创建CFG的技术

Question

I have to create a CFG which generates

{a^n (ab)^nc^md^le^k | n>0, k, l, m>=0, k<m, m=l+k}

The first part is easy enough, I came up with

S -> aS2abS3 S2 -> aS2ab | epsilon

However, the second part is very confusing. So far I have

S3 -> S4 | epsilon

The problem I have is how do I possibly keep track of all of these variables? K has to be less than m, m has to be equal to l + k, and l must be at least 1 by extension. Can someone give me some general tips for approaching these CFG's?

Answer 1

Think inside out (because that's the way CFGs work) and don't get confused by extraneous details.

Here's the tip: CFGs are push-down automata (PDAs) which mean that they have a stack. PDAs are good at symmetry but they can't do antisymmetry. So they can do palindromes, but not repetitions. That's the nature of a stack.

So you always need to look for the mirror images. For example, a m b m is a trivial form of mirror iamge, where the mirror turns a s into b s.

This one is just a little more complicated, but if you focus on looking for the symmetry, you'll find it.

Having said that, here's the solution:

Since m > l ,

cmdlek

can be rewritten as:

cm-lcldlek

And since k = m - l , that's the same as:

ckcldlek

From there it's trivial:

Sinner → cd | c Sinner d
Souter → Sinner | c Souter e