We study two simple yet general complexity classes, which provide a unifying framework for efficient query evaluation in areas like graph databases and information extraction, among others. We investigate the complexity of three fundamental algorithmic problems for these classes: enumeration, counting and uniform generation of solutions, and show that they have several desirable properties in this respect. Both complexity classes are defined in terms of non deterministic logarithmic-space transducers (NL transducers). For the first class, we consider the case of unambiguous NL transducers, and we prove constant delay enumeration, and both counting and uniform generation of solutions in polynomial time. For the second class, we consider unrestricted NL transducers, and we obtain polynomial delay enumeration, approximate counting in polynomial time, and polynomialtime randomized algorithms for uniform generation. More specifically, we show that each problem in this second class admits a fully polynomial-time randomized approximation scheme (FPRAS) and a polynomial-time Las Vegas algorithm (with preprocessing) for uniform generation. Remarkably, the key idea to prove these results is to show that the fundamental problem #NFA admits an FPRAS, where #NFA is the problem of counting the number of strings of length n (given in unary) accepted by a non-deterministic finite automaton (NFA). While this problem is known to be #P-complete and, more precisely, SpanL-complete, it was open whether this problem admits an FPRAS. In this work, we solve this open problem, and obtain as a welcome corollary that every function in SpanL admits an FPRAS.

中文翻译：

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用于枚举、计数和统一生成的高效日志空间类

我们研究了两个简单但通用的复杂性类，它们为图形数据库和信息提取等领域的高效查询评估提供了统一的框架。我们研究了这些类的三个基本算法问题的复杂性：枚举、计数和统一生成解决方案，并表明它们在这方面具有几个理想的属性。两种复杂度类别都是根据非确定性对数空间换能器（NL 换能器）定义的。对于第一类，我们考虑了明确的 NL 传感器的情况，我们证明了常数延迟枚举，以及多项式时间内解决方案的计数和统一生成。对于第二类，我们考虑不受限制的 NL 换能器，我们获得多项式延迟枚举，多项式时间内的近似计数，和用于均匀生成的多项式时间随机算法。更具体地说，我们展示了第二类中的每个问题都允许使用完全多项式时间随机逼近方案 (FPRAS) 和多项式时间拉斯维加斯算法（带有预处理）来进行统一生成。值得注意的是，证明这些结果的关键思想是证明基本问题 #NFA 承认 FPRAS，其中 #NFA 是计算非确定性有限自动机接受的长度为 n（以一元形式给出）的字符串数量的问题(NFA)。虽然这个问题已知是#P-complete，更准确地说，是 SpanL-complete，但这个问题是否承认 FPRAS 是开放的。在这项工作中，我们解决了这个未解决的问题，并得到一个受欢迎的推论，即 SpanL 中的每个函数都承认一个 FPRAS。