On Invariance and Convergence in Time Complexity theory

The present paper intends to introduce a principle of invariance of "complexity hierarchies" under any "reasonable" changing of the input's dimension measure function. For a Turing machine that takes as input a word of a given length (say n) the input's dimension is n, so in fact the "dimension measure" is the identity function. We state that the set inclusion relation between the complexity class C and it's nondeterministic extension class NC is invariant under any monotonic crescent injective measure function. Apart from this there comes another statement of invariance saying that the inclusion relation is kept even if any set of Turing machine transitions, determined by some general property in all conceivable Turing machines, are not counted when determining the computation's length. As an example, we make reference to the set of transitions that move the head left or right without reading or writting -in a computer program accesing the first or the last element of an array is usually independent of the size of the array, i.e is constant, thus not depending on the size of the array ( moving through the array is not counted ). We must state that this paper was developed thinking only time complexity, although we see no reason why it couldn't be ported to space complexity also. Finally, we’ll prove that for any language there is an infinite sequence of languages from O(n) that converges to that language.