In this paper we characterize normal sequences and finite-state dimension in terms of the automatic Kolmogorov complexity and finite-state a priori probability. We show that many known results about normal sequences and finite-state dimension, including the equivalence between aligned and non-aligned normality, Wall's theorem, Piatetski–Shapiro's theorem, Champernowne's example of normal number and its modifications, equivalences between different definitions of finite-state dimension, Agafonov's and Schnorr's results about finite-state selection rules, become easy corollaries of this characterization. For that we use notions of automatic (finite-state) complexity and finite-state a priori probability that are the natural counterparts of the notions of Kolmogorov complexity and Solomonoff–Levin a priori probability in the algorithmic information theory. We also give a machine-independent characterization of normality and finite-state dimension in terms of superadditive calibrated functions. We compare our approach with previous results and notions relating finite automata and complexity.

在本文中，我们根据自动Kolmogorov复杂度和有限状态先验概率来刻划正常序列和有限状态维。我们显示出许多关于正态序列和有限状态维的已知结果，包括对齐和不对齐正态之间的等价性，沃尔定​​理，Piatetski–Shapiro定理，Champernowne正数示例及其修改，不同定义之间的等价性状态维，关于有限状态选择规则的Agafonov和Schnorr结果，很容易成为这种表征的推论。为此，我们使用自动（有限状态）复杂度和先验概率有限状态的概念，它们与算法信息论中的Kolmogorov复杂度和Solomonoff-Levin先验概率的概念自然对应。我们还根据超加性校准函数给出了与机器无关的正态性和有限状态维特征。我们将我们的方法与以前的结果和有关有限自动机和复杂性的概念进行了比较。