8
votes
Accepted
Are LR(k) languages and DCFLs equivalent?
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 ...
- 274k
6
votes
Accepted
Is there a LL(K) Grammar which is not LALR(K) Grammar?
Every $LL(k)$ grammar is $LR(k)$, but there are $LL(k)$ grammars which are not $LALR(k)$.
There's a simple example in Parsing Theory by Sippu&Soisalon-Soininen
$$\begin{align}S &\to a A a \...
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5
votes
Accepted
can every LR(k) grammer be transformed to LR(1)
Remember that the point of parsing (in almost all practical applications) is not simply to recognize whether or not a sentence is in a language, but rather to find the parse tree corresponding to the ...
- 11.7k
4
votes
Accepted
Item lookaheads versus dot lookaheads for $LR(k)$ with $k \gt 1$?
I think you are mistaken, they are needed but the dot look-ahead there is so obvious that you have not paid attention to the fact it is used.
First, let's remark that there are three kinds of items:
...
- 3,024
4
votes
Accepted
How much bigger can an LR(1) automaton for a language be than the corresponding LR(0) automaton?
The grammar
$$\begin{array}{l}
S \rightarrow T_0 \\
T_n \rightarrow a \; T_{n+1} \\
T_n \rightarrow b \; T_{n+1} \\
T_n \rightarrow b \; T_{n+1} \; t_n \\
T_N \rightarrow t_N
\end{array}
$$
has the ...
- 3,024
4
votes
When will SLR(1) parser fail but CLR(1) will not
No, the authors meant that there is no right-sentential form of the grammar that starts with R =. That is, the = in that ...
- 11.7k
4
votes
What is handle in bottom up LR parsing?
"When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean—neither more nor less."
Charles Dodgson -- who wrote those words under ...
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3
votes
DCFL with prefix property have LR(0) grammar?
Your grammar is $\mathrm{LR}(0)$ by adding a $\mathrm{\$}$ symbol to the alphabet and a starting deduction rule $S'\rightarrow S\$$ when constructing the $\mathrm{LR}(0)$ automaton (and it is a ...
- 2,245
3
votes
Accepted
Need for an augmented grammar
Yes, if you follow the canonical conversion of augmented grammars, a first step is adding a new start symbol $S'$ and the production $S' \to S$, since the algorithm may later "change" $S$.
However, ...
- 20.6k
3
votes
Is LR(1) closed under union?
A simple example is
$$L_1 = \{a^i b^i c^j \mid i,j\ge 0\}$$
$$L_2 = \{a^i b^j c^j \mid i,j\ge 0\}$$
Clearly, both languages are $LR(1)$. (Indeed, they are $LL(1)$.) But their union is inherently ...
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2
votes
Accepted
An intuitive explanation for LR grammars?
A language is $LR(k)$ if looking at the right side of a production, and looking $k$ symbols ahead, one can determine the left hand side of the production. This is quite incomplete, but intuitively ...
- 13.8k
2
votes
Accepted
Is THEADS the same as FIRST?
I recently had the same question in mind when reading Pager's work. I found the paper The Edge-Pushing LR(k) Algorithm by Chen and Pager, which seems to say that the two terms are indeed the same. ...
- 137
2
votes
Accepted
How to remove empty strings a.k.a (lambda, epsilon) from a LR(0) grammar
Find all nullable variables. In this case only E' is nullable.
Let me illustrate the second step with an example:
Replace a production A -> BCD with a family of productions like this (assuming B, C &...
- 3,459
2
votes
Accepted
Is $\{a^nb^n\}\cup\{a^nb^{2n}\}$ LR(k)?
Propbably you should read the citation as: "The problem is that the c (which separates the two parts of the language) is at the extreme right and not at the extreme left, where we start parsing."
...
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2
votes
Are shift and goto moves for all LR parsers ( LR(0), SLR(1),CLR(1),LALR(1) ) same?
Every reduce action corresponds to a production; the action "undoes" the production by replacing its right-hand side with the non-terminal which produced it.
That can only happen in one way; the ...
- 11.7k
2
votes
Accepted
Is following grammar LR(0)?
You haven't actually augmented the grammar. The augmented grammar has the production $$start\to SL\;\$$$
With that change, state 1 is not a reduction state and there is no conflict.
If you did not ...
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2
votes
Accepted
Merging states in LALR parser
No, you can't do that. The definition of LALR parsers is that it has combined(union of lookaheads) states having the same cores but different lookaheads.
So what would happen if you were to combine ...
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2
votes
Accepted
How does LALR(1) parser behave compared to LR(1) paser?
The explanation is given in the following paragraphs. Given a correct input, the LALR parser produces exactly the same sequence of reduce and shift actions as would an LR parser for the same grammar. ...
- 11.7k
2
votes
Accepted
How is the lookahead for an LR(1) automaton computed?
$\text{FIRST}(\beta)$ is the set of possible first tokens in a derivation from β. (It's usual to also add ε to the first set if β could derive the empty sequence.)
So $\text{FIRST}(\mathbf{+}\mathit{T}...
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1
vote
What is the time complexity of SLR and LALR parsers?
They all run linear time with regards to input length ($O(n)$). The different algorithms have different memory trade offs for making it easier to write grammar for. That is not all grammars that work ...
- 227
1
vote
Accepted
Reduce-reduce conflict in SLR vs LALR
Note: I'm assuming that when you wrote $SLR$ and $LALR$ that you actually meant $SLR(1)$ and $LALR(1)$.
It is certainly the case that if a grammar shows a conflict in an $LALR(1)$ automaton, that ...
- 11.7k
1
vote
Accepted
Why can't a LR(0) parser have GOTO and Reduce Items in the same canonical set?
This isn't what you asked, but let me start by saying that I suspect the answer they're looking for in the question you link is (D), "A LR(0) parser can parse any regular grammar".
That ...
- 11.7k
1
vote
Accepted
What are cases where we cannot perform LR parsing without augmented grammar?
Once we have a parsing table, we can parse (or reject) any sentence without any reference to the grammar. So at that point, the fact that the grammar was or was not augmented is essentially moot. (It ...
- 11.7k
1
vote
Accepted
How to create a LR(k) grammar for an arbitrary k
Here's one pattern for an LR(k) grammar which is not LR(k-1).
I didn't fill in the definition of $A$; there's nothing particularly special about it. It might have an empty right-hand side, or it might ...
- 11.7k
1
vote
Accepted
Analysis of a LALR shift-reduce conflict
This conflict arises because when the parser is at the beginning of a line and the lookahead is VAR, it can't tell whether or not to reduce an empty ...
- 11.7k
1
vote
Accepted
Wikipedia says this grammar is LR(0), but Grammophone says it is not; is it?
Before applying the LR algorithm to a grammar, the grammar must be "augmented" by adding the rule
$S' \to S \$$
where $S$ is the original start symbol and $S'$ and $\$$ are symbols not in the ...
- 11.7k
1
vote
LR parsers and ambiguous and non deterministic grammars
The grammar is really ambiguous.
$\textbf{ if } expr \textbf{ then if }expr \textbf{ then } stmt \textbf{ else } stmt$
has two possible parses with that grammar. (The $ \textbf{ else }$ can ...
- 11.7k
1
vote
Can I mix LL and LR parsing?
Using recursive descent in combination with an operator precedence variant for expressions is a very common approach. You might also want to search for Pratt parsing.
An older, now uncommon ...
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1
vote
Is this grammar LR(1)?
This grammar (which isn't ambiguous) isn't LR(k) due to the right-recursion, thus it isn't LR(1).
- 669
1
vote
Use DFA minimization on LR(1) states
As shown here, minimizing the states in an LR(1) parser is NP-hard. The basic idea is that given an arbitrary graph, you can construct a grammar such that for every node in the original graph, the LR(...
- 404
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