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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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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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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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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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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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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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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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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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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23 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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27 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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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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307 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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32 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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51 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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43 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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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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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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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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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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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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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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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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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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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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54 views

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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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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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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28 views

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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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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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 ...
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Help with formulating a mathematical logic formula

I need to write a precise mathematical expression to formulate an algorithm that could be implemented in software. It has the following simple logic: An Internet user of the software in a ...
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59 views

Natural definitions of families of subgraphs

Cluster analysis is a vibrant area of applied mathematics, "used in many fields, including machine learning, pattern recognition, image analysis, information retrieval, and bioinformatics". A subfield ...
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52 views

Z notation - Does operation refinement make an operation more deterministic or even more non-deterministic?

I've stumbled about the following statement: An operation $b$ refines an operation $a$ correctly if and only if $a$ is more deterministic than $b$ As I would guess, it's exactly the other way ...
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Define the set of valid algebraic expressions ALEX as follows:

I am trying to find set of valid algebraic expressions and I know how to do them in math way and my question is how would I be able to do them in syntax rules ? This is the question. ...
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Prove that $\overline{L}$ is not recognizable by showing that $B_{TM} \le_m L$

$\textbf{Problem}:$ $L$ = $\{\langle M \rangle$ | $M$ is a Turing machine over $\{0, 1\}$ such that for some $x \in \{0,1\}^*$, $M$ does not halt on input $x\}$. $B_{TM}$ = $\{ \langle M \rangle$ | ...
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showing language that is non-regular using pumping lemma

I am looking over pumping lemma and the author is using it to show that the language is non-regular. {a^n b^n a^n} = {aba aabbaa aaabbbaaa........} Is there ...
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transition graph that accepts only Λ and language a*

I am trying to have a transition graph that accepts only Λ and also one that accepts language a* ...is this ok ??? transition graph that accepts only Λ transition graph that accepts language a*
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Build a Transition Graph that accepts the language L…

I need to build a transition graph that accepts the language L of all words that begin and end with the same double letter, either of the forms aa......
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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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Which strings does L language produce?

Let $L = 1 \cup (01 \cup 10)(00 \cup 11)^*0$. Which are the strings $L$ produces? I thought the ones that have even number of zeroes and odd number of $1$. But, you can not produce $111$. Then I ...
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Give context-free grammars that generate these languages

Give context-free grammars that generate these languages {a^(2i) b^(3k) c^(4i) | i => 1, k => 1} {a^(i) b^(k) c^(k) a^(i) | i => 1, k => 1} I am seriously stuck here. Esp for the 1st one. For the ...
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Let $L_{UIUC}$ = $\{ \langle M \rangle$ : $L(M)$ contains the string $UIUC\}$. Prove that $L_{UIUC}$ is undecidable.

Been stumped as to why the following proof works. Note: I have taken this proof directly from here. Proof by reduction from $A_{TM}$. Suppose that $L_{UIUC}$ were decidable and let $R$ be a Turing ...
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What does it mean for a Turing machine $M$ to accept $\epsilon$

Suppose $B_{TM}$ = $\{ \langle M \rangle$ | $M$ is a Turing machine over $\{0, 1\}$ and $M$ accepts $\epsilon\}$. I do not understand what it means for $M$ to accept $\epsilon$. Can someone explain ...
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machine accepting words using Transition Graphs

I am preparing for an exam next week, and I am practicing some Transition graphs problems so I would like to know if I am doing this right or not. As you can see here I have machines to see which one ...
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Proving a language is not recognizable

I have the following question that I just want to verify I have done correctly. Let $L$, $L_1$, $L_2$ $\subseteq \Sigma^*$ such that $L = L_1 \cup L_2$, and $L_2$ is decidable. Prove that if $L$ is ...
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Prove the language $\{a^k b^l : k \neq l \}$ is not regular

Prove that the following language is not regular: $$L=\{a^k b^l : k,l \ge0, k\ne l\}$$ The problem is that I should use "distinguished states" not the pumping lemma, which is usually used for such ...
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59 views

Proving that a language is not context-free

Given the language $$L = \{ a^p \mid p\, \text{IS NOT prime} \}$$ is $L$ Context free? If not, prove that it's not. May I have some suggestions on how to use the pumping lemma to prove this, ...