Formal languages are studied in computer science and linguistics. They are usually defined using various types of grammars (e.g. regular, context-free) and automata (e.g. deterministic and pushdown automata, Turing machines). There is a hierarchy of formal languages, which is based on the type of ...

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Regular Expressions Help

I need a little help with Regular Expressions. The allowed operations are obviously + (union) , * (Kleene star) and concatenation. I have to write Regex for the following 2 examples. I have tried a ...
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how to a draw a turing machine that has the same number of a's ,b's and c's

how to a draw a turing machine that has the same number of a's ,b's and c's SOMELANGUAGE = {abc acb bac bca cab cba aabbcc aabcbc}
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DFA Construction with three strings in language

Draw DFA that recognizes the following language, with the alphabet {0, 1} {0011, 11, 0101} I'm having a lot of trouble with this, because I know DFA have to have a determined path from each state ...
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Finding the CFG (Context Free Grammar) of a language

Can we write a CFG (Context Free Grammar) for the set of all non-empty string whose length are multiple of 3 on the alphabet $ \Sigma = \{A,R,G,C\} $
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1answer
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Constructing PDA for a language

I want to prove or disprove that for a given two PDA's (Pushdown Automata) $M_1$ and $M_2$, we can build a PDA $M$ such that $$L(M) = \{w \in L(M_1) \mid w\text{ contains some string in ...
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Show L is not context free using the CFL pumping lemma

I am trying to use the pumping lemma to show this language is not context free: $L = a^nb^{n+1}c^{2n} : n \ge 0$ So I took $z = a^mb^{m+1}c^{2m}$ where $|z| = 4m+1 > m$. We can decompose $z = ...
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4answers
533 views

Why isn't there a pumping lemma for recursively enumerable languages?

I'm studying the theory of computation, and I know there are pumping lemmas for regular and context-free languages, but why not for recursively enumerable languages? Is there something about a Turing ...
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If $P=NP$, prove that $L' \in NP$

I think I'm overthinking this problem and need some hints in the right direction. The goal of this question is to show that if $P=NP$ then for every language $L \in NP$ via a polynomial time verifier ...
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(L1* ∩ L2*) = (L1 ∩ L2)* for all languages L1 and L2 over the alpabet Σ={A,B} Is it true or false and why?

plz answer me Determine whether each of the following statements is true or false. If a statement is false, give a counterexample..... 1- $(L_{1}^{*} \cap L_{2}^{*}) = (L_{1} \cap L_{2})^{*}$ for ...
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1answer
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Is this function (Strings and Languages) True or false, and why? plz help me

plz answer this two question if u know an give me the reason Determine whether each of the following statements is true or false. If a statement is false, give a counterexample. 1- uv=vu for all ...
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1answer
20 views

Prove a language is Context Free

I am working on the following problem and I do make a little progress. Let me state the question first: Prove that the set of strings consisting of $a$'s and $b$'s with an equal number of $a$'s and ...
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1answer
58 views

Prove 2-HamiltonianCycle $\in \textbf{NP}$

Just want to verify that I have the right idea here with this hamiltonian cycle question. $HC$ = $\{\langle G \rangle$ | $G=(V,E)$ is an undirected graph such that there is a simple cycle (no vertex ...
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2answers
42 views

Is it possible to formalize areas such as image processing and computer vision?

Is it possible to formalize areas such as image processing? By formalize I mean setup axioms, then derive theorems, and reason about image processing concepts and methods formally. I would say now ...
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0answers
56 views

Determine the language that corresponds to the following automata.

I want to determine that language that corresponds to the following automata Note: $q_{6}$ have arrow to $a$ to himself. I started with the minimal words: $aaabb$ $aaba$ $aaaba$ $bababa$ the ...
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0answers
14 views

What is the correct notation for A∆B?

For the alphabet Σ = {0, 1}, let A, B, C ⊆ Σ∗ be the languages below. i. A={1,0,00,11,000,111,0000,1111} ii. B={w∈Σ∗|||w||≥2} iii. C = {w ∈ Σ∗|||w|| ≤ 2} Note: ||w|| denotes the length of the word ...
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1answer
31 views

Determine the following languages of Σ∗

For the alphabet Σ = {0, 1}, let A, B, C ⊆ Σ∗ be the languages below. i. A={1,0,00,11,000,111,0000,1111} ii. B={w∈Σ∗|||w||≥2} iii. C = {w ∈ Σ∗|||w|| ≤ 2} Note: ||w|| denotes the length of the ...
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1answer
23 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
31 views

Name for grammars with rules $A \to uA$

Recall that a right-linear grammar is a grammar that consists of rules of the form $A\to uB$, where $A$ and $B$ are non-terminals and $u$ is a (possibly empty) word of terminals. Similarly for ...
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1answer
33 views

Finite state machine

I am doing discrete math, and we are studying Finite State Machines. But i am a little confuse on how to do this. Here is a question, Write a regular expression for the language, and define a finite ...
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0answers
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Eliminate Useless Productions

How do I eliminate useless productions from the grammar: $S \rightarrow a|aA|aaB|abC$ $A \rightarrow aB|\lambda$ $B \rightarrow Aa$ $C \rightarrow cCD$ $D \rightarrow ddd|aC$
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1answer
33 views

Prove or disprove whether the following is a regular language

I'm given the regular language L, and w being an element of L. If we remove the w from the language L, will the resulting language be still regular? Well I thought to be true. Since initially is a ...
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1answer
14 views

Eliminating Unit Productions

Eliminate all unit-productions from the grammar: $S \rightarrow abA\:|\:A\:|\:B$ $A \rightarrow B\:|\:ba\:|\:aBA$ $B \rightarrow A\:|\:aa\:|\:aA$ An article I was reading said that a unit ...
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3answers
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Prove that this language is not regular (Pumping Lemma)

Prove that the following language is not regular. I have no clue where to start. $$L = \{ a^n b^n c^n \mid n \geq 0 \}.$$
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1answer
44 views

Is the language $L=\{ww^f|w\in \{0,1\}^*\}$ CFL?

Where $w^f=$flipping the bits of w. For example, $(0010)^f=1101$, $(010111)^f=101000$ I tried to prove that $L$ is not CFL using the pumping lemma, with no succeed. In addition, I need to prove ...
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1answer
18 views

Finding a length of a set?

Let $\Sigma=\{a,b,c\}$ and let $\Sigma^*$ be the set of all words $w$ using letters from $\Sigma$. Define $$L(w)=\text{length}(w)$$ for all $w\in\Sigma^*$. How to calculate the L(w) for the words ...
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1answer
18 views

Prove L is not a regular language (A Finite State Automaton cannot accept it)

$$\mathscr L = \{x \in \{0,1\}^* \mid \text{there is a } y \in \{0,1\}^* \text{ such that } x = yy\}$$ How can I prove that this is not a Regular language? I tried using proof by contradiction but ...
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2answers
33 views

How to solve pumping lemma questions?

I am trying to prove that L = { aNbMaN-M|N>=M>=0} is not regular using the pumping lemma. I am pretty confused how to solve this. What I have so far (which I am not sure is right) is: Assume L is ...
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1answer
57 views

Build a deterministic turing machine to decide L = { ww }

As the title says. w is in {a, b}^*.Note that I am not looking for the non-deterministic one. Use a Turing machine of one tape and "pointer". An idea: I thought that I would do something like ...
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1answer
21 views

Recursively Enumerable Languages and Turing Machines

L1 = { M | Turing Machine M terminates for at least 637 inputs} L2 = { M | Turing Machine M terminates for at most 636 inputs} One of them is recursively enumerable, which one?
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LL(1) grammar for boolean language

Is there a LL(1) grammar for this language? Here are some words of this language. It is a boolean logic, which uses negation, binary operators and braces (redundant braces are allowed too): A ...
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1answer
31 views

Accepting/rejecting states in Turing Machine

In language decidability problems, a TM halts with a halting state, an accepting state or a rejecting state. My understanding is that when a TM machines halts on accepting state, it removes everything ...
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1answer
37 views

Checking some Regular Expression problems

I'm given the alphabet $$ \Sigma = {\{a,b}\} $$ I tried to write a regular expressions for presenting the following sets: All strings in $$\Sigma ^ *$$ with: a-) number of 2s divisible by 4 b-) ...
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1answer
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How to prove the following related with regular languages

How can we prove the following. If $$\sum$$ is any alphabet and L is any language $$L \subset \sum*$$ Then L*L* = L* ?
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Prove the following by using mathematical induction

If we define the alphabet such that $$ \Sigma = {\{a,b}\} $$ and let $w$ be a string over it. I'd like to prove $$ ( \operatorname{comp}(w))^R = \operatorname{comp}(w^R) $$ where $$ w^R$$ and ...
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1answer
43 views

Pure Lambda Calculus: Call-by-value Free Variable Argument Application Reduction

In pure lambda calculus, under the call-by-value reduction strategy, a term of the form $(\lambda x. x)y \rightarrow y$ implies that the free variable $y$ is a value. However, only abstractions are ...
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1answer
28 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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Is there an unambiguous CFL whose complement is not context-free? [migrated]

I'm doing a little bit of research about context-free languages. A question that's popped up is whether or not there exists an unambiguous context-free language whose complement is not a context-free ...
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Question on Proof that the Fibonacci Word is Sturmian

I am currently reading a text where it is proved that the infinite Fibonacci Word $u$ defined as the limit of the sequence $$ u_n = \varphi^n(0) $$ where the morphism is given by $\varphi(0) = 01, ...
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2answers
26 views

Prove by induction on a string

I want to practice proving the following: Given a binary string s, I want to prove $s$ has the same number of substrings 01 and 10 $\iff$ the first and last character of $s$ is the same. For ...
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0answers
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Recursive and Recusively enumerable

{1^n | n finite integer and >1}, is this language recursive? I'm not sure how to prove that a language is recursive, I only know that there should be a TM that accepts any finite input and then ...
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1answer
30 views

find a regular expression and FA that each define L1 ∩ L2

from the following pairs I am trying to find a regular expression and FA that each define ...
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1answer
20 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
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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
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Prove $L$ = $\{\langle M \rangle$ | $M$ is a TM over $\{0,1\}$ and $\langle M \rangle \langle M \rangle \notin \mathcal{L}(M)\}$ is undecidable.

Was stuck on this for a bit so I need to know if I am on the right track. To show that $L$ is undecidable we will show that $\overline{L}$ is undecidable instead. Suppose $\overline{L}$ is decidable ...
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0answers
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This proof in my textbook involving the pumping lemma appears incorrect - is it?

It states Let $B$ be the language $\{0^n1^n2^n | n \geq 0\}$. We use the pumping lemma to prove that $B$ is not regular. The proof is by contradiction. Assume to the contrary that $B$ is regular. ...
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5answers
181 views

The deep structure of logical formulas

A long-standing question to which I never found a concise answer is: Is there something like an unambiguous deep structure of a formula of propositional logic, opposed to its comparingly ...
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Proving a language is nonregular with Myhill-Nerode theorem

I am trying to figure out how to prove a language that is nonregular with using Myhill-Nerode theorem. Here I have x^​n y^​n x^​n = {xyx xxyyxx xxxyyyxxx ...} How would I be able to do this ? ...
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52 views

What does this language mean?

$$L=\left\{w\in\{0,\,1\}^*\;:\; n_1(w)\not\equiv 0\;(\mathrm{mod}\; 3) \right\}$$ It doesn't make any sense. It says any language with a string of any length containing $0$ or $1$ such that the ...
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2answers
58 views

Sigma hierarchy of logical formulae

In some papers on mathematical logic I've found references to hierarchy like $\Sigma_1^0$-sentence and so on. What does it mean? What is $\Sigma_n^m$, what is $n$ and $m$ here?
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How does one generally use partial function in logical statements?

How does one generally use partial function in logical statements? How it's done in practice? Specifically, let $M$ by a Turing machine, $f_M:\{0,1\}^*\to\{0,1\}$ the characteristic function which ...