Questions related to formal languages, grammars, and automata theory

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1answer
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Are constituency grammars and dependency grammars two different types of context free grammars?

From http://en.wikipedia.org/wiki/Parse_tree A concrete syntax tree or parse tree or parsing tree[1] or derivation tree is an ordered, rooted tree that represents the syntactic structure of a ...
0
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2answers
45 views

Show that the language of words with even sum of positions of a letter is regular

Let $\Sigma=\{a,b\}$, and let $S(a)$ be sum of the positions of $a$ of string $S$. I want to prove $$L=\{S\in \Sigma^{*} \mid S(a)=0(\bmod 2)\}$$ is regular. What I was thinking is to do somehow keep ...
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0answers
36 views

What is regular about regular languages? [duplicate]

I am new to automata theory. I am well aware of the definition of regular language in automata, that is "a language is called a regular language if some finite automaton recognizes/accepts it" [MS]. ...
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0answers
28 views

How to create instances in a post production system? [on hold]

I attempted to solve the question below and got stuck. Create a Post Production system that generates language instances which Are made up of A’s and B’s Odd numbers of A’s Have even ...
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1answer
84 views

Why is the language of even-length non-palindromes context-free?

We know $L_1=\{w_1 w_2 \in (a+b)^*\mid |w_1|=|w_2|, w_2 \neq w_1^{\;\mathrm{R}}\}$ is a context-free language. Can anyone help me produce a PDA or give me any hint how I can quickly understand why ...
-3
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1answer
58 views

Language of a grammar

What's the language of following grammar? $G: S \to S_1B$ $S_1 \to aS_1b$ $bB \to bbbB$ $aS_1b \to aa$ $B \to \lambda$ any hint or solution?
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0answers
14 views

$L=${$a^n ww^R b^n$ | $ w \in (a+b)^+ $} [duplicate]

I read that $L=${$a^n ww^R b^n$ | $ w \in (a+b)^+ $} is Context free. any hint or idea for draw a PDA or CFG? thanks to all.
0
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1answer
153 views

What could 'two characters are terminals' mean?

In the context of this statement, what does 'a & b are terminals' mean? Stacks and queues can be used for determining whether a particular input string is in the language or not. L = ...
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2answers
52 views

Is string matching and replacement considered in formal languages?

Is string matching and replacement, as an operation on strings or on formal languages, considered in formal languages? For example, the family of regular languages, or the family of context free ...
6
votes
1answer
182 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 ...
1
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1answer
49 views

Expressive power of lexer + parser

Most modern compilers split their syntax analysis into a lexical phase that is followed by a parsing phase. The lexical phase is given by a regular expression, while parsing is guided by a ...
2
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3answers
319 views

Context-free grammar for language with unequal numbers of a and b

I've been trying to get a CFG for the language of all words with unequal numbers of a and b, i.e. $$\{u \in \{a, b\}^* \mid \text{number of occurrences of $a$ and $b$ in $u$ are unequal} \},$$ but ...
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3answers
80 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
49 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 ...
1
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1answer
39 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 ...
-2
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1answer
81 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? ...
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1answer
38 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
50 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
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0answers
68 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
158 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
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1answer
65 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
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3answers
419 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 ...
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2answers
119 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
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1answer
56 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 ...
1
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1answer
65 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 ...
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1answer
33 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
41 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
69 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
68 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
39 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
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1answer
236 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
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3answers
62 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
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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
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1answer
93 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
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2answers
95 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
472 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
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0answers
43 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
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2answers
146 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
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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. ...
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3answers
108 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 ...
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2answers
41 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
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2answers
281 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
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1answer
43 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 ...