图灵完整的语言可以使用CFG吗？

Question

Does turing completeness preclude a language from having a CFG? I couldn't find any paper saying that.

I found this: "TeX can only be parsed by a complete Turing machine (modulo the finite space available), which precludes it from having a BNF."

Answer 1

We are often imprecise with these terms, but a correct answer to your question requires that we be very precise with how we are using terms.

Two computation systems are equivalent if they can simulate each other. A computation system is Turing-equivalent if it is equivalent to Turing machines .

A computation is complete with respect to a computation system if it requires all capabilities of that system to be computed in that system; that is, any change to the computing system which causes it not to be capable of performing at least the same computations as before will cause it not to be able to perform this computation. A computation is Turing-complete if it is complete with respect to Turing machines .

BNF grammars describe context-free languages, and the least capable computing system capable of parsing such languages are the pushdown automata . This computing system cannot simulate Turing machines in that there are computations a Turing machine can perform which a pushdown automaton cannot; therefore, pushdown automata are not Turing-equivalent .

The article says that TeX is a Turing-complete language, that is, deciding the language of valid TeX strings requires all capabilities of Turing machines. Any system not capable of simulating a Turing machine cannot possibly parse decide membership in the language of valid TeX strings.

The article is NOT saying that TeX is Turing-equivalent (maybe it is, maybe it isn't; I have no idea). As pointed out in the comment, Turing-completeness of a computation system's representation is completely unrelated to that computation system's Turing-equivalence. Even Turing machines themselves can be represented using strings of a regular language (in fact, extend the interpretation of any language so that otherwise invalid programs compile to the program which halts without doing anything, and suddenly ALL strings are valid, and the language of all strings is certainly regular).