have been obtained either for the Minsky machines themselves or for computing devices close to the Minsky machines [6,7]. For these purposes, no other computing devices have been used. However, during the last forty years new computing devices which can be used for investigation of the class E 2 were suggested, and it was found that some known devices could be also used.Below we define a series of computing devices with the use of which it is possible to represent the Grzegorczyk class E 2 . We observe that for each of these devices the