Unambiguous and Co-Nondeterministic Computations of Finite Automata and Pushdown Automata Families and the Effects of Multiple Counters

TL;DR

This paper explores how unambiguous and co-nondeterministic finite automata and pushdown automata with multiple counters influence complexity class collapses, using inductive counting techniques.

Contribution

It introduces new results on complexity class collapses for automata with multiple counters and provides technical methods for counting reachable states.

Findings

Complexity classes collapse under certain automata constraints.

Multiple counters impact the computational power of automata.

Inductive counting is key to analyzing automata behaviors.

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, where a counter is a stack using only a single non-bottom symbol. As immediate 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.