# Algorithm to test whether a language is context-free

Is there an algorithm/systematic procedure to test whether a language is context-free?

In other words, given a language specified in algebraic form (think of something like $L=\{a^n b^n a^n : n \in \mathbb{N}\}$), test whether the language is context-free or not. Imagine we are writing a web service to help students with all their homeworks; you specify the language, and the web service outputs "context-free" or "not context-free". Is there any good approach to automating this?

There are of course techniques for manual proof, such as the pumping lemma, Ogden's lemma, Parikh's lemma, the Interchange lemma, and more here. However, they each require manual insight at some point, so it's not clear how to turn any of them into something algorithmic.

I see Kaveh has written elsewhere that the set of non-context-free languages is not recursively enumerable, so it seems there is no hope for any algorithm to work on all possible languages. Therefore, I suppose the web service would need to be able to output "context-free", "not context-free", or "I can't tell". Is there any algorithm that would often be able to provide an answer other than "I can't tell", on many of the languages one is likely to see in textbooks? How would you build such a web service?

To make this question well-posed, we need to decide how the user will specify the language. I'm open to suggestions, but I'm thinking something like this:

$$L = \{E : S\}$$

where $E$ is a word-expressions and $S$ is a system of linear inequalities over the length-variables, with the following definitions:

• Each of $x,y,z,\dots$ is a word-expression. (These represent variables that can hold any word in $\Sigma^*$.)

• Each of $a,b,c,\dots$ is a word-expression. (Implicitly, $\Sigma=\{a,b,c,\dots\}$, so $a,b,c,\dots$ represent a single symbol in the underlying alphabet.)

• Each of $a^\eta,b^\eta,c^\eta,\dots$ is a word-expression, if $\eta$ is a length-variable.

• The concatenation of word-expressions is a word-expression.

• Each of $m,n,p,q,\dots$ is a length-variable. (These represent variables that can hold any natural number.)

• Each of $|x|,|y|,|z|,\dots$ is a length-variable. (These represent the length of a corresponding word.)

This seems broad enough to handle many of the cases we see in textbook exercises. Of course, you can substitute any other textual method of specifying a language in algebraic form, if you like.

• Wouldn't it be easier to start with regularity of languages? – Yuval Filmus Nov 11 '13 at 20:05
• @YuvalFilmus, sure would! Now that you mention it, that's a great idea. Do you think the problem is feasible for regular languages? I'd be happy to ask a corresponding about regular languages, if you think that might be valuable. – D.W. Nov 11 '13 at 20:46
• It would certainly be easier for regular languages. By the way, the general non-decidability doesn't necessarily apply to languages of the form you mention. – Yuval Filmus Nov 11 '13 at 23:50
• I'm afraid this problem is probably open, at least a specific case is: cstheory.stackexchange.com/questions/17976. There might be a way to get undecidability for your more general problem, but I don't see it. – sdcvvc Nov 12 '13 at 14:55
• it would be helpful to give some example words in the language. suggest further research/ collaboration in Computer Science Chat – vzn Sep 18 '14 at 19:10

• How would you represent $\{a^nb^nc^n : n \in \mathbb{N}\}$ as a grammar? Also, it is probably undecidable to tell whether an unrestricted grammar generates a context-free language, so I doubt JFLAP can do that. – Yuval Filmus Nov 2 '18 at 16:45