In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) is a set of production rules for strings in a formal language. A formal grammar is a set of rules for rewriting strings, along with a "start symbol" from which rewriting starts. Therefore, ...

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Construct grammar $\ a^i b^j c^{i+j} b^j a^i $

I've been going through old exams at my college and I found this problem that I haven't yet been able to solve. Construct grammar defined on the alphabet $\ \{{a, b, c}\} $ which generates strings of ...
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33 views

Context-free language or not

It is language: $L = \left\lbrace a^ib^jc^kd^l \mid i+k < j+l+3 \right\rbrace$ Is it context-free or not? I have two versions: 1)Pumping lemma(refute): get a word $uvxyz = aabcd$ if $i=2$ we get ...
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Formal writing in math: equations

What is the formally correct way to solve a bunch of equations in math? Is it \begin{align} 42x = 4324 \\ x = 4324/42 \end{align} or \begin{align} 42x = 4324 \\ \Rightarrow x = 4324/42 \end{align} or ...
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1answer
40 views

Is this language context-free or not?

I have a problem to solve, the problem is: Is the language of strings $$L=\{0^x1^y:x\nmid y\}$$ context free? I suspect it isn't, I spent some time trying to make a grammar that could ...
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1answer
45 views

Write grammar for given language

I'm trying to write a grammar for the language below: $$(a+b)^*βˆ’\{𝑀𝑀𝑀 ∢ 𝑀\in\{a,b\}^*\}$$ could anyone help me?
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59 views

How to show $L = \{a^n b^m c^p: n\neq m\} \cup \{a^n b^m c^p: m\neq p\}$ is ambiguous?

I'm recently familiar with this site and prefer to ask a very hard problem :) How can we prove that the following language is inherently ambiguous? $$L = \{a^n b^m c^p: n\neq m\} \cup \{a^n b^m ...
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1answer
25 views

Context free grammar and regular expressions

Consider a grammar $G = (V, \Sigma, R, S)$ where $V = \{S\}$, $\Sigma =\{A, B\}$ and $R$ has two production rules, namely $S \to S^+ AS $ and $S \to B$. Is $G$ context-free? The $^+$ symbol is Kleene ...
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47 views

Is it possible to write an Unambiguous Grammar for Two Hard Language ?!?

I came across a very hard interview exam. It was asked wrote an unambiguous grammar for two following language, Who can hint it to solve it? 1) $L = \{a^n b^{2n} c: n\geq 0\} \cup \{a^{2n} b^n d: ...
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1answer
85 views

Languages and their relation : help

This is not a homework question though, this is something I wish to know to add to my own knowledge. While reading one of the texts on automata (K.L.P. Mishra, "Theory of Computer Science : Automata, ...
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53 views

$L_{a}= \{w_{1}cw_{2} : w_{1},w_{2} \in \{a,b\}^{\ast}, w_{1} = w_{2}\}$ [closed]

Why the following is not context free? Anyone could describe it for me. $L_{a}= \{w_{1}cw_{2} : w_{1},w_{2} \in \{a,b\}^{\ast}, w_{1} = w_{2}\}$
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109 views

Context Free Grammar and some details [closed]

Why this grammar shows a Context Free and Linear Language, (i.e, not context sensitive or non-context free). S -> SBA | a BA -> AB aA -> aaB B -> b
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1answer
31 views

Eliminating epsilon-productions in grammar

I am wondering how to eliminate epsilon-productions in grammar: S β†’ S0 S β†’ 1 S β†’ AB B β†’ AC A β†’ Ξ΅ C β†’ Ξ΅ I know that because of ...
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1answer
36 views

Formal Grammar generating $0^p$

Exercise: Find the formal grammar generating the language ${0^p}$ in the binary alphabet for $p$ prime. I have absolutely no clue where to start, nothing of the 'usual' construction strategies seem ...
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1answer
33 views

Construct a grammar that generates this language

This is a homework problem. The problem is: Find a grammar that generates this language: L = {w: |w| mod 3 β‰  |w| mod 2} over alphabet Ξ£ = {a}. The transitions I came up with are: S -> Baa B -> ...
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2answers
118 views

Find a grammar that generates this palindrome language

This is a homework problem. The problem is: Find a grammar that generates this language: L = {wcw^R: w ∈ {a,b}+ } over alphabet Σ = {a, b, c}. I have tried many different transitions, but can't ...
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152 views

Determine type of grammar?

Let $V = \{S, A,B, a, b\}$ and $T = \{a,b\}$. Determine whether $G = (V, T, S, P)$ is a type $0$ grammar but not a type $1$ grammar, a type $1$ grammar but not a type $2$ grammar, or a type $2$ ...
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1answer
23 views

Given the following grammars with start symbol $\langle S \rangle$, specify the type ($0$, $1$, $2$ or $3$)

So I'm working on this problem set and I'm having some trouble figuring out what type each one of these are. I think (a) is type $0$ and really can't tell for (b). I know the difference between each ...
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1answer
108 views

Turing Machine Problem

We know, A Turing machine is a hypothetical device that manipulates symbols on a strip of tape according to a table of rules I Draw a TM for input $x=(0+1)^*$ i want to implement ...
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0answers
38 views

Smallest grammar problem on a single character.

Let the alphabet be $\Sigma = \{a\}$. Say $s = a^6 = aaa aaa$. If the repeated variable $A = aa$ appears $k$ times in the expanded starting rule of a smallest grammar $G_s$ for $s$. Then that ...
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1answer
55 views

Regular Language and Unknown Operation

I read the $L$ is a Regular language. Define $L'$ as following: $$L'=\{a_2a_1a_4a_3\ldots a_{2n}a_{2n-1}\mid a_1a_2\ldots a_{2n} \in L\}.$$ why $L'$ is Regular? any hint or idea would be highly ...
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30 views

Algebra on trees?

Given a forest of trees, I am interested in tree recombination operations that produce me a sub-forest spanned by some selected trees. To me, it somehow resembles space and spanned subspace in ...
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0answers
27 views

Do automatons and formal grammars belong to formal systems

A formal system consists of a formal language, a formal grammar that generates the language, a set of axioms, and a set of inference rules. Are automatons also formal systems? Is an automaton ...
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35 views

Predictability of a grammar

I've encountered this in a book: The language $ \{ a^n0b^n \mid n β‰₯ 1\} βˆͺ \{a^n1b^{2n} \mid n β‰₯ 1\}$has no predictive grammar. (A predictable grammar is that in which no two rules of production for ...
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1answer
219 views

Context free grammar $L=\{a^ib^jc^k|j=i+k-2\}$

$L=\{a^ib^jc^k|j=i+k-2\}$ This expression surprise me a lot and put me into deep thinking. what i am doing by solving the expressions: ...
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0answers
32 views

What about generlizing grammars?

Say we came up with a finite grammar (specifies a finite number of strings) given a set of input strings. How then do we generalize the grammar so that it works for larger input strings and also ...
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1answer
48 views

Chomsky Normal Form for Integer Recognition

If I have the following CFG, which is just the regex [0-9]+: ...
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39 views

There is a sequence of operations on grammars of a string that strictly decreases the size of grammars down to the smallest grammer.

I'm trying to figure out the smallest grammar problem, which yes I know is impossible since it's such a hard problem, but humor me for a sec. Let $g$ be a smallest grammar for the string $s$ over the ...
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1answer
78 views

Why use *Ξ»x.x* instead of *f(x)*?

In my semantics class, we're learning about using (abusing?) lambda calculus. So far the professor hasn't imparted any reason for using Ξ»x.x instead of using f(x). Why use lambdas instead of basic ...
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1answer
212 views

Is well formed XML context-sensitive grammar?

Solution Copy language is noncontracting, so it's context-sensitive. Look at https://en.wikipedia.org/wiki/Noncontracting_grammar for transforming noncontracting grammar to explicitly $\alpha ...
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362 views

How to denote this in game theoretic notation

I'm writing a paper that demonstrates that linguists can use the concepts in game theory to infer what interlocutors naturally infer when the literal meaning of their utterances doesn't ostensibly ...
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1answer
71 views

Palindromes with 2 symbols and $3|l(u)$

The following grammar generates palindromes with 2 symbols. $$G=\{\{S\}, \{a,b\}, \{S\rightarrow\epsilon|a|b|aa|bb|aSa|bSb\}, S\}$$ So if I'm right, each $u$ in the language $L$ generated by $G$ is a ...
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1answer
29 views

How can I show ithat a language is regular?

I have a very quick question about regular languages, I think $\{a^{2n}| n\geq 1\}$ is regular. I do know that pumping lemma can be used to show something that is not regular. I am wondering what ...
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1answer
40 views

Question About Pumping Lemma used on a FA

I am learning about pumping lemma and I am trying to solve a problem. I need to use pumping lemma to show that: the Language L(M) defined by the following machine is infinite. Here is the dfa: ...
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1answer
34 views

How to notate truth conditional functions

My semantics professor uses functions to teach her class. She wrote the following sentence and three examples in one of her slides. [[smile]] is a function that takes something, let’s call it x, and ...
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1answer
72 views

Creating a Push Down Automaton from a Grammar

I have the following grammar, but I'm not sure what exactly it is that it does: $\qquad\begin{align} S &\to S \vee T \mid T \\ T &\to T \wedge F \mid F \\ F &\to p \mid \thicksim p ...
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1answer
399 views

Eliminating Epsilon Production for Left Recursion Elimination

Im following the algorithm for left recursion elimination from a grammar.It says remove the epsilon production if there is any I have the following grammer ...
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1answer
93 views

Context free grammar for a language

I have this context free language $L = \{x^{3i} y^j z^k\ |\ i \ge 0 \land k \gt 2j \gt 0\}$ I'm working out a ...
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1answer
71 views

Shortest word possible

In a give context-free grammar language $L = \{z^{3x} y^i w^j \ |\ x \ge 0 \land j \gt 2i \gt 0\}$ The shortest possible word that does not belong to the ...
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1answer
292 views

Converting Context Free Grammar to Chomsky Normal Form

This is an exercise that I had to complete in my class and I struggled a lot with it $$\begin{align*} &S\to 0A0\mid 1B1\mid BB\\ &A\to C\\ &B\to S\mid A\\ &C\to S\mid\epsilon ...
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1answer
96 views

Grammar outside the Chomsky Hierarchy

This grammar describes a language that may fall outside the Chomsky Hierarchy (CH): \begin{array}{l} S \to abAbba \\ A \to abA \mid bbaB \\ B \to aab \\ \lambda \to Aab \mid aB \\ \end{array} Going ...
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1answer
49 views

Consider this grammar

Consider this grammar: \begin{array}{l} S \to aabBba \mid aAb \mid aab \\ bBb \to bCa \mid aaa\\ aA \to aC \mid bba\\ C \to aab \mid Cb \end{array} This is clearly context-sensitive (CS). It's not ...
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1answer
135 views

Help with context sensitive grammar [closed]

I am not able to understand writing a context sensitive grammar for this language. Can anyone please help me out? $L = \{ a^p | p \text{ is a non-prime integer}\}$
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214 views

Write a grammar that generates the strings over {a,b} starting with a

The answer is: S -> aA, A -> aA, A -> bA, A -> a, A -> b, S -> a Any idea how they got this?
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1answer
1k views

Build regular grammar from regular expression

Is there an algorithm for creating a regular grammar directly from a regular expression? All the discussions and notes I found so far go through an intermediary step of creating an FA for the reg ex ...
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1answer
72 views

NFA from grammar productions

Based on this grammar: \begin{align} G = (\{S,A,B\}, \{a,b, c\}, S, P) \end{align} \begin{matrix} \\P: \\S β†’ abaS | cA \\A β†’ bA | cB | aa \\B β†’ bB | cA | bb \end{matrix} I created this NFA: ...
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1answer
214 views

Right-linear grammar from regular expression

I made a right-linear grammar that from this regular expression: The alphabet is: $Ξ£ = \{a, b, c\} $ Regular expression: $r = cc^*(ba)^*bb$ My solution, it seems a little too short like I'm ...
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1answer
53 views

NFA from regular expression

I'm trying to make an NFA from the following regular expression. I'm not sure about the edges between nodes $q2$ and $q4$, maybe someone can point out where everything went wrong.
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1answer
30 views

Language made by a regular expression

I created a language from this regular expression but I'm not sure about it, especially where I wanted to use the $w$ to express a sequence of terminals. The expression: $r = a a ^{*} (b + bb + bbb) ...
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1answer
27 views

Regular expression for a language

I made a regular expression to match this language but I'm not sure it's right. Perhaps someone can show me where it deviates. The language: $L = {a^{n} c b^{m} (cc)^{p} : n \geq 1, m \leq 1, p\geq ...
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2answers
77 views

Constructing regular grammar

I'm trying to make a regular grammar for this language: $$ L = \{ a^ncb^m(cc)^p : n\ge 1, m\le 1, p\ge 0\} $$ Where the alphabet is $ \Sigma = \{a,b,c\}$ It seemed like right-linear. This may be ...