🤖 AI Summary
This paper investigates the computational power and complexity boundaries of families of nonuniform polynomial-size nondeterministic finite automata (NFA-poly) with respect to partial counting functions, gap functions, and their associated promise decision problems. Using state-complexity analysis and counting-complexity frameworks, it provides the first systematic characterization of the complexity hierarchy for NFA-poly without relying on unproven hardness assumptions. Key contributions include: (i) a strict separation between the counting class #NFA-poly and the gap class GapNFA-poly; (ii) an exact simulation equivalence between these classes and polynomially stack-bounded pushdown automata (PDA); and (iii) identification of novel conditions under which complexity collapses occur for promise problems. These results unify the theoretical connections between counting-based finite automata and resource-restricted PDAs, yielding a foundational stratification of automata-based complexity classes.
📝 Abstract
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.