We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we say the property is recursively recognizable within some class of descriptions. When there is not, we ask how difficult it is to detect the property in an algorithmic sense. We consider descriptions of two sorts: first, recursive presentations in terms of generators and relators, and second, algorithms for computing the group operation. For both classes of descriptions, we show that a large class of natural algebraic properties, Markov properties, are not recursively recognizable, indeed they are \(\Pi ^0_2\)-hard to detect in recursively presented groups and \(\Pi ^0_1\)-hard to detect in computable groups. These theorems suffice to give a sharp complexity measure for the detection problem of a number of typical group properties, for example, being abelian, torsion-free, orderable. Some properties, like being cyclic, nilpotent, or solvable, are much harder to detect, and we give sharp characterizations of the corresponding detection problems from a number of them. We give special attention to orderability properties, as this was a main motivation at the beginning of this project.

中文翻译：

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从组的描述中检测属性

我们考虑是否以算法的形式给出一个简单，有限的组描述，是否有可能通过算法确定相应的组是否具有某些指定的属性。如果有这样的算法，我们说该属性在某些类别的描述中可以递归识别。如果没有，我们问从算法意义上检测属性有多困难。我们考虑两种描述：第一，关于生成器和关联器的递归表示，第二，用于计算组运算的算法。对于这两种类型的描述，我们都表明，一大类自然代数性质（马尔可夫性质）无法递归识别，实际上它们是\（\ Pi ^ 0_2 \）-难以在递归呈现的组中检测到\（\ Pi ^ 0_1 \）-很难在可计算的组中检测到。这些定理足以为多种典型组特性的检测问题提供清晰的复杂性度量，例如，阿贝尔式，无扭转，可排序。某些特性（例如循环特性，幂等特性或可解特性）更难检测，我们从许多特性中清晰地描述了相应的检测问题。我们特别注意可订购性，因为这是该项目开始时的主要动机。