How can I create the context free grammar for this language? [duplicate]

I need help finding the context-free grammar for this language.

$ $ L = \{a^ib^jc^k \in \{a,b,c\}^* \mid \text{$ i,j,k \geq 1$ , and $ i=j$ or $ i=k$ or both}\}. $ $

I’ve found a way to satisfy $ i = j$ or $ i = k$ or both, and $ i,j,k \ge 0$ .

But I am lost at the $ i,j,k \ge 1$ part. Any tips or help would be greatly appreciated.

How to calculate the redundancy of a language [closed]

I’m trying to calculating the unicity distance for a cipher applied to a language I wrote and I’m having trouble understanding the concept of redundancy in a language.

From this book

The redundancy of a language severely reduces the amount of information conveyed with each character and the rate of a language is defined as the average number of bits of information contained in each character of a message, i.e. H(X)/N where N is the number of characters in the message.

You need to generate a frequency table for a language to calculate H(X) but an accurate frequency table requires a long message. Won’t this cause all redundancies to tend towards 0 since N can be arbitrarily long? How exactly is the redundancy for English, 1 to 1.5 given by this book, calculated? What value of N do they use?

Prove the following language is regular?

Assume $ L_1$ is a regular language, and define:

$ $ L = \{wcv ∈ \{a, b, c\}^* \mid |w|_a + 2|v|_b ≡ 3 \bmod 5, w, v ∈ L_1\}.$ $

Show that $ L$ is regular.

I first tried to prove by showing that the pumping lemma holds true, then learned that it was not a double implication and can only be used to prove languages are not regular.

Then I tried to draw an NFA, but didn’t make any progress.

What’s a good way to prove that a language like this is regular?

Proving a language is not regular using Myhill Nerode Theorem

Let $ L = \{\alpha\in\{a,b,c\}^{*} \mid \alpha \text{ is palindrome}\}$ , show that $ L$ is not regular using Myhill-Nerode relation.

I don’t know how to show that $ L$ has infinite equivalence classes because $ \alpha$ is a palindrome. I tried to use something like this, but I don’t know if its correct:

$ \alpha \equiv_{L} \beta \iff \alpha (aba)^k \in{L} \iff \beta (aba)^k \in{L}$ $ \forall k \in \mathbb N$ which implies that for every k there exists an equivalence class because the repetition of aba k times.

What is the earliest use of the “this” keyword in any programming language?

I understand the this (or self or Me) is used to refer to the current object, and that it is a feature of object-oriented programming languages. The earliest language I could find which has such a concept was Smalltalk, which uses self but was wondering where and when (which programming language) the concept was first implemented?

What are the terminals, non-terminals and set of production rules for the following language of valid first order formula?

variables: w x y z constants: C D predicates: P[2] Q[1] equality: = connectives: \land \lor \implies \iff \neg quantifiers: \exists \forall formula: \forall x ( \exists y ( P(x,y) \implies \neg Q(x) ) \lor \exists z ( ( (C = z) \land Q(z) ) \land P(x,z) ) ) 

How would I go about deriving the set of terminals, non-terminals and production rules for this language of valid First order formula?

A method to derive the solution would be more appreciated than just the solution.

Prove that a language is regular

I’m working on an example which says that a string x is obtained from a string w by deleting symbols if it is possible to remove zero or more symbols from w so that just the string x remains. For example, the following strings can all be obtained from 0110 by deleting symbols:

λ, 0, 1, 00, 01, 10, 11, 010, 011, 110, and 0110.

Let Σ = {0, 1} and let A ⊆ Σ ∗ be an arbitrarily chosen regular language.

Define B = {x ∈ Σ ∗ | there exists a string w ∈ A such that x is obtained from w by deleting symbols}.

In words, a string is in B if you can obtain that string by first choosing a string from A and then deleting zero or more symbols from that chosen string. Prove that B is regular.

I’m not able to prove it. Any help would be appreciated.