This paper is dedicated to studying decidability properties of theories of regular languages with classical operations: union, concatenation, and the Kleene star. The theory with union only is a theory of some Boolean algebra, so it is decidable. We prove that the theory of regular languages with the Kleene star only is decidable. If we use union and concatenation simultaneously, then the theory becomes both Σ₁- and π₁-hard over the one-symbol alphabet. Using methods from the proof of this theorem we establish that the theory of regular languages over one-symbol alphabet with union and the Kleene star is as hard as arithmetic. Then we establish that the theory with all three operations is reducible to arithmetic also, hence, it is equivalent to arithmetic. Finally, we prove that the theory of regular languages over any alphabet with concatenation only is equivalent to arithmetic also. The last result is based on our previous work where an analogous theorem was proved for one-symbol languages.

本文致力于研究具有经典运算的普通语言理论的可判定性：联合，串联和Kleene星。仅具有并集的理论是一些布尔代数的理论，因此它是可判定的。我们证明，仅带有Kleene星的常规语言理论是可以判定的。如果我们同时使用工会和连接，然后将理论变成两个Σ ₁ -和π ₁-难过一个符号的字母。使用该定理的证明方法，我们可以确定正则语言在带有并集和Kleene星号的单符号字母上的理论与算术一样困难。然后我们确定具有所有三个运算的理论也可以归结为算术，因此它等同于算术。最后，我们证明了仅对任何带连接字母的常规语言的理论也等同于算术。最后的结果是基于我们以前的工作，其中证明了一种符号语言的类似定理。