this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject

We extend the definition of dynamic ordinals to generalised dynamic ordinals. We compute generalised dynamic ordinals of all fragments of relativised bounded arithmetic by utilising methods from Boolean complexity theory, similar to Krajíček in [14]. We indicate the role of generalised dynamic ordinals as universal measures for implicit computational complexity. I.e., we describe the connections between generalised dynamic ordinals and witness oracle Turing machines for bounded arithmetic theories. In particular, through the determination of generalised dynamic ordinals we re-obtain well-known independence results for relativised bounded arithmetic theories.

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a formalized no-gap theorem

Definable functions Language of Bounded Arithmetic (BA) Language of first order arithmetic similar to Peano Arithmetic Non-logical symbols: {0,1,+, ·, ≤} + {|.|,#,...} |x | = length of binary representation of x x#y = 2 |x|·|y | produces polynomial growth rate Arnold Beckmann (joint work with Klaus Aehlig)