Questions tagged [formal-grammars]

Questions about formal grammars, generative descriptions of formal languages.

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Constructing a DFA of strings that are in A but not in B

I am tasked with creating a DFA for the regular language L = A/B, which are the strings that are in A but not in B. The alphabet is Σ = {a,b,c} I am not really sure where to even start with this one, ...
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Pushdown Automata - constructing a PDA to recognise a language with at least as many as as bs

I am trying to construct a 3-state PDA to recognise (I need to create a transition diagram for this question) ...
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1answer
30 views

Context-free grammar how to have unequal number of a's on either side of b

I have been trying to create a CFG for the set $\{w=a^iba^j \mid i \neq j\}$. To my understanding, there are essentially 2 scenarios, one where there are more $a$s on the left side of $b$, and one ...
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How to split a context-free language into three sub-languages?

I try to split the language $$ L = \{a^ib^j \mid i \neq 2j, i \neq 3j\} $$ into three languages \begin{align} L_1 &= \{a^ib^j \mid i < 2j\} \\ L_2 &= \{a^ib^j \mid 2j < i < 3j\} \\ ...
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48 views

Proving that the set of grammars generating L or L complement is undecidable

Let $X$ be a regular language, I need to prove that either $\{G \mid L(G) = X\}$ or $\{G \mid L(G) = \overline{X} \}$ is undecidable using the following hint: Use reduction to absurdity supposing that ...
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1answer
78 views

LL(k) language and not LL(k) grammar

I have nonambiguous and not LL(k) grammar which defines some language. How can I prove that I can't build some LL(k) grammar for this language? Grammar: S -> a b X c d | a X f X -> b X c | ε
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32 views

I want to design a context free grammar for the following [closed]

This below language $$L = \{ w \in \{a, b\}^n : \lvert w\rvert \text{ mod } 3 = 0 \}$$ where $n \geq0$.
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Formal definition for strong equivalence of grammars/parse trees

The Wikipedia article on equivalence of formal grammars defines weak and strong equivalence as follows: In formal language theory, weak equivalence of two grammars means they generate the same set ...
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1answer
33 views

Are you allowed to use index variables in grammars?

If we say that the grammar for the language $L = \{ww \mid w \in \{a,b\}^*\}$, is: $\begin{align} S &\rightarrow A_1A_2S \mid B_1B_2S \mid Z_2\\ A_2A_1 &\rightarrow A_1A_2\\ A_2B_1 &\...
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Ambiguos Grammar detection algorithm

I tried to solve if a grammar is ambiguous or not. I know that I have to find a word that can be generate by two different leftmost derivations. Is there any algorithm to find a word that, if the ...
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2k views

How to find whether a grammar is ambiguous?

$$S\rightarrow aAB$$ $$A\rightarrow bBb$$ $$B\rightarrow A|\epsilon$$ I'm choosing the string abbbb. First left most derivation is the following: $$S\Rightarrow aAB\Rightarrow abBbB\Rightarrow ...
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31 views

What is the density of a regular language $L$ over an alphabet $\Sigma$ in $\Sigma^n$?

In other words, what is the likelihood that a recognizer of a given regular language will accept a random string of length $n$?   If there is only a single non-terminal $A$, then there are only ...
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Acyclic Manhattan turtle

There is a grammar that describes the walks of a turtle around Manhattan, such that the turtle always returns home. It is described in the book "Parsing Techniques" by Dick Grune and Ceriel J.H. ...
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24 views

Conversion of left-recursive context-free grammars to strongly equivalent ones without left-recursion

It is a well-known problem that many top-down parsers have problems parsing a context-free grammar with left recursive rules. There exist algorithms to convert grammars with direct or indirect left-...
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43 views

Is Python's Grammar in a known Category between CFG and CSG?

I have a high level understanding of formal languages and grammars, and I'm familiar with the four major types of grammars in Chomsky hierarchy. I was interested in knowing the classification of ...
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41 views

Show that $L:=\{(a^{k}b)^{i}|i,k \epsilon \mathbb{N}_{+} \}$ is context-sensitve. (With context-sensitive/noncontracting grammar)

I am studying for an upcoming exam and this is an old exam question from two years ago (all exams were made available through our lecturer): Show that $L:=\{(a^{k}b)^{i}|i,k \epsilon \mathbb{N}_{+} \}...
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308 views

Shift-resolve parsing - questions

I've recently came across a paper describing the parsing technique mentioned in the title. Unfortunately, the terminology used in said paper is somewhat beyond my comprehension, so I've been ...
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1answer
30 views

Is following grammar has language which is inherently ambiguous?

Grammar is as follow: $S \rightarrow aaAb | aab | A$ $A \rightarrow aaAb | aAb | \epsilon$ I think that this grammar has equivalent unambiguous grammar as follow. Let’s first rewrite the grammar ...
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35 views

Language containing all unambiguous grammars

Suppose $L$ is the language of the unambiguous grammars. That is, a sentence $w\in{}L$ if it is a string that describes an unambiguous context-free grammar. Considering that deciding whether a ...
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52 views

Whether following language is linear or not?

I have a language $L= \{a^nb^nc^m : n, m \ge 0\}$. Now, I wanted to determine whether this language is linear or not. So, I came up with this grammar: $S \rightarrow A\thinspace|\thinspace Sc$ $A \...
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16 views

With a grammar, does precedence and associativity change the accepted language?

When parsing, certainly precedence and associativity effect the AST. But, do precedence and associativity alter the set of accepted sentences? In other words, can precedence and associativity be "...
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36 views

Create automata from non regular grammar

I have two grammars: L → ε | aLcLc L → ε | aLcLc | LL This two grammars are equals but the first one is regular, so it produces a regular language and a ...
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1answer
70 views

Unrestricted grammar which generates $\{ a^1\#a^2\#a^3\#\dots \#a^k \mid k >0 \}$

I am looking for an unrestricted grammar which generates the following language: $\{ a^1\#a^2\#a^3\# \dots \#a^k \mid k >0 \}$ That is, words like $a\#aa\#aaa\#aaaa\# \dots \# \text{$k$ times '$a$...
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1answer
182 views

Proof that the grammar is LL(2)

I am given the following grammar: $ S \rightarrow AabAba \\ A \rightarrow a | \epsilon $ and I have to prove it is LL(2). I know what LL(k) means - one can choose a production based on k characters ...
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199 views

Negative lookahead in LR parsing algorithm

Consider such a rule in grammar for an LR-family parsing generator (e.g YACC, BISON, etc.): Nonterminal : [ lookahead not in {Terminal1, ..., TerminalN} ] Rule ; ...
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3k views

Simplification of CFG

Recently i was studying removal of useless symbols in productions given in Ullman Hopcroft. The grammar goes as follows S-> aAa | aBC A -> aS | bD B - > aBa | b C-> abb | DD D -> aDa In the ...
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CFG - Left factoring in recursive nested productions

I'm attempting to convert a CFG into an LL(1) grammar for predictive parsing in a compiler. I've been able to left factor and eliminate left recursion and ambiguity for every case in the grammar, with ...
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2k views

Difference between substitution, morphism, and homomorphism

In closure properties, I got confused between substitution and morphism. 1) According to Wikipedia, string substitution means to map letters in a set of alphabets to languages (possibly in a ...
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2answers
117 views

How do i tell if a grammar is regular or not?

I know that a regular grammar has a definition $$\begin{align}S &\to aS\\ S &\to \lambda \end{align}$$ But I dont really know how to apply this information to check whether or not a grammar ...
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Why are CFL not closed under set difference, and complementation? [duplicate]

I was wondering why CFL are not closed under set difference, and complementation can anyone explain? I tried searching, but no luck.
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7k views

What is rightmost sentential forms?

I'm solving some past job interview problems. I met an embarrassing question about compilers. The question is : Consider the following grammar, with start symbol $E$: \begin{align*} E &\...
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3answers
434 views

Context-free grammar of the concatenation of a string S and subsequence of reversed S

I have to find a Context-Free grammar that generates the language: $L_1 = \{x\#y\ |\ y$ is a subsequence of $x^R$, and $x\in\{a,b\}^*\}$, $\Sigma=\{a,b,\#\}$ The concatenation of two mutually ...
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2answers
350 views

Context-free grammar for binary words

I am supposed to create CFG for this languague: $L= \{w : w \in \{a, b\}^*, |w_b| = 3k, k \geq 0 \}$ where $|w_b|$ is count of terminals $b$ in $w$. For example: aa - OK, no 'b' abb - wrong, only ...
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1answer
22 views

Calculating LL(1) grammar

I am trying to calculate the First and Follow of the following grammar S->ABC A->Aa A->aB B->Bb B-> epsilon C-> Cc C-> Epsilon I have ...
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grammar for the no.of a's one more than no. of b's [duplicate]

write the grammar for the following L = {w : na (w)= nb (w) + 1}. * no. of a’s is one more than no. of b’s *
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202 views

Prefix/suffix property of language containing only empty word

Does language $L ={\varepsilon}$, where $\varepsilon$ - empty word has suffix/prefix property? The definition says that language has prefix/suffix property requires that there is no code word in the ...
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1answer
28 views

Why $P$ cannot have NULL string in Arden's Theorem?

Arden's Theorem says that in the equation $R=Q+RP$, the $P$ cannot have NULL string. In this respect,the theorem will not be valid for the expression $R=Q+R(NULL+01)$. Am I correct? If so, then what ...
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21 views

Is $LR(k) \subset SLR(k+1)$ for $k=1,2,…$? [duplicate]

I know that: Point 1: Set of languages accepted by $LR(0)$ parsers $\subset$ Set of languages accepted by $SLR(1)$ parsers Does this logic hold for higher $k$'s? That is, does following fact hold?...
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30 views

Does SLR(0), LALR(0) exists?

I read about LL(1), LR(0), SLR(1) and LALR(1) in many online sources and even in dragon book. However I found that no one talks about LL(0), SLR(0) and LALR(0). So I googled and come up against these ...
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Proving correctness of LR parser facts

I have came across following facts while reading some compilers related text. However I did not find them in any standard reference book (mainly dragon book). Are they correct? If yes, how can we ...
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1answer
29 views

How making non LL, non LR grammar a valid LL grammar, also makes it a valid LR grammar? Is there any connection between LL and LR conflicts?

I might unncecessarily overthinking here, but I had this weird possibly meaning less doubt: When grammar is neither LL nor LR, it means, both LL and LR parsing tables involve conflicts. LL parsing ...
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1answer
56 views

Confusion with first condition of Chomsky Normal Form

I had a very quick question when it comes to CFG (more specifically the attributes of CNF). I've been browsing over some examples and I've come across a few that confuse me. One such example is this: ...
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75 views

Does left factoring CFG make it unambiguous?

I came across following problem: If the CFG is left factored then it must be Unambiguous and Not left Recursive. TRUE/FALSE? I have many thoughts about this. But I feel they are somewhat ...
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96 views

How to give a context-sensitive grammar for a^nba^nba^nb?

I am struggling on this problem since days: $L = \{a^nba^nba^nb \mid n \in \Bbb N\}$. I have to give for this language a context-sensitive grammar.
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1answer
60 views

Why are four context sensitive grammar (CSG) rules needed to represent AB -> CD?

In Wikipedia of Kuroda normal form, it says A straightforward technique attributed to György Révész transforms a grammar in Kuroda's form to Chomsky's CSG: AB → CD is replaced by four context-...
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1answer
40 views

Wikipedia says this grammar is LR(0), but Grammophone says it is not; is it?

E -> E * B . E -> E + B . E -> B . B -> 0 . B -> 1 . I am confused because Wikipedia cites this grammar as an example of an LR(0) grammar ...
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1answer
47 views

Is there a LL(K) Grammar which is not LALR(K) Grammar?

It is easy to know that there are LALR(K) grammars which are not LL(K) because any grammar with left recursion which is LALR(K), is not LL(K) because all LL(K) grammar must be left recursion free. And ...
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1answer
50 views

Construct a grammar for the language

How can I achieve this. Write a grammar for the language consisting of strings built only of the letters a and b. The strings may have any number of these letters, but the letter combinations bab must ...
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16 views

FOLLOW of LL(1) grammar [duplicate]

Please consider following grammar: E -> P : P P -> X R R -> . P | ε X -> num Y Y -> num Y | ε I am searching FIRST/FOLLOW. First: ...
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62 views

Connection between non determinism and LL(1) conflicts

I am trying to understand connection between non determinism of grammar and LL(1) conflicts introduced by it. As per my understanding non deterministic context free grammar is a context free grammar ...