## CNF2 = { φ | φ is a satisfiable CNF-formula in which each variable appears at most 2 times}. Show CNF2 is in P

CNF2 = { φ | φ is a satisfiable CNF-formula in which each variable appears at most 2 times}. Show CNF2 is in P.

I found this solution:

We use the method of resolution to take the variables out one by one. First, observe that if the variable x appears only as x or only as ¬x in the formula, or if it appears only once, we can satisfy whatever clauses it takes part of and thus exclude x and these clauses. On the other hand, if x and ¬x are both clauses for some variable, we know our formula is not satisfiable. So the interesting case is when we have x in one clause, c+, and ¬x in another, c−, such that (c+, c−) 6= (x, ¬x). Call the remaining part of the formula r. We claim that c+ ∧ c− ∧ r is satisfiable if and only if c ∧ r is satisfiable, where c is the disjunction of all literals from c+ and c− that are not x or ¬x. Indeed, given a satisfying assignment of c+ ∧ c− ∧ r, some literal in either c+ or c− has to be satisfied, because we can’t have both x and ¬x be satisfied. Thus, forgetting the assignment of x, we get a satisfying assignment of c ∧ r. Conversely, given a satisfying assignment of c ∧ r, some literal different from x and ¬x will be satisfied in either c+ or c−; the other one can be satisfied by assigning x in an appropriate way. Thus, we have a procedure to decide satisfiability of a CNF2 formula φ efficiently: reduce the number of variables by the methods described above; if there is no contradiction along the way and you get to the empty formula, then φ is satisfiable. Otherwise, it is not.

Why is there not a control about to check if each variable appears at most 2 times?

## limit of two variable function at zero

How can I show that the limit of the following function when the variables {x,y} tend to zero, is zero? Mathematica gives the error “{x,y} is not a valid variable”.

Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]] 
Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] -    y Cosh[x/Sqrt[y]], {x, y} -> {0, 0}] 

Any comment is welcomed.

## Deliver Your Direct Mail Campaign With Impact by Using Variable Data Printing

Variable Data Printing (VDP) by means of definition, is a form of on-call for printing that attracts records from a particular database, to personalize text and photographs from one printed piece to the following, without preventing or slowing down the printing method.

Sounds like a mouthful, but actually a easy concept whilst it comes down to it.

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Really, the options have grow to be infinite.

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## Problem of code Linkedlist while using struct variable

Code for linkedlist using memory allocation:https://ideone.com/loH6Ev.

Code for linkedlist using struct variable:https://ideone.com/YrZmW9. When i use 2nd code it doesn’t work correctly.It print just two value.In while loop of create() i use struct node type variable declaration.that’s why it should it would declare a new struct node type link variable and a new memory(struct node link) address should be seen in every execution .So what’s the problem?I am a newbie learning data structure using c.Please explain in simple language. Thank you sir.

## Variable derives itself

Can anyone explain me how is this true? T =>* T with this grammar

R -> XRX | S
S -> aTb | bTa
T -> XTX | X | $$\epsilon$$
X -> a | b
Reference: introduction to the theory of computation 3ed – sipser

## Coding the Javascript part of a TinyMCE plugin: the IDE can’t resolve the tinymce variable

I’ve copied a very basic plugin for WordPress which adds a button to TinyMCE.

The code is correct and works. But while I’m coding (in PHPStorm), the IDE can’t resolve the javascript objects. How can I configure PHPStorm (or a generic IDE) to see the code of TinyMCE included in WordPress?

## How does a variable caster level work for spells?

There are a fair few spells that have more powerful results with a higher Caster Level. Magic Missile creates more missiles, Fireball uses more damage dice, Cure spells restore more hitpoints, etc. In addition, any spell that can be resisted/countered/dispelled uses Caster Level to determine how likely this is to happen.

But I’m unclear where exactly this caster level comes from. Lets take Magic Missile as our example. It is a first level spell, so as a level one wizard I can prepare it in a first level spell slot just fine. But once I become a fifth level Wizard, get a caster level of three and have access to third level spell slots, I’m not sure what happens.

If I prepare Magic Missile in a first level spell slot, does it get its second missile because I personally have Caster Level three, or does the spell itself need to be in a third level spell slot to be considered a Caster Level three spell? Or is it some third thing I’m just unclear on?

## solution of a coupled differential equation with variable

I have written a code of a set of a differential equation and for one of the variables in the differential equation I have written set of another differential equation how should I call it in the first set of differential equation in Mathematica.

## Are polynomials in one variable logspace-computable?

Given a polynomial function $$p\quad=\quad \lambda\, x\in\mathbb{N}_0.\ \sum_{i=0}^n a_i x^i$$ with $$a_i\in\mathbb{Z}$$ for all $$i\leqslant n$$, can you compute $$p$$ by a logspace transducer if the input and the output are in binary, the working tape is semi-infinite, and there are no stationary N rules (i.e., all the transducer rules are either L (move left) or R (move right) rules)?