We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. All previous algorithms solving this problem have doubly exponential complexity. Our algorithm thus represents an exponential acceleration over state-of-the-art algorithms for all input data outside a set that vanishes exponentially fast.

中文翻译：

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计算半代数集的同调性。I：宽松的公式

我们描述和分析了一种算法，该算法用于计算由布尔公式给出的闭合半代数集的同质性（贝蒂数和扭转系数），而对松弛多项式不等式没有求反。该算法在较弱的指数时间内工作。这意味着在数据量较小的子集之外，算法的成本在数据大小上为单指数。解决该问题的所有先前算法具有双倍的指数复杂度。因此，对于集外的所有输入数据，我们的算法代表了超过现有技术算法的指数加速，而指数加速很快消失了。