We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. 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. This extends the work in Part I to arbitrary semialgebraic sets. All previous algorithms proposed for this problem have doubly exponential complexity.

中文翻译：

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计算半代数集的同调性。II：通用公式

我们描述和分析一种数值算法，用于计算由布尔公式给出的半代数集的同源性（贝蒂数和扭转系数）。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。这将第一部分的工作扩展到任意半代数集。针对该问题提出的所有先前算法都具有双指数复杂度。