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Questions tagged [formal-grammars]

Questions about formal grammars, generative descriptions of formal languages.

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25 views

How to check whether a language is regular or not? [duplicate]

I am given expressions such as \begin{align} L_2 &= \{ a^n b^{n!} \}, \\ L_3 &= \{ abcva^n \mid v \in \{a,b,c\}^*, n \in \mathbb{N}, n \text{ is even}, |v|=n/2 \}. \end{align}
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1answer
22 views

Does this grammar accept this words?

I made this grammar: $S \rightarrow ASa$ $S \rightarrow c$ $A \rightarrow a|b$ And I want to check that it accepts words like $aacaa$, $abcaa$, $babcaaa$, I formed the grammar by thinking about the ...
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1answer
21 views

Prove that grammar accepting arithmetic expressions is not regular

I created a grammar which accepts all arithmetic expressions consisting of $+,-,*,/, (, )$. I created the following grammar: $S \rightarrow M+-M$ $+-M \rightarrow +M+-M$ $+-M \rightarrow -M+-M$ $+-M \...
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29 views

Grammar for all words $0^n1^m$ such that $n \ge m+2$

Given grammar $$L(G) = \{ 0^n1^m | n \ge m + 2 \}$$ What is the grammar for this? I know the grammar for the following language: $$ L(A) = \{ 0^n1^m | n = m + 2 \} $$ We can divide any string in $L(A)$...
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2answers
53 views

What is appearance checking in the context of formal grammars?

As I did not find any definition of the term "appearance checking" although it is widely used, I am eager to ask as what it can be defined. Perfect would be an example using a context free ...
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2answers
45 views

Is there a grammar for this language? $w^{m-1}aca^m$?

I have to form a free context grammar for this language $w^{m-1}aca^m$ where $w \in \{a,b\}$, so what I have been able to do is this: $X \rightarrow SacA$ $S \rightarrow aS|bS$ $A \rightarrow aA$ But ...
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1answer
16 views

How to demonstrate unambiguous CFG and CNF?

I have to show that if G is an unambiguous CFG, the transformed grammar G' in CNF is also unambiguous. But couldn't come up with something concrete. I could only visualize the case where the grammar G ...
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0answers
28 views

Unambiguous formal grammars for a specific class of languages

Suppose that $w \in \{0; 1\}^*$ is a binary word. Let's denote the number of $0$-s in $w$ as $\#_0(w)$ and the number of $1$-s in $w$ as $\#_1(w)$. Now suppose that $q \in \mathbb{Q}$ is a positive ...
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0answers
43 views

How LR parser works with grammar containing epsilon productions

In Dragon book for compiler design there is this example of SDT in Example 5.16 $L\rightarrow En$ $E\rightarrow \{ print (' +') ; \} \space E_1 + T$ $E\rightarrow T$ $T\rightarrow \{ print (' *' ) ; ...
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24 views

Prove $L =\{0^{2^n}\mid n \geqslant 0\}$ is not context free [duplicate]

Here $0^j$ means $0$ repeated $j$ times e.g. $0^2$ is $00$. So to prove this I was asked to use the pumping lemma. So let $m$ be the pumping length and assume $L$ is a CFL by contradiction. We can ...
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1answer
45 views

Finding the language generated by this grammar

I'm having problems with this. Can someone help me please. Find the language generated by this grammar over the alphabet $\{0,1\}$: $S\rightarrow BAB\mid CAB$ $BA \rightarrow BC$ $CA \rightarrow AAC$ ...
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1answer
30 views

Myhill-Nerode - Prove irregularity for $\{a^{n^3}\}$

I need to prove that the following language is not regular by showing there are infinite pairwise distinct equivalence classes: $$ L = \{a^{n^3} \mid n \geq 1\} \subseteq \{a\}^* $$ Looking at a ...
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44 views

Checking correctness of grammar for $L = \{w \in \{a, b\}^* \text{ }| \text{ } w \text{ has } n_a(w) = 2n_b(w)\} $

I have written a CFG that supposedly generates $L$ below. $$L = \{w \in \{a, b\}^* \text{ }| \text{ } w \text{ has } n_a(w) = 2n_b(w)\}$$ Where $n_a(w)$ is the number of $a$'s in $w$ and similarly for ...
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19 views

Algorithmically find a formal grammar for a recursively enumerable formal language

The algorithmic problem is as follows. The input is the source code of a program accepting an integer as input and outputting a finite binary sequence. This program defines a recursively enumerable ...
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2answers
57 views

Are decidable set/languages EQUIVALENT to type 1 grammars (non-contracting)?

Suppose a Turing Machine (TM_G) that generates natural numbers following < or, equivalently, it generates words in lexicographical order. Then, that language/set is decidable. Because it is trivial ...
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1answer
60 views

Existence of a CFL $L$ such that $\sqrt{L}$ is not CFL

Does there exist a CFL L such that the language defined as $L' = \sqrt{L} = \{w | ww \in L\}$ is not CFL? I feel that there is no such $L$ but obviously, I am unable to prove it. I am sorry but I have ...
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How do bottom up parser evaluate things that need an inherited attribute?

I learned that Bottom up parsers use only synthesized attributes to evaluate semantics. Which makes sense considering that it would be very hard to evaluate an inherited attribute in bottom up ...
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1answer
23 views

Context free grammar transformation to Normal Form

I found a task where you need to transform context free grammar to normal form. I'm a High Shcool student at this moment. But my Brother learning this at the university. He don't have much time to ...
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1answer
36 views

Grammar for $\{a^n b^n c^m d^m \mid n \geq 1, m \geq 0\}$

I'm trying to understand how the construction of simple grammars works. In my textbook, there's the following example I am supposed to find a grammar for: Let $L_1= \{a^n b^n c^m d^m \mid n \geq 1, m ...
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1answer
745 views

Can a language be context free and not have a BNF grammar?

Leslie Lamport claims that TLA+ is too complex to be described in BNF. Does that mean TLA+ is not a context free language? What is the relationship between the set of context free languages and the ...
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1answer
67 views

Grammar for $L=\{a^{i+1}b^{i}c^{2j}d^je^{2j}|i,j>0\}$

I'm supposed to write grammar for this language: $$L=\{a^{i+1}b^{i}c^{2j}d^je^{2j}\mid i,j>0\}$$ This is what I have so far: $$\begin{align} S &\to aXbY \; \\ X &\to aXb \;|\; a \\ Y &\...
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2answers
36 views

Proving that a language defined by a regular expression is equivalent to a right linear grammar

After thinking for a bit, I am not able to prove a double inclusion proof for the following problem. It seems very interesting to me. Consider the regular expression $r= ((1(00)^∗1 + 0)1)^∗$ and the ...
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2answers
53 views

Proving that $X\to aX|Y$, $Y \to Yab|b$ is unambiguous

Prove that the following grammar is unambiguous: $$X \to aX | Y$$ $$Y \to Yab | b$$ I know that I must prove that the strings produced by this grammar have only one parse tree, but how can I do this?...
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1answer
22 views

BNF syntax for a recursive function?

I'm to write a syntax that will allow for a recursive function, i.e. f(x) = if x == 0 then x else f(x+1) Here's one attempt at creating the grammar: But I don't ...
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2answers
669 views

Why does the Java grammar have a StatementExpression that resolves to just Expression? Why have this and other redundant rules in the grammar?

I'm looking at the following grammar rules for the Java language described on the Oracle docs: ...
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1answer
42 views

CFG for $L=\{a^m b^n c^k | m,n,k > 0, k\neq m+n\}$

I started learning CFG and I'm trying to find CFG for this language, but I have no idea where to start and I can't seem to find this one online anywhere. It would be great help, if someone could show ...
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0answers
12 views

What to do with operators with the same precedence in an unambiguous grammar?

I'm trying to create an unambiguous grammar for a calculator that uses $+$, $-$, $*$, $/$ and $()$. From watching videos and reading articles online, I understand how to create the grammar with $+$, $*...
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1answer
35 views

Difficulty in understanding the proof of “Every context-sensitive language L is recursive” as given in the Peter Linz text

I was going through the automata text by Peter Linz. There I came across the proof the theorem below. I could not quite get the portion of the proof in bolds. Every context-sensitive language L is ...
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11 views

Difficultly in understanding the construction corresponding how any Turing machine can be mimicked by an unrestricted grammar

I was going through the automata text by Peter Linz where I came across the construction below. To show the converse, we describe how any Turing machine can be mimicked by an unrestricted grammar. We ...
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40 views

Eliminating ambiguity in $A \to AA \mid (A) \mid a$

I'm trying to solve this complier design problem related to ambiguity in CFG the given grammar is \begin{align} &A → AA \\ &A → (A) \\ &A → a \end{align} I was able to find that this ...
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2answers
47 views

Converting a regular expression to a context-free grammar

Does this conversion look right? I am learning conversion from RE to CFG. RE: $$(a \cup b)^* \cup ab(a \cup b)^*$$ CFG: Terminals: $$ S_1 \to a \\ S_2 \to b $$ This is for the first $(a + b)^*$: \...
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0answers
29 views

Generating a recursive descent parser for grammar having Kleene star

From what I have been taught, I cannot use left-recursive, nondeterministic, or ambiguous grammars in recursive descent parsers. So, here is the grammar: \begin{align} &E \to E+T \mid T \\ &T \...
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2answers
166 views

Arguing that $L= \{a^n | n\geq 0\} \cup \{a^nb^n| n\geq 0\}$ is not $LL(k)$ for any $k$

Consider the language $L= \{a^n \mid n\geq 0\} \cup \{a^nb^n\mid n\geq 0\}$ and the following statements. $\quad\quad\text{I. }L$ is deterministic context-free. $\quad\quad\text{II. }L$ is context-...
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2answers
43 views

Language of regular grammar

What is the regular grammar of the language: $$L=\left\{a^nb^nc^md^m\left|n,m\ge 1\right|\right\}\:above\:\Sigma =\left\{a,\:b,\:c,\:d\right\}$$ My attempt: $$S\rightarrow aAbcBd|aXd$$ $$A\rightarrow ...
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1answer
30 views

Grammar for all words other than $wq,qw$

I want to generate a grammar that can't generate the words $qw$ and $wq$ but can generate the word $qwwq$. In other words, $L(G)=\{m ∈ \{q,w\}^* \mid m \neq wq,qw \}$. My grammar: \begin{align} &S ...
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1answer
24 views

Words which, cyclically shifted twice, equal their reverse

Let the alphabet be $Σ = \{0, 1\}$. For any string $w ∈ Σ^*$ of length at least 2, define the operation $C_2(w)$ to be a cyclic shift of size 2 on $w$. That is, if $w = w_1w_2 \cdots w_n$ with $n ≥ 2$ ...
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1answer
59 views

Construct a grammar for $\{a^n(bc)^m : m,n \ge 1, m < n/2\}$

I'm new to writing languages in context-free or regular grammar, so I'm struggling how to do this one. It is a bit more complicated that simpler ones I've practiced doing. The problem is to construct ...
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1answer
30 views

How do I represent this regular expression in regular grammar?

Question: Is the regular expression and regular grammar equivalent? I've look on some examples of regular grammar however I don't think I fully understand how to convert regular expression to its ...
1
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1answer
112 views

Constructing a context-free grammar

I want to design a context-free grammar that generates words that either both start and end with $c$, or contain the same amount of $a$-s and $b$-s. Here is what I have. The nonterminals are $S,X,Y$, ...
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1answer
31 views

Regular set of the “does not contain” kind

Given a language $L$ and a set of strings $\Sigma^* = \{0, 1\}^*$, how can I find a regular set that expresses $L = \{ w \in \Sigma^* \mid w$ ends with $00$ and does not contain $11\}$? Well, the part ...
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0answers
23 views

How to change a grammar so that it can be unambiguous?

The original grammar is $$ S \to SaS \mid SbS \mid ScS \mid d $$ My answer is $$ S \to daS \mid dbS \mid dcS \mid d $$ Is that correct?
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L= ${ a^mb^nc^pd^q: m+n<>p+q }$ context free? [duplicate]

I cant find the grammar to prove it is context free but. I also tried a PDA but couldnt make it. Can someone suggest a grammar for this?
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1answer
58 views

Which of the following words is in the language of the grammar G?

This is taken from a practice quiz by my university. I ruled out that aabbbaab is not part of the grammar: S → aSb → aaSbb... This shows that I can't make this word because it would have to have ...
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16 views

Why can't I evaluate this L-Attributed SDD with a pre-order traversal?

My powerpoints for a compiler class says "an L-Attributed SDD can be evaluated with a pre-order (root, left, right) traversal", and to be L-Attributed the nodes need to have either ...
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2answers
37 views

How to evaluate a Kleene's Closure through CFG and attribute grammars

For a CFG with the production rules that can represent a regular expression. How can one calculate all the set of strings that regular expression would produce. For T = {a, b,*,(,)} and an arbitrary ...
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1answer
48 views

How can I make the following grammar unambiguous

Given the below ambiguous grammar how can I make it inambiguous and how can I prove the new modified unambiguous grammar is unambiguous? S -> S + S | S − S | S ∗ S | S / S | (S) | x | y My attempt: ...
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1answer
27 views

Formal Grammar: derivation form posted on Wiki?

Wiki describes the binary relation $\underset{\mbox{G}}{\implies}$ as "G derives in one step". I have a question on the condition when there are multiple productions for a single non-...
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1answer
59 views

Is it possible to make a grammar LL($1$) which recognizes palindroms?

Is it possible to make an algebraic grammar LL($1$) which recognizes palindroms for an alphabet $\{a,b\}$?
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18 views

generating strings from this formal grammar [duplicate]

Hey guys I am having trouble generating strings from this language, I haven't seen a grammar that looks like this and can't figure how to generate strings from this grammar, is this Context Sensitive ...
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0answers
52 views

A Formal Grammar: defining English counting numbers?

I would like to define a grammar that produces and recognizes the counting numbers of the English language. I created the production rules below based on the assumption this is context-free, but I am ...

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