The idea of counting the number of satisfying truth assignments (models) of a formula by adding random parity constraints can be traced back to the seminal work of Valiant and Vazirani showing that NP is as easy as detecting unique solutions. While theoretically sound, the random parity constraints used in that construction suffer from the following drawback: each constraint, on average, involves half of all variables. As a result, the branching factor associated with searching for models that also satisfy the parity constraints quickly gets out of hand. In this work we prove that one can work with much shorter parity constraints and still get rigorous mathematical guarantees, especially when the number of models is large so that many constraints need to be added. Our work is motivated by the realization that the essential feature for a system of parity constraints to be useful in probabilistic model counting is that its set of solutions resembles an error-correcting code.

中文翻译：

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带有纠错码的模型计数

通过添加随机奇偶校验约束来计算公式的满意真值分配（模型）数量的想法可以追溯到Valiant和Vazirani的开创性工作，表明NP与检测唯一解一样容易。尽管从理论上讲是合理的，但是在该构造中使用的随机奇偶约束具有以下缺点：每个约束平均而言包含所有变量的一半。结果，与搜索还满足奇偶性约束的模型相关的分支因子很快就失控了。在这项工作中，我们证明了可以使用更短的奇偶校验约束，并且仍然获得严格的数学保证，尤其是当模型数量很大从而需要添加许多约束时。