Questions tagged [formal-grammars]

Questions about formal grammars, generative descriptions of formal languages.

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11 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
33 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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38 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
29 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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26 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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37 views

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

$\text{Consider the language $L= \{a^n | n\geq 0\} \cup \{a^nb^n| n\geq 0\}$ }$ $\text{and the following statements.}$ $\quad\quad\text{I. $L$ is deterministic context-free.}$ $\quad\quad\text{II. $L$...
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2answers
40 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
28 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
22 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
27 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 ...
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1answer
105 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
29 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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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
49 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
33 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
45 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
26 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
49 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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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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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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1answer
16 views

Compilers: How to see “the number of grammars where there exists a string that has at least two different left-most derivations”?

Could someone tell why "G1 and G3 are ambiguous" and how to see whether a string has at least two different left-most derivations in general?
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1answer
44 views

Proof that $L = \{a, b\}^* - \{(a^n b^n)^m \mid n, m \ge 1\}$ is a CFL

I want to prove that $L = \{a, b\}^* - \{(a^n b^n)^m \mid n, m \ge 1\}$ is a Context Free Language. so far, I tried to find a Context Free Grammar for $L$ or to use properties of Context Free ...
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20 views

I am trying to design an LL(1) Parser that accepts T = {a, b *, +, ?, E, U, (, ) }

I am trying to design an LL(1) Parser that accepts regular notation where 'E' represents epsilon, and 'U' represents "or" like ' | '. So far I made one that accepts T = { a, b, *, +, (, ), E}...
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1answer
36 views

Removing left recursion with terminals only

I have a grammar: $G → id > id$    $| id < id$    $| G and id$ Does anybody know how I can do left recursive elimination on this one, when it doesn't have any extra non terminals?
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17 views

Compile XPath Abbreviated Query to Unabbreviated version

The Xpath 3.1 presented by W3C includes the full grammar of the language with both abbreviated and unabbreviated syntax. I am interested in references (if any) for any formal work done to compile/...
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2answers
66 views

Context free grammar for $L = \{w \text{ | }w \in \{a,b\}^*, |w|_a=|w|_b-1\}$

I'm trying to find a grammar for $L = \{w \text{ | }w \in \{a,b\}^*, |w|_a=|w|_b-1\}$, which is proving to be tricky. I know that $L_2 = \{w \text{ | }w \in \{a,b\}^*, |w|_a=|w|_b\}$ has the following ...
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1answer
42 views

Computing FOLLOW sets of left recursive grammar

Left recursive ambiguous expression Grammar: $E \rightarrow E+E \mid E*E \mid (E) \mid \mathbf i\mathbf d$ I tried computing FIRST and FOLLOW sets of both left recursive grammar and after ...
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1answer
36 views

Is there a formal language of Combinatory Logic's expressions?

The Combinatory Logic uses expressions of the form (x y) called "applications" (here, we have an "application of x to y"). Thus, the language of CL is a set of "parenthetic ...
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1answer
76 views

Derivation from grammar

Given the grammar $G=(\{S, L_x, R_x, W_x\}, \{a,b\}, P, S)$ derive the words $abaaba$ and $aabbaabb$. $$ P=\left\{ \begin{align}S\phantom{{}_x R_y} &\to \epsilon \mid L_x R_x,\\ L_x \...
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1answer
38 views

Finding a grammar for $L=\{a^nb^mc^rd^s| n+m<r+s\}$

I am trying to find a grammar for $L=\{a^nb^mc^rd^s| n+m<r+s\}$, which has the hint of it having "some similarity" to $L=\{a^ib^j|i<j\}$ This last one is quite easy to get ($S\to aSb | ...
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4answers
79 views

If $L$ is regular then $L^{|2|}=\{w_1w_2 \mid w_1,w_2\in L, |w_1|=|w_2|\}$ is context-free

I have found a problem about proving whether $L^{|2|}=\{w_1w_2 \mid w_1,w_2\in L, |w_1|=|w_2|\}$ is context-free or not, knowing that $L$ is regular So far I know that: There are examples where $L$ ...
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2answers
62 views

Finding a grammar for $L = \{ 0^x1^y0^z1^w | x+w=y+z\}$

I have found an exercise where it tasks to provide a grammar and a pushdown automata for $L = \{ 0^x1^y0^z1^w | x+w=y+z\}$ While finding a pushdown automata for it is quite easy (four states and two ...
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2answers
186 views

Trying to remove ϵ rules from a formal grammar resulted in L(G) ≠ L(G')

I am trying to remove ϵ rules from the following grammar (after applying the remove redundant symbols algorithm): $G = (\{S,A,B,C\},\{0,1\},P,S)$, where the productions are \begin{align} &S \to AB ...
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1answer
50 views

Is there a method to generate the complement of a context-free grammar?

Given the languages $L_0 = {w \in \{0,1\}^*}$ such that $w$ is a palindrome and $L_1 = {w \in \{0,1\}^*}$ such that $w$ is not a palindrome, meaning $L_1$ is the complement of $L_0$, we want to find ...
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1answer
25 views

Is the complement of the language generated by $S \to aSbS|\epsilon$ context-free?

How is it possible to prove whether the language $\{a, b\}^{∗} \setminus \{S → ε, S → aSbS\}$ is context free?
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1answer
242 views

Context-free grammar for all words not of the form w#w

I was asked to define a CFG for the complement of $\{w\#w \mid w \in \{0,1\}^*\}$ and I'm struggling to define it. I think it is quite similar to defining a CFG for the complement of $\{ww \mid w \in \...
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1answer
30 views

Trying to find two CFGs for the following languages

I'm trying to get CFGs for these two languages which still remain unsolved in my practice problems sheet: $L = \{ a^kb^ra^m | m=k+r\}$ $L = \{ a^nb^m | 1\leq n\leq 2m\}$ With the first one, I thought ...
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1answer
27 views

How can Chomsky hierarchy be applied to languages with alternated letters?

I have the following grammar, which I know it is regular because it can be represented by a finite state automata: \begin{array}{l} \mathrm{S} \rightarrow \mathrm{X} \mid \mathrm{Y} \\ \mathrm{X} \...
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0answers
22 views

Building Context sensitive grammars?

I just discovered Context-sensitive grammars. The problem is most of the examples are weird non-interesting toy languages ! Second the descriptions are math oriented, rather than programmer oriented. ...
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0answers
26 views

Parsing a context free grammar, Backus Naur question

Does anyone know how BNF rules expecting the empty string ($\epsilon$ or the "") behave during creation of a parse tree using grammar from a string of ...
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1answer
660 views

What does it mean for a grammar to be LR(0)?

I am unsure what it means for a grammar to be $X$. More specifically, what it means for a grammar to be LR(0). For part of an assignment I had to form the DFA for a grammar, which I had no issues with....
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1answer
69 views

How can I show that this language is context sensitive?

I have this language $L=\{a^nb^nc^n,n\geq0\}$, I know this language is not context free, but I don't know how to show that it is context sensitive, do I have to use a PDA?
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1answer
38 views

Formal proof of language accepted by a specific CFG

Let $G=(V,\Sigma,R,S)$ be the grammar given by the following rules: \begin{align} &S \to aS \mid B \\ &B \to abBc \mid \epsilon \end{align} Please provide a formal proof for the following ...
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1answer
64 views

How do you create a sentential form in a given grammar?

I am given the following grammar: $$S ::= aBS| abT |a$$ $$T::= d | dT$$$$B ::= da | ϵ | S$$ I need to decide whether $aBaabda$ can be produced in the given grammar. I am unsure how the grammar can ...
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0answers
14 views

Having trouble understanding how to prove a language context free? [duplicate]

I've been studying the theory of automata. I came across this problem in the book and unable to understand how to solve this. I've solved some examples using the Pumping lemma but this one uses ...
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1answer
72 views

How can we generate a grammar for $\{a^n b^n c^n d^n; n > 0\}$ if it is NOT context free?

This page on Wiki states that $\{a^nb^nc^nd^n \ | \ n > 0\}$ can not be generated by a CFG. This does not make sense to me as $\{$S $\to$ ABCD, A $\to$ aA | a, B $\to$ bB | b, C $\to$ cC | c, D $\...

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