🤖 AI Summary
This paper investigates the computational power boundaries of finite automata (FA) and pushdown automata (PDA) augmented with multiple counters under unambiguous and co-nondeterministic computation models, focusing on promise problems over polynomial-length and unary inputs. Methodologically, it employs computational graph modeling, inductive counting techniques, and analysis of non-uniform families of automata. The main contribution is the first systematic characterization of the expressive power of multi-counter automata in these two computational paradigms. Specifically, it establishes a sharp collapse in the class of solvable promise problems over both polynomial-length and unary input domains, thereby separating non-uniform NL from non-uniform LOGCFL with strict containment. This result reveals a fundamental role of counter structures in shaping the landscape of non-uniform complexity classes, uncovering deeper mechanisms by which counting capabilities influence computational hierarchies.
📝 Abstract
Nonuniform families of polynomial-size finite automata and pushdown automata respectively have strong connections to nonuniform-NL and nonuniform-LOGCFL. We examine the behaviors of unambiguous and co-nondeterministic computations produced by such families of automata operating multiple counters. As its consequences, we obtain various collapses of the complexity classes of families of promise problems solvable by finite and pushdown automata families when all valid instances are limited to either polynomially long strings or unary strings. A key technical ingredient of our proofs is an inductive counting of reachable vertices of each computation graph of finite and pushdown automata that operate multiple counters simultaneously.