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### Is there a simple way to construct a Boolean formula that is true if and only if at most $k$ of the input variables are true? [duplicate]

I could of course construct a truth table for the function $$f(x) = \left(\sum_i x_i\right) \leq k$$ Where $x$ is an assignment and I'm slightly abusing notation to count Booleans. And then I could ...
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### Reducing k Vertex Cover to SAT (last clause problem)

I am working on a transformation from k Vertex Cover to SAT and I have some issues regarding the last clause in the boolean formula. Here is my approach: \forall \text{ nodes } n_i \in V, \text{...
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### Efficient algorithm for simple constraint satisfaction problem

There are $k$ Boolean variables $x_1, x_2, \dots, x_k$. $m$ arbitrary subsets of these variables such that sum of each set equals to $1$ (i.e., only one variable is $1$, the others are $0$). E.g., ...
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### CNF form of variable assignment problem

There are n variables $x_1$, $x_2$,..., $x_n$ and each one of them takes values from 1 to k (k>= n) and all are distinct. How can I represent this in the CNF form? (I tried the trivial way of trying ...
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### Alternative representations for the zebra puzzle?

All of the solutions for the zebra puzzle have a variable for each of the properties and a domain with the possible values. For instance A for Nationalities, B for pets, ... Ai with i = 1..5 and the ...
We are given a finite set of propositional atoms $\{x_1, \dots, x_n\}$ and an integer $k$. Can we capture through a propositional formula $\varphi$ (built from the standard connectives \$\neg, \wedge, \...