Lately, there have been intensive studies on strengths and limitations of nonuniform families of promise decision problems solvable by various types of polynomial-size finite automata families, where "polynomial-size" refers to the polynomially-bounded state complexity of a finite automata family. In this line of study, we further expand the scope of these studies to families of partial counting and gap functions, defined in terms of nonuniform families of polynomial-size nondeterministic finite automata, and their relevant families of promise decision problems. Counting functions have an ability of counting the number of accepting computation paths produced by nondeterministic finite automata. With no unproven hardness assumption, we show numerous separations and collapses of complexity classes of those partial counting and gap function families and their induced promise decision problem families. We also investigate their relationships to pushdown automata families of polynomial stack-state complexity.

翻译：近来，对于由各类多项式大小有限自动机族可解的承诺判定问题的非均匀族的能力与局限性，已有大量研究，其中"多项式大小"指有限自动机族的态复杂度受多项式界约束。在本研究脉络中，我们进一步将研究范围扩展至部分计数函数与间隙函数族，这些函数基于非均匀多项式大小非确定性有限自动机族及其相关的承诺判定问题族定义。计数函数具有对非确定性有限自动机产生的接受计算路径数目进行计数的能力。在不依赖未证明的困难性假设下，我们证明了这些部分计数函数与间隙函数族及其诱导的承诺判定问题族的复杂性类之间存在大量分离与包含关系。此外，我们还探究了它们与具有多项式栈-态复杂度的下推自动机族之间的关联。