Questions tagged [formal-grammars]

Questions about formal grammars, generative descriptions of formal languages.

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26
votes
2answers
25k views

How to prove that a language is context-free?

There are many techniques to prove that a language is not context-free, but how do I prove that a language is context-free? What techniques are there to prove this? Obviously, one way is to exhibit ...
22
votes
1answer
5k views

How to show that L = L(G)?

Specifying formal languages by giving formal grammars is a frequent task: we need grammars not only to describe languages, but also to parse them, or even do proper science. In all cases, it is ...
68
votes
1answer
11k views

Language theoretic comparison of LL and LR grammars

People often say that LR(k) parsers are more powerful than LL(k) parsers. These statements are vague most of the time; in particular, should we compare the classes for a fixed $k$ or the union over ...
25
votes
4answers
21k views

How to prove that a grammar is unambiguous?

My problem is how can I prove that a grammar is unambiguous? I have the following grammar: $$S → statement ∣ \mbox{if } expression \mbox{ then } S ∣ \mbox{if } expression \mbox{ then } S \mbox{ else } ...
7
votes
2answers
5k views

Converting to CFG from a CFL? [duplicate]

I am trying to learn CFG. Now to make a CFG from a CFL it is really difficult for me. Is there any simple rule or steps so that I can easily find a CFG for a CFL. I am trying to solve one problem ...
7
votes
2answers
11k views

Context Free Grammar for {a^ib^j | i,j ≥ 0; i ≠ 2j}

Can someone help with this: $L=\{a^ib^j \mid i,j \ge 0 \text{ and } i \ne 2j\}$ I'm trying to write a grammar for this language? I don't know how to do this. I tried this: $S \rightarrow aaAb \...
15
votes
1answer
21k views

Intersection of context free with regular languages

The intersection of a context free language L with a regular language M, is said to be always context free. I understood the cross product construction proof, but I still don't get why it is context ...
6
votes
3answers
1k views

Context-free grammar for $L = \{a^{2^k}, k \in\mathbb{N}\}$

In an exercise, I am asked to find a context free grammar for language $L = \{a^{2^k}, k \in \mathbb{N}\}$. I have been trying to use a "doubling" variable. If $a^{2n} \in L, n\in\mathbb{N}$ then use ...
36
votes
2answers
5k views

Are there inherently ambiguous and deterministic context-free languages?

Let us call a context-free language deterministic if and only if it can be accepted by a deterministic push-down automaton, and nondeterministic otherwise. Let us call a context-free language ...
9
votes
3answers
510 views

Context-sensitive grammar for the language of words concatenated with themselves

I'm looking for a context-sensitive grammar that describes the following language: $L = \{ ww \mid w ∈ \{a,b\}^{\ast}, |w| ≥ 1\}$ . I've got problems with the fact that no rules such as $X \to \...
2
votes
1answer
1k views

Context Free Grammar for language L

Can someone help with this: $L=\{a^ib^j \mid i,j \ge 1 \text{ and } i \ne j \text{ and } i<2j\}$ I'm trying to write a grammar for this language? I tried this: $S \to S_1 \mid S_2 \\ S_1 \to aXb ...
2
votes
2answers
20k views

How to convert PDA to CFG

I learned how to convert context-free grammar to pushdown automata but how can I do the opposite? to convert PDA to CFG? For example: to write CFG for the automata My attempt: $S=A_{03}$ because $...
1
vote
1answer
75 views

Is it possible for a Turing machine to be able to reduce a grammar and tell where it fits in chomsky hierarchy?

For example: This looks like a context free grammar: 𝑆 → 𝑄𝑅𝑇 𝑄 → 𝑎𝑄 | 𝑎 𝑅 → 𝑏𝑅 | 𝑏 𝑇 → 𝑐𝑇 | c but it can be reduced to this regular language: 𝑆 → 𝑎𝑆 | 𝑎𝑅 𝑅 → 𝑏𝑅 | 𝑏𝑇 𝑇 → 𝑐𝑇...
13
votes
2answers
14k views

Is the complement of { ww | … } context-free?

Define the language $L$ as $L = \{a, b\}^* - \{ww\mid w \in \{a, b\}^*\}$. In other words, $L$ contains the words that cannot be expressed as some word repeated twice. Is $L$ context-free or not? I'...
19
votes
1answer
2k views

How can I convert the Turing machine the recognizes language $L$ into an unrestricted grammar?

According to this Wikipedia article, unrestricted grammars are equivalent to Turing machines. The article notes that I can convert any Turing machine into an unrestricted grammar, but it only shows ...
10
votes
3answers
901 views

Decidable languages and unrestricted grammars?

Turing machines and unrestricted grammars are two different formalisms that define the RE languages. Some RE languages are decidable, but not all are. We can define the decidable languages with ...
5
votes
2answers
2k views

Unrestricted grammar to generate $a^{n^2}$

I have been asked to find a grammar that will generate the language $\{a^{n^2}:n \ge0\}$ in an exercise. So far I tried to replicate the previously written characters with my grammar rules but it didn'...
16
votes
2answers
4k views

Are regular expressions $LR(k)$?

If I have a Type 3 Grammar, it can be represented on a pushdown automaton (without doing any operation on the stack) so I can represent regular expressions by using context free languages. But can I ...
15
votes
2answers
2k views

Decidable non-context-sensitive languages

It is arguable that most languages created to describe everyday problems are context-sensitives. In the other hand, it is possible and not hard to find some languages that are not recursive or even ...
19
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1answer
1k views

Is language equality for linear context-free grammars decidable?

Let's consider two context-free grammars $G_1$ and $G_2$ and ask the following question: Is $L(G_1) = L(G_2)$, that is, are the two grammars equivalent? In general, this problem is undecidable. ...
12
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2answers
5k views

Is there any way to distinguish between LL(k) and LR(k) grammar?

I am recently studying about Compilers designing. I came to know about two types of grammar one is LL grammar and other is LR grammar. We also know the facts that every LL grammar is LR that is LL ...
4
votes
2answers
3k views

unambiguous grammar that produce equal number of a and b

is there any unambiguous grammar on alphabet={a,b} that can produce strings which have equal number of a and b (e.g. "aabb" , "baba" , "abba") ?
10
votes
5answers
1k views

Is there a known method for constructing a grammar given a finite set of finite strings?

From my reading it seems that most grammars are concerned with generating an infinite number of strings. What if you worked the other way around? If given n strings of m length, it should be possible ...
1
vote
1answer
992 views

Context sensitive grammar for $\{a^{2^n}\mid n\geq 0\}$

I want to build a context sensitive grammar for the language $\{a^{2^n}\mid n\geq 0\}$. I think it should be something like this \begin{align*} S &\to aA \mid a\\ aA&\to aaaA \mid aa \end{...
0
votes
1answer
198 views

Example Context free grammar

Is there a nice way to give context free grammar for $$\{a^nb^ma^kb^l:n+m=k+l\}?$$ From PDA point of view it seems we just push + on stack if we see a, push + on stack if we see b, pop + from stack ...
11
votes
2answers
4k views

How can I prove this language is not context-free?

I have the following language $\qquad \{0^i 1^j 2^k \mid 0 \leq i \leq j \leq k\}$ I am trying to determine which Chomsky language class it fits into. I can see how it could be made using a context-...
2
votes
1answer
579 views

Is this unambiguous grammar equivalent?

I have a simple context free grammar $(\{A\}, \{(,)\}, R, A)$, which consists of this one production rule: $A \rightarrow AA\, \vert\, (A)\, \vert\, \epsilon$ I believe this is ambiguous! For ...
7
votes
1answer
17k views

Left recursion and left factoring — which one goes first?

if I have a grammar having a production that contains both left recursion and left factoring like $\qquad \displaystyle F \to FBa \mid cDS \mid c$ which one has priority, left recursion or left ...
6
votes
2answers
3k views

In context-free grammar (CFG), what is the importance of doing both leftmost and rightmost derivations?

I am a beginner learning theoretical computer science. While learning about CFG, I found that doing both leftmost and rightmost derivations gave me the same parse tree. So, my question is: Why is it ...
22
votes
4answers
1k views

Are there other ways to describe formal languages other than grammars?

I'm looking for mathematical theories that deal with describing formal languages (set of strings) in general and not just grammar hierarchies.
13
votes
3answers
1k views

What would you get if you add parameters to context free grammars?

I was thinking of grammars for indendation-sensitive languages and it looks like CF grammars would do the trick if combined with parameters. As an example, consider this fragment for simplified Python ...
7
votes
3answers
15k views

Regular Expression to Context-Free Grammar

Anyone knows if there is an algorithm for directly write the context-free grammar that generates a given regular expression?
7
votes
3answers
2k views

How to generate a context sensitive grammar for www

I am trying to solve for my exam coming up and have no clue how to generate the grammar for Context sensitive languages for example how do i proceed on this kind of question. Give a context-...
3
votes
2answers
4k views

Defining a context-free grammar for $\{w \in \{0, 1\}^* : \#_0(w) = \#_1(w)\}$ [duplicate]

I have a language where each string in the language has even amount of $0$'s as $1$'s (e.g., $0101$, $1010$, $1100$, $0011$, $10$ are all in the language). I was hoping to define a context-free ...
18
votes
2answers
2k views

Can there be 'dead states' in a context-free grammar?

Can a context-free grammar include "dead states" from an automaton, such as $$G = \big(\{a, b, c\}, \{A, B, C\}, \{A\to aB, B\to b, B\to C, C\to cC\}, A\big)\,?$$ The production rules $B\to C$ and $...
9
votes
1answer
22k views

Convert CFG to PDA

Is there any set of rules or methods to convert any context free grammar to a push down automata? I already found some slides online but I wasn't able to understand them. In slide 10 he speaks ...
3
votes
1answer
943 views

Unambiguity of Reverse Polish Notation

Lets say I have given following grammar which generates arithmetic expressions in reverse polish notation: $G=({E},{a,+,*},P,E)$ $P={ E \rightarrow EE+ | EE* | a }$ I know this grammar is ...
14
votes
2answers
1k views

Are the Before and After sets for context-free grammars always context-free?

Let $G$ be a context-free grammar. A string of terminals and nonterminals of $G$ is said to be a sentential form of $G$ if you can obtain it by applying productions of $G$ zero or more times to the ...
11
votes
2answers
8k views

Finding the language generated by a context-free grammar

This is a question from the Dragon book (I apologize for translation mistakes, I don´t have the English version on hand): What language is generated by this grammar? $S \rightarrow a S b S \...
3
votes
2answers
241 views

Figuring out the language of a non-linear CFG

I have the CFG G with the following production rules: $$ S \to aSaS \mid b $$ Is it possible to find $L(G)$? I have no idea how describe it by any pattern. I use grammophone to check example words, ...
3
votes
0answers
89 views

What kind of formal language is generated by Parsing Expression Grammars?

I've been unable to find what class of languages is recognized by PEGs. The original paper [1] only conjectures that there are some Context-Free Grammars that are unrecognizable by PEGs. It also ...
3
votes
2answers
494 views

How to check for ambiguous grammar if you don't know the string

Let's say I have a CFG grammar $G$ which describes some rules for language generation. How can you tell that grammar can generate ambiguous results for a string if you don't know that string? I know ...
2
votes
1answer
319 views

Inducing a context free grammar [closed]

I have a file containing a subset of possible strings from a context free language. I am looking for a mechanism to induce the grammar from this information. Is that possible?
1
vote
2answers
3k views

CFG for $\{a^ib^jc^k \mid i \neq j+k\}$

I am trying to design a context-free grammar for the language $L = \{a^ib^jc^k \mid i\neq j+k\}$ over the alphabet $\Sigma = \{a,b,c\}$. I know that I can split this up into the union of two cfg's $...
-1
votes
1answer
1k views

CFG for the language “number of a's = number of b's + 2”

How can I construct a context-free grammar for the following language? $$ L = \{ w \in \{a,b\}^* : \#_a(w) = \#_b(w) + 2 \}. $$ Please help me out in this. I am not sure how to approach this ...
6
votes
4answers
378 views

Why does $A(L)= \{ w_1w_2: |w_1|=|w_2|$ and $w_1, w_2^R \in L \}$ generate a context free language for regular $L$?

How can I prove that the language that the operator $A$ defines for regular language $L$ is a context free language. $A(L)= \{ w_1w_2: |w_1|=|w_2|$ and $w_1, w_2^R \in L \}$, where $x^R$ is the ...
4
votes
2answers
228 views

Context free grammar construction

My problem with CFG is, I am to generally create ones that don't have requirements such as: $\qquad \{a^m b^n \mid m \le n \le 2m \}$ I have no clue where to begin, and how to approach it. I was ...
4
votes
3answers
237 views

Designing a CFG that produces as many c's as the difference of numbers of a's and b's

The question is to design a CFG for the language of words that have as many c's as the difference of numbers of a's and b's, that is $\qquad\displaystyle L = \{(a^l)(b^m)(c^n) \mid l, m \in \mathbb{N}...
3
votes
1answer
75 views

Are there any specific mechanical ways to reduce a regular expression 'equation' to a more simple one?

So if we have a complex equation in regular algebra we can use properties like distrbuivity, associativity and commutativity to make an equation simper or more compact. Can we use some sort of ...
2
votes
3answers
6k views

How does one make an unambiguous context-free grammar for arithmetic expressions?

Say I have a context-free grammar defined by the following rule. $$ \langle EXPR\rangle \rightarrow \langle EXPR\rangle + \langle EXPR\rangle~|~\langle EXPR\rangle \times \langle EXPR\rangle~|~(\...