6
$\begingroup$

In the familiar book of Theory of Computation by M. Sipser, the author proved that for endmarked context-free languages, the set of languages having a LR(k) grammar for a predefined $k \in \mathbb{N}$ (denoted as LR(k) languages) is the set of deterministic context-free languages (denoted as DCFL).

My question is also about the relation of those two sets, but in the broader field of all context-free languages. Specifically, are LR(k) languages and DCFLs equivalent? And, in which book can I find a proof?

For now, I just have some surrounding facts as followed. Also in the book, the author proved that LR(0) languages strictly belong to DCFLs, and LR(k) languages belong to DCFLs for all $k \in \mathbb{N} \setminus \{0\}$. In addition, it's obvious that LR(a) languages belong to LR(b) languages for all $0 \le a < b$. Recently, I've got an unchecked fact from here: LR(1) languages and DCFLs are equivalent.

$\endgroup$
1
  • 2
    $\begingroup$ This question resp. its answer may be interesting for you. $\endgroup$
    – Raphael
    Oct 17, 2016 at 12:51

1 Answer 1

8
$\begingroup$

According to Wikipedia:

  • For every fixed $k \geq 1$: A language has an LR($k$) grammar iff it is DCFL.
  • A language has an LR(0) grammar iff it is DCFL and has the prefix property (no word is a prefix of another word).

The first property is proved in Knuth's original paper, in Section V on page 628.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.