下推式自动机是否可以识别给定的常规语言是否可以确定？

两个下推自动机是否识别相同语言的问题尚不确定。下推式自动机是否识别空语言的问题是可以确定的，因此也可以确定是否可以识别给定的有限语言。下推式自动机接受的语言是否正常是不确定的。但是...

...是否可以确定下推式自动机是否可以识别给定的常规语言？

如果答案是否定的，如果给定的常规语言的星星高度为 $1$，是否可以确定问题？

不能确定PDA是否识别出，即输入字母上所有字符串的集合。$\Sigma^*$

添加。由于TM的“无效”计算可以被编码为CFG的字符串，因此无法确定。这是Hopcroft和Ullman撰写的《自动机理论导论》的引理8.7。作者将此结果参考Hartmanis（1967），无上下文语言和图灵机计算。$L(G)=\Sigma^*$

$M$$M$$xpy$$uv$$p$$ucpav \vdash uqcbv$$(p,a,q,b,L)$$ucpav \vdash ucbqv$$(p,a,q,b,R)$

$w_0\#w_1^R\#w_2\#w_3^R\# \dots$ 哪里 $w_0 = q_0x$ codes the initial configuration on string $x$, and we have proper steps $w_i \vdash w_{i+1}$. The last configuration in the string should be final, i.e., have a halting/final state.

It is now an exercise to verify that the strings that are not valid computations can be generated by a CFG $G_M$ (or accepted by a PDA). The strings that do not consist of configuration sequences are even regular. Otherwise one non-deterministically guesses a position where not $w_i \vdash w_{i+1}$. This part of the string is generated by a grammar that is similar to one for $\{ x\#y^R \mid x,y\in \{a,b\}^* , x\neq y\}$.

If the TM $M$ has no accepted strings, it will have no valid computations, and all strings are generated by the grammar $G_M$.