Questions related to formal languages, grammars, and automata theory

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2
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0answers
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Is Post's Correspondence Problem decidable with fixed word size?

So, it's known that PCP is undecidable even when we fix the number of tiles to $n \geq 7$. I'm wondering, can anything similar be said for when there is a fixed word length? To be precise, here's ...
-1
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0answers
18 views

What is a productive set of all natural numbers

I'm trying to come up with a recursive language with a non-recursive subset. Many if not all of the examples I've found describe all natural numbers as recursively enumerable and their productive set ...
1
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3answers
76 views

Unable to understand an inequality in an application of the pumping lemma for context-free languages

The problem Prove that the language $\qquad L = \{a^n b^j \mid n = j^2\}$ is not context free using pumping lemma. Approach taken by the book To prove such statements, the book takes the ...
1
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1answer
46 views

Non Deterministic PDA accepted language not clear

This is a PDA from the lecture slides I'm using: They say it accepts all words that contain double a's. While it makes some sense it's not full proof. What prevents the second a to be read in the ...
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0answers
28 views

Does a NFA accept context free languages? [closed]

A NFA accepts regular languages (Type-3). So because of the chomsky hierarchy every regular language is also context-free. So a NFA accepts context free languages. Am I wrong? Moreover a NFA accepts ...
1
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1answer
37 views

Are all Chomsky-Type3 grammars LL(1)?

Referring to this Question, where an answer is stating that all Type 3 languages are LL(1), I'd like to know if all Type 3 grammars are possibly LL(1). If not, why is it so? Are there maybe ambiguous ...
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1answer
78 views

Is a language closed under string concatenation, repetition, and/or taking substring regular?

Is a language $L$ regular, context-free, context-sensitive, recursively enumerable, or ..., if $L$ is closed under string concatenation, and/or string repetition, and/or taking substring? ...
-2
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1answer
33 views

Generative grammars and analytic grammars?

What are a generative grammar and an analytic grammar? How are they different from a formal grammar? Is the recursive definition of the language of a propositional calculus, a first order logic ...
0
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1answer
48 views

Can the definition of regular languages be simplified?

Wikipedia says The collection of regular languages over an alphabet Σ is defined recursively as follows: The empty language Ø is a regular language. For each a ∈ Σ (a belongs to Σ), ...
4
votes
0answers
64 views

Counting the number of words accepted by an acyclic NFA

Let $M$ be an acyclic NFA. Since $M$ is acyclic, $L(M)$ is finite. Can we compute $|L(M)|$ in polynomial time? If not, can we approximate it? Note that the number of words is not the same as the ...
10
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1answer
157 views

The number of different regular languages

My question is: Given an alphabet $\Sigma = \{ a,b \}$, how many different regular languages are there that can be accepted by an $n$-state nondeterministic finite automaton? As an example, let us ...
7
votes
1answer
64 views

Smallest NFA accepting concatenations of two words of the length $k$ which are different at all positions

Let $k\in \mathbb N$ I'm looking for a small NFA build for the language of concatenation of two words of the length $k$ which are index-wise different, i.e. $$L_k=\{u\cdot v \in \Sigma^* : ...
3
votes
3answers
411 views

Does a logical system have semantics?

From Wikipedia: A logical system or, for short, logic, is a formal system together with a form of semantics, usually in the form of model-theoretic interpretation, which assigns truth values to ...
1
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2answers
113 views

What are the definitions of syntax and semantics?

For a formal language $L \subseteq \Sigma^*$ over an alphabet $\Sigma$. From https://proofwiki.org/wiki/Definition:Syntax The syntax of a formal language is its structure, and is specified by a ...
2
votes
1answer
55 views

Neural network: noisy temporal sequence converter (transducer?producer?) on demand?

I start to suspect this problem is very hard now that I cannot find a single relevant literature on the subject, but it's too late to change the class project topics now, so I hope any pointers to a ...
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0answers
31 views

Is a formal grammar an element in a metalanguage?

Is it possible that for a formal language $L$ over an alphabet $\Sigma$, an element $s \in L$ is a set of something? For example, from here, I assume that the grammar itself is a sentence in a ...
1
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1answer
51 views

What are the meanings of metalanguage and metasyntax and EBNF?

I am trying to understand what BNF, metalanguage, and metasyntax are. From https://proofwiki.org/wiki/Definition:Metalanguage A metalanguage of a formal language is a formal language used to ...
6
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1answer
166 views

Can this CFG be written into an equivalent LL(1) grammar?

I have the following CFG which I suspect cannot be rewritten to one which is LL(1): $S \rightarrow \epsilon\ |\ aSbS\ |\ bSaS\ |\ cSdS\ |\ dScS$ I've thought about it for a while, and can't seem to ...
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0answers
33 views

Are context-free languages ​​complement? [duplicate]

I've these languages: $$ \overline L = \left\{a^{n^2} \big| n\geq0 \right\} $$ $$ \overline L = \left\{a^n b^n c^n \big| n\geq0 \right\} $$ $$ \overline L = \left\{ww \big| w\in \{0,1\}^* \right\} ...
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1answer
30 views

What is the language generated by the following grammar? [closed]

Could please tell me the language generated by this grammar? S->iS |iSeS|ε
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1answer
36 views

How to write this regular expression

Consider the language over the alphabet $\sum= \{a\}$ containing strings whose length is either a multiple of 2 or 3 (including the empty strings). Writing a regular expression for this language
5
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1answer
67 views

Infinite non-regular decompositions of regular languages

The title pretty much says it: I'm interested in examples of infinite families of non-regular, pairwise disjoint languages whose union is regular. When is this the case? Or, from a different ...
0
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1answer
67 views

A DFA recognizing my name

How can I know if my DFA is implemented correctly? For example, I need to build a DFA, and then minimize it which will recognize my name. Language which describe my name is: L = {pustai, marius} I ...
2
votes
1answer
29 views

Describing explicitly the MyHill-Nerode classes created by a language

I want to practice proving a language is regular or not using the MyHill-Nerode theorm, but for that I need to be able to describe the classes. Here's my practice attempt: For the language ...
0
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1answer
37 views

Is there a PDA for every Type 3 Grammar?

we learned that for every type 2 grammar G exists a PDA A with L(A) = L(G). But does for every type 3 grammar G exist a PDA A_G with L(A_G) = L(G)? I think it does, because type 2 grammar is a subset ...
3
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3answers
59 views

generate possible inputs valid for automata

I find lots of solution where you have an Automata and a input string , you can validate whether input string is accepted by automata or not. Can we do the reverse ? I am looking for solution which ...
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2answers
79 views

Grammar for a language with 1/3 of a's

I have this language: $$ L = \left\{ w \in \{a,b,c\}^* \;\big|\; |w| / |w|_a = 3 \right\} $$ where $|w|_a$ is the number of occurrences of $a$. How can I find a grammar that generates it?
3
votes
1answer
232 views

How to convert a non-embedding context free grammar to regular grammar?

Please note that I am aware the undecidability of the conversion of context-free grammar to regular grammar. But given the non-embedding property of the input context-free grammar, is there any ...
0
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1answer
36 views

CFL, pumping lemma

I have difficulty with proving that the language $ L = \{ a^p b^q | p \ge 1 , q \ge 1 , p \ge q^2 \vee q \ge p^2\}$ $ w = uvxyz $ I've chosen word $ w = a^{N^2} b^N $ where $ N $ is a constant ...
1
vote
3answers
61 views

Precedence in regular expressions

I'm having trouble finding the language represented by the following: (AA|BB)* Should the expression be read as... ( A (A|B) B ) * or... ( (AA) | (BB) )* If that isn't clear, should this produce ...
24
votes
1answer
511 views

Asymptotics of the number of words in a regular language of given length

For a regular language $L$, let $c_n(L)$ be the number of words in $L$ of length $n$. Using Jordan canonical form (applied to the unannotated transition matrix of some DFA for $L$), one can show that ...
2
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0answers
306 views

Are regular languages closed under such an operation? [duplicate]

Given a string, take all of its substrings (including the empty string). For example, given $abc$, we can form a set $\{\emptyset, a, b, c, ab, bc, abc\}$. Given a regular language, take all the ...
1
vote
1answer
90 views

Resolve left-rescursion

Can anybody give me a hint on how to get rid of the left recursion in the following grammar? $$A \rightarrow B \mid a$$ $$B \rightarrow b \mid C \mid D \mid E \mid F \mid G$$ $$C \rightarrow c \mid A ...
3
votes
2answers
93 views

Correspondence between automata and formal grammars?

From Wikipedia Since there is a one-to-one correspondence between linear-bounded automata and such grammars, no more tape than that occupied by the original string is necessary for the string ...
5
votes
2answers
468 views

does every CFL have an ambiguous CFG?

some questions have been popping up recently on ambiguity in CFLs/CFGs which can have subtleties (eg languages vs grammars & ambiguity vs inherent ambiguity). wikipedia states: Many [context ...
19
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2answers
2k views

Are there inherently ambiguous and deterministic context-free languages?

Let us call a context-free language deterministic if and only if it can be accepted by a deterministic push-down automaton, and nondeterministic otherwise. Let us call a context-free language ...
3
votes
0answers
41 views

Automatic tool for resolving left-recursion within CFG [closed]

Though facing the fear that someone might not like my question but does somebody know a useful tool to either resolve left recursion or to simplify a context-free grammar automatically ? I need ...
5
votes
2answers
144 views

Which language families admit inductive definitions?

I am self-learning about formal languages. I learned that the family of the regular languages can be defined inductively, in terms of the operations they are closed under (namely the smallest ...
3
votes
1answer
125 views

Generators of families of langauges?

From Wikipedia's definition of regular langauges The collection of regular languages over an alphabet $Σ$ is defined recursively as follows: The empty language $Ø$ is a regular language. ...
1
vote
3answers
106 views

What happens with trios, full trio, (full) semi-AFL, (full) AFL if we require closure under intersection?

Wikipedia says: A trio is a family of languages closed under e-free homomorphism, inverse homomorphism, and intersection with regular language. A full trio, also called a cone, is a trio ...
0
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2answers
38 views

Can a language be the one recognized by more than one automatons?

The language recognized by an automaton is defined as the set of strings that are accepted by the automaton. I wonder if it is possible that the languages recognized by two automatons are the same? ...
7
votes
2answers
268 views

How to prove that a language is context-free?

There are many techniques to prove that a language is not context-free, but how do I prove that a language is context-free? What techniques are there to prove this? Obviously, one way is to exhibit ...
0
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1answer
47 views

Definition of prefixes of a string

From Wikipedia: The prefixes of a string is the set of all prefixes to a string, with respect to a given language: $$ \operatorname{Pref}_L(s) = \{t \mid s=tu \mbox { for } t,u\in ...
3
votes
1answer
42 views

Definition of the cyclic shift of a formal language?

Wikipedia says, the context-free languages are closed under the cyclic shift of $L$ (the language $\{vu : uv \in L \}$) So I am looking for the definition of the cyclic shift operation on formal ...
2
votes
1answer
28 views

Differences between substitution and rewriting?

I am continuing with my self-study of formal languages. Given two alphabets $\Sigma$ and $\Delta$, a string substitution is a mapping from $\Sigma$ to $\mathcal P(\Delta^*)$, which induces a mapping ...
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1answer
230 views

Are regular languages closed under sort (Parikh image)?

Assume $L$ is a regular language over an ordered alphabet. Is the language built by taking every word in $L$ and sorting it always a regular language?
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2answers
99 views

Syntax and formal grammar of a formal language

For a formal language, I wonder what differences and relations are between its syntax and its formal grammar. A formal grammar is a set of formation rules that describe how to generate the strings ...
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1answer
63 views

Formal language inverse

How can you specify the "inverse" of a word, so: let's say a word consists of a's and b's the language is: $ww^{-1}$ the second word is the same as the first but every a is replaced by b and every b ...
1
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1answer
184 views

Why is this language over {a,b,c} regular?

The language of all words over the alphabet {a,b,c} such that the number of as in the word minus the number of cs in the word is divisible by three. How is this language regular? Lecturer ...
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2answers
136 views

Using the Pumping Lemma to show that the language $a^n b a^n$ is not regular

I've seen a lot couple of questions regarding the pumping lemma that are pretty similar to each other and this one is unfortunately not the exception. Most likely will be this question marked as a ...