Tagged Questions

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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Size of automata or regular expressions avoiding cross patterns

Let $\Sigma$ be an alphabet of finite size $k$, and $n$ some integer. I am interested in the language of words of size $n$ that do not contain $abab$ as a subword, for any pair $(a,b) \in \Sigma$ (I ...
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0answers
14 views

find an algorithm to find terminal string [duplicate]

I would like to know an algorithm which, given a cfg, finds those variables A that derives atleast one terminal string. I can show it by giving some production rules and say that particular variable ...
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0answers
21 views

how to state which words the right congruence class contains.

I would like to know what words does a right congruence class contains given a condition. ie., if there is a class [ab], then the words that come under this class are string with any number of a's and ...
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2answers
44 views

How to prove Kleene star to be uncounably infinite?

Hi I have a language $L = \{a, b\}$. How can I prove that the Kleene star (set of all words over the language) of this language is uncountably infinite or countably infinite?
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2answers
23 views

How to show if a language is infinite, then there is no upper bound on the length of words in L?

L is a language over a finite alphabet. How to show that if L is infinite, then there is no upper bound on the length of the words within L? Can someone help me prove this.
3
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0answers
67 views

Intuition for the choice of background (set) theory

Problem From the formalist point of view, any mathematical statement should ultimately be an assertion about the derivability of a certain formula in a certain formal system, call it the background ...
0
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1answer
21 views

How do I generate regular expression from this deterministic finite state automaton?

I want to create a regular expression from the following deterministic finite automaton: "abb" substring search How do I generate regexp from above dfa and what are the steps for doing so? ...
0
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1answer
32 views

Question about formal languages, quotient operator L1/L2.

I come across this problem: Consider the following regular languages: L1={a*b*c*} over the alphabet A={a,b,c} L2={( a b | b b | a )*} the alphabet is the same as above. Find the shortest strings ...
3
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2answers
46 views

If $L$ is regular, must the language $L_1 = \{w : w^Rw \in L\}$ be regular, or may it be non-regular?

The reverse, $w^{R}$, of a string $w = w_1w_2...w_n$ is the string $w_n...w_2w_1$. Suppose that L is a regular language. Must the language $L_1 = \{w : w^Rw \in L\}$ be regular, or may it be ...
1
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1answer
16 views

Give an example of a language whose Myhill-Nerode equivalence relation is such that if $x,y \in \Sigma^*$ with $x \neq y$, then $[x] \neq [y]$

Suppose $\Sigma = \{0,1\}$. Provide an example of a language $L \subseteq \Sigma^*$ with the property that its associated Myhill-Nerode equivalence relation, $R_L$, is such that every one of its ...
0
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1answer
23 views

Push Down Automata that recognizes language

I'm struggling on how to use the stack for this push down automata problem. The problem is to design a PDA that recognizes the language: $$A = \{a^ib^{2i}|\,i>0\}$$ So, we will be pushing a's onto ...
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2answers
54 views

Designing a deterministic finite automata

How would I go about designing a deterministic finite automata to recognize the language L = {λ, ab, abab, ababab, . . . } consisting of strings that start with ‘a’, end with ‘b’, and alternate in ...
0
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1answer
45 views

Deterministic finite automata [closed]

For this question about Deterministic finite automata: Is this answer: bbbb, bbba, bbab, bbaa, b, a correct?
7
votes
1answer
128 views

Challenge on Some Language and PDA

Suppose We have Some language as follows: $L_1=\{w^* | w=x \text{ and } x \in \Sigma^*\}$ $L_2=\{ww^R ww^R | w \in ( \Sigma + \Sigma)^*\}$ $L_3=\{w | w=xy, x,y \in \Sigma^*, y \text{ is a ...
0
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3answers
83 views

Prove that the set of palindromes are not regular languages using the pumping lemma.

Firstly I pick a language $xyz$ where $x = \epsilon$, $y = (abb)^{k}$, $z = (bba)^{k}$ where $|y| \ge$ the number of states in the automaton representing my language. Then $xyz = (abb)^k(bba)^k$ is a ...
2
votes
0answers
121 views

The congruence $\{(a^m, a^{m+r})\}^\#$ on $a^+$.

I've spent a bit too long on this exercise. It's time to ask for help. This is Exercise 1.20 of Howie's Fundamentals of Semigroup Theory. Let $\rho_{m, r}$ (for $m, r\ge 1$) be the congruence ...
0
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1answer
37 views

How to prove a language is decidable

I would like to see some proofs to show if particular string in machine M which dfa is decidable or not. I am trying to find some results on this but those are not appropriate. Any examples or proofs ...
1
vote
1answer
44 views

equivalence class for language in Theory of automata

we say x,y is equivalent to language L, if for any $z \in L$ we have: $xz \in L \Longleftrightarrow yz \in L$. for $ L= (ab \cup aab)^* $, what is the equivalence class for L? my professor ...
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0answers
39 views

Prove that if $L$ is regular, then $\mathcal{L}(L)$ must be regular

Let $\Sigma_n = \{0, 1, ... , n-1\}$. Suppose $L \subseteq$ $\Sigma^*_n$, and let $\mathcal{L}(L) = \{ x \in L : x \text{ is the lexicographically largest among all strings of length } |x| \text{ ...
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votes
1answer
31 views

CFG with reverse strings

I've been trying to figure this out for a while, and I'm at a total loss: Write a context-free grammar that generates the language $\{x y\ |\ x$ is a string over $\{a,b,c\},\ y$ is a reverse of ...
2
votes
2answers
27 views

Prove that a regular language $L$ exists satisfying $L_1 \subseteq L \subseteq L_2$ and $L - L_1$ and $L_2 - L$ are both infinite.

Let $L_1, L_2$ be regular languages, with $L_1 \subseteq L_2$ and $L_2 - L_1$ infinite. Prove that a regular language $L$ exists satisfying $L_1 \subseteq L \subseteq L_2$ and $L - L_1$ and $L_2 - L$ ...
0
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0answers
28 views

contradiction of pumping lemma on even palidrome language's completion

I have the language of all the words that are not even length palindromes, or more formally: $L= \{ w: \forall u\in\Sigma^*, w\ne uu^r \}$. And I need to prove that the Pumping Lemma for regular ...
0
votes
1answer
30 views

Find CFG for language

I'm trying to find CFG of language where $L = \{ a^x b^y a^z \}$ where x,y,z = 1 2 3.. .and y = 2x + 2z I have no idea, I'm completely stuck. Any help would be very appreciated thank you
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vote
3answers
60 views

Deriving contradiction from $a\Leftrightarrow\neg a$

Recently I've been trying to prove some things by strictly following deduction rules. I've been trying to derive incononsistency from unrestricted comprehension axiom via Russell's paradox. I have ...
2
votes
3answers
47 views

propositional language. don't understand the definition?

I'm taking a mathematical logic class, and I don't understand this definition of the $propositional$ $language$, as given by my book: "The propositional language $\mathscr{L}_0$ is the smallest set ...
1
vote
2answers
59 views

Definability and the Separation and Replacement Axiom Schemata

I am beginning to study Set Theory, so naturally one might begin with the axioms. When reading the schemata of separation and replacement, my initial thought was, "Why would we need separation if we ...
0
votes
2answers
20 views

Regular Expression Similarity Check

I was solving Formal Language and Automata Theory for a competitive exam, whence I came upon this following question: The regular expression 0*(10*)* denotes same set as: 0(0+10)* (0+1)10(0+1) ...
0
votes
0answers
36 views

Number of states of a finite automaton recognizing all words beginning with some fixed string $x$

For a string $x \in \{a,b\}^\ast$ with $|x| = n$, how many states are required for an FA accepting the language of all strings in $\{a,b\}$ that begin with $x$? For each of these states, describe the ...
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2answers
55 views

Induction Proof - Regular Expressions and Languages

Find an example of languages $L_1$ and $L_2$, where neither $L_1$ nor $L_2$ are subsets of each other, but $$L_1^* \cup L_2^* = \left(L_1 \cup L_2\right)^*$$ Prove Correctness.
0
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0answers
31 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 ...
1
vote
1answer
53 views

PDA and Some language Grammar inference

L1={$w^* $| w=x and $ x \in \Sigma^*$} L2={$ww^R ww^R $| $ w \in ( \Sigma + \Sigma)^*$} L3={$w | w=xy, x,y \in \Sigma^*$, y is a substring of x} a) there is a PDA(push down automata) that accept ...
0
votes
4answers
58 views

recursive definition odd length strings

Given the alphabet {aaa bbb}, give a recursive definition for the language that only contains odd length strings. must be constructive definition we are suppose to treat aaa as one letter and bbb as ...
1
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1answer
66 views

Countable Set & Formal Grammar

We know set A is countable if A is finite or in a one-to-one mapping to natural numbers. I try to summarize my though. I think the following proposition is true. suppose $\Sigma$ is arbitrary ...
1
vote
1answer
106 views

Why is the Kleene star of a null set is an empty string?

The articles and textbooks mention that, $\emptyset^\star = \{\epsilon\}$ The star operation puts together any number of strings from the language to get a string in the result. If the language ...
2
votes
1answer
62 views

Challenge on Property of Complement in Language

We have: and M be a finite automata. Suppose d(M) be a deterministic automata that equivalence to M. if $M_1$ and $M_2$ be two finite automata, $M_1 + M_2$ is the finite automata such that the ...
4
votes
2answers
155 views

What's the difference between a logic, an internal logic (language) of a category, an internal logic of a topos and a type theory?

maybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all). ...
1
vote
1answer
23 views

Can Type 0 grammar generate Type $1,2,3$ languages? [closed]

Can Type $0$ grammar generate Type $1,2,3$ languages? And can Type $1$ grammar generate Type $2,3$ languages? I am confused with the Chomsky hierarchy.
0
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0answers
28 views

Design an finite automata for 010 and 01? [duplicate]

I have tried below question for designing automata but didn't success . ...
0
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1answer
38 views

Programming string in math

It seems that everything in programming is relatated to some math theory even regular expressions, but what about strings. Is there simple "thing" in math that can be equivalent of a string in ...
0
votes
1answer
35 views

Formulating the Twin Prime Conjecture as a Language Recognition problem.

I'm trying to figure out how to formulate the Twin Prime Conjecture as a language recognition problem. I've got: A = {p: p is the largest prime such that p + 2 is prime} B = {p: p and p+2 are both ...
0
votes
0answers
12 views

What are the classes in A.N. Maslov hierarchy of indexed languages corresponding to Chomsky Hierachy?

As we know,that classes in A.N. Maslov hierarchy of indexed languages of level 2 is in sensitive languages of Chomsky hierarchy. What are the classes in A.N. Maslov hierarchy of indexed languages ...
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0answers
20 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 ...
0
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1answer
36 views

finding right quotient of languages

Can someone enumerate in detail, steps to find right quotient of languages ie L1/L2. Using an examle will be great. Having a hard time to understand the same.
0
votes
1answer
42 views

What are the differences between a collation and a rule of formation?

I'm a beginner in mathematical logic, and currently studying(myself, without any colleague, which is sad and so asking in here) basics of formal system. Before asking a question, I'll introduce my ...
0
votes
3answers
45 views

Formal Notation - A simple example

I'm taking Formal Languages and Automata Theory course and i couldn't understand the notation below. Can someone explain me this please? $\{ d \mid d \in \{ b f \mid b, f \in \{ a, c, e \} \} \}$
0
votes
1answer
56 views

How do I define a string in formal language by means of a definition of tuple?

I'm constructing mathematical notions and definition from the bottom of the mathematical structure. So whenever I learn, or encounter new concepts, I try to define it step by step, without using any ...
3
votes
2answers
34 views

Is my proof of 2 languages equality correct?

I started to self-study the formal languages theory, and tried to solve the following problem: Problem Prove, that these 2 languages are equivalent: $$ \Sigma = \{a,b,c\}\\ L_1 = \{(abc)^na | n \ge ...
-1
votes
1answer
25 views

Context Free-Language [closed]

Is it true that if L1 are CFLs and L1 subset of L subset of L2, then L must be context-free? if not true, give a counterexample. Otherwise, explain intuitively why
4
votes
2answers
103 views

Mathematical concept for formal languages

A formal language is defined as a subset of finite-length strings over an alphabet. It is similar to the mathematical concept "relation", but the lengths of its strings are not fixed. Since the name ...
0
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0answers
18 views

Is this set $L$-distinguishable $M = \{a^i b^k a^j \mid i\#k\#j\}$? Why (not)?

Is set $M$ $L$-distinguishable? Why (not)? Here $L$ is the language that accepts the $bababaa$ string, so $L = \{a,b\}^*\{bababaa\}\{a,b\}^*$, and $M = \{a^i b^k a^j \mid i\#k\#j\}$. Please would ...