Tips for creating “Context Free Grammar”

Question

I am new to CFG's, Can someone give me tips in creating CFG that generates some language

For example

$L =\{ w v w^R \mid v,w\in \{a,b\}^*\wedge|v| \text{ is even } \}$, where $w^R$ is the reverse of string $w$.

I'm having trouble understanding the language .. I want to build DFA to convert it

Welcome to SE Computer Science. What is your idea of the language, even if you have trouble understanding it? What can a DFA have to do with your problem? — babou, May 27 '15 at 18:08
I would build a DFA to convert it to CFG .. but the language not clear fo me — Abdallha dagga, May 27 '15 at 18:13
use the search engine to find similar questions in this site, and get some intuition. See, for example, this question — Ran G., May 27 '15 at 18:41
also check our reference questions, and in particular this one — Ran G., May 27 '15 at 18:45
I think you are confusing DFA (Deterministic Finite Automaton) with PDA (Push-Down Automaton) — babou, May 27 '15 at 18:48
Converting a PDA to a CFG is not particularly simple, and in this case it's not hard to construct a CFG directly, so that's a better option. Construction a PDA is also not too difficult, so you can take that as an additional exercise. — Yuval Filmus, May 27 '15 at 20:22

Hendrik Jan · Accepted Answer · 2015-05-28T17:18:03.753

3

Mmmm.

Let me note that in fact $L =\{ w v w^R \mid v,w\in \{a,b\}^*\wedge|v| \text{ is even } \}$ equals $\{ v \mid v \in \{a,b\}^*\wedge|v| \text{ is even } \}$.

Al strings in $L$ are of even length, and conversely all strings of even length can be otained by taking $w=\varepsilon$.

That should help.

edited May 28 '15 at 17:18

answered May 27 '15 at 23:35

Hendrik Jan

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I wonder whether this was expected ! – babou May 27 '15 at 23:43
@babou That is a good question. Perhaps the DFA confusion is related to this? – Hendrik Jan May 28 '15 at 00:20
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Cute! It might help the OP even more if you included a sentence of explanation. (Too bad I can't upvote twice.) – Rick Decker May 28 '15 at 01:12
It is really a great psychological example of the way we work, and what may block us. Science is largely a question of looking at thing the right way. Hence the importance of conceptualization and notation. But in such a simple example ... Great for teaching. – babou May 28 '15 at 08:40
I think this question should not be closed. The answer is indeed too cute. What do you think? cc @RickDecker – babou May 28 '15 at 13:59
Yes, I didn't see it -- subtle problem -- I was too hasty to answer. Looks line ww^R, but adding the even length string in the middle changes everything since ww^R is always even and you can always let w = empty string. – wcochran May 28 '15 at 14:07
@babou. I agree that this question shouldn't be closed, especially in light of the fact that nothing in the reference question comes close to this answer. – Rick Decker May 28 '15 at 14:18
@RickDecker if Hendrick Jan agrees, we can take this to chat, I am afraid that is the only effective way. – babou May 28 '15 at 14:22
There's a very similar question here. – Rick Decker May 28 '15 at 14:36
Indeed, the question Rick links is much closer to a reason for closing as duplicate than the canned how-to-context-free question. – Hendrik Jan May 28 '15 at 17:16

score 2 · Answer 2 · answered May 27 '15 at 20:21

2

Hints:

Construct a non-terminal $V$ and appropriate productions so that $V$ generates all strings of even length.
Create productions so that your starting symbol $S$ generates $wVw^R$ for all $w$. To accomplish that, modify a grammar for $ww^R$.

answered May 27 '15 at 20:21

Yuval Filmus

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Tips for creating “Context Free Grammar”

2 Answers2

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