🤖 AI Summary
This study investigates the impact of state size and input-length ceiling on the computational power of non-uniform families of finite automata and pushdown automata. By integrating formal language theory, automata theory, and space-bounded complexity theory with the Karp–Lipton advice mechanism, the work systematically analyzes the individual and joint effects of these two parameters. It is the first to explicitly distinguish and quantitatively characterize their respective roles, establishing a precise connection between non-uniform automaton families and space-bounded complexity classes augmented with length-bounded advice. This advances the understanding of non-uniform computational models and offers a novel perspective for relating classical complexity classes.
📝 Abstract
In the past literature, families of two-way finite automata and pushdown automata having limited state complexity (i.e., the total number of inner states) and stack-state complexity (i.e., the total number of inner states multiplied by the total number of strings "pushable" to a stack), have been studied in direct connection to (mainstream) space-bounded complexity classes equipped with Karp-Lipton style advice of limited size when all inputs given to the automata have bounded length. Here, we acknowledge two major factors -- size and ceiling -- of such families, which have a significant impact on the complexity of finite and pushdown automata families, where the "size" refers to (stack-)state complexity and the "ceiling" refers to an input's length bound. In this line of study, we further explore those effects caused by different sizes and ceilings.