Straight-line programs are a central tool in several areas of computer science, including data compression, algebraic complexity theory, and the algorithmic solution of algebraic equations. In the algebraic setting, where straight-line programs can be interpreted as circuits over algebraic structures such as semigroups or groups, they have led to deep insights in computational complexity. A key result by Babai and Szemerédi (1984) showed that finite groups afford efficient compression via straight-line programs, enabling the design of a black-box computation model for groups. Building on their result, Fleischer (2019) placed the Cayley table membership problem for certain classes (pseudovarieties) of finite semigroups in NPOLYLOGTIME, and in some cases even in FOLL. He also provided a complete classification of pseudovarieties of finite monoids affording efficient compression. In this work, we complete this classification program initiated by Fleischer, characterizing precisely those pseudovarieties of finite semigroups that afford efficient compression via straight-line programs. Along the way, we also improve several known bounds on the length and width of straight-line programs over semigroups, monoids, and groups. These results lead to new upper bounds for the membership problem in the Cayley table model: for all pseudovarieties that afford efficient compression and do not contain any nonsolvable group, we obtain FOLL algorithms. In particular, we resolve a conjecture of Barrington, Kadau, Lange, and McKenzie (2001), showing that the membership problem for all solvable groups is in FOLL.

翻译：直连程序是计算机科学多个领域的核心工具，包括数据压缩、代数复杂性理论以及代数方程的算法求解。在代数背景下，直连程序可被解释为半群或群等代数结构上的电路，这为计算复杂性理论带来了深刻见解。Babai与Szemerédi（1984）的关键结果表明，有限群可通过直连程序实现高效压缩，从而为群设计出黑箱计算模型。基于这一成果，Fleischer（2019）将有限半群特定类别（伪簇）的凯莱表成员问题归入NPOLYLOGTIME复杂度类，部分情形甚至属于FOLL类。他还对允许高效压缩的有限幺半群伪簇进行了完整分类。本研究完成了Fleischer发起的分类工作，精确刻画了可通过直连程序实现高效压缩的有限半群伪簇特征。在研究过程中，我们还改进了关于半群、幺半群及群上直连程序长度与宽度的若干已知界。这些成果为凯莱表模型中的成员问题带来了新的上界：对于所有允许高效压缩且不包含任何不可解群的伪簇，我们得到了FOLL算法。特别地，我们证实了Barrington、Kadau、Lange与McKenzie（2001）的猜想，证明所有可解群的成员问题均属于FOLL类。