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I am asking for help to explain some crucial points of the central lemma and it's proof of famous paper NP is as easy as detecting unique solutions by L.Valiant and V.Vazirani.

The proof can be found in the section 2. The proof of the paper.

The idea of the lemma: lemma shows that there is a randomized polynomial - time reduction from SAT to Unique-SAT. Lemma uses GF[2] inner product with polynomial few {0,1} vectors.

Problem in general: I have some difficulties in understanding the definition of the lemma and in usage of GF[2] for proving the lemma.

Specific questions:

Following is the number of citations from the paper with relevant question to the citation.

Lemma 2.1. If $f$ is any CNF formula in $x_1,...,x_n$ and $w_1,...,w_k \in \{0,1\}^n$,

Q:I think $x_1,...,x_n$ are literals, but what are $w_1,..,w_k$? Are they literals or something else? If yes, why to distinct between them and $x$ literals.

A:$w_1,..,w_k$ are the truth assignments.

then one can construct in linear time a formula $f'_k$ whose solution $v$ satisfy $f$ and the equations $v\cdot w_1=...=v\cdot w_k=0$. Furthermore, one can construt a polynomial-size CNF formula $f_k$ in variables $x_1,...,x_n,y_1,...y_m$,

Q: $x_1,...x_n$ are unknown literals, how can we construct $f_k$ with unknown literals and where $y_1,...,y_m$ came from?

for some $m$ such that there is a bijection between solutions of $f_k$ and $f'_k$, defined be equality on the $x_1,...,x_n$ values.

Proof: It is sufficient to show the lemma for $k=1$. Then, $f'_k$ is

$f \wedge (x_{i_1} \bigoplus x_{i_2} \bigoplus ... \bigoplus x_{i_j} \bigoplus 1)$

Q: this is the most vague point, how do we come to the such construction of $f'_1$ and why showing only $f'_1$ sufficient for the proof.

I hope you forgive me for my naivety. I will appreciate any hint in accordance to the above questions.

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1 Answers1

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Q: $x_1,...x_n$ are unknown literals, how can we construct $f_k$ with unknown literals and where $y_1,...,y_m$ came from?

$x_1, \ldots, x_n$ are variables, each of which can appear as a positive literal ($x_i$) or a negative one ($\lnot x_i$). To construct an equivalent formula $f_1$ of another formula $f_1'$, you don't have to know the values (truth assignments) of $f_1'$. (A simple example is $x_1 \to x_2 \Leftrightarrow \lnot x_1 \lor x_2$.)

The variables $y_1, \ldots, y_m$ are unimportant (at least in this Lemma). The Lemma define the "equivalence" of the two formulas $f_1$ and $f_1'$ by equality on the $x_i$ values (leaving $y_i$ values unspecified).

Q: How do we come to the such construction of $f'_1$ and why showing only $f'_1$ sufficient for the proof?

Each $f_i$ (and $f_i'$) is independent and similar. Therefore, the case of $f_1'$ is sufficient for the proof. To justify the construction of $f_1' = f \land (x_{i_1} \oplus x_{i_2} \oplus \cdots \oplus x_{i_j} \oplus 1)$, you should prove the following two points:

  • Each solution $v$ to $f_1'$ satisfies $f$.

This is trivial by the definition of $f_1'$.

  • Each solution $v$ satisfies the equation $v \cdot w_1 = 0$.

This is the tricky part. We argue as follows:

First, $$v \cdot w_1 = x_{i_1} + x_{i_2} + \cdots + x_{i_j} \text{ (here `+` is over GF[2])} = x_{i_1} \oplus x_{i_2} \oplus \cdots \oplus x_{i_j}$$ The first equality follows from the fact that other $w_l$ with indices $l \neq i_1, i_2, \cdots, i_j$ are all 0 (according to the definition of all $x_{i_j}$).

Because $v$ is a solution to $f_1'$, we have $$x_{i_1} \oplus x_{i_2} \oplus \cdots \oplus x_{i_j} \oplus 1 = 1$$ Thus, $$v \cdot w_1 = x_{i_1} \oplus x_{i_2} \oplus \cdots \oplus x_{i_j} = 0$$

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