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

TL;DR

This work investigates unambiguous and co-nondeterministic computations within nonuniform polynomial-size automata enhanced by multiple counters, connecting these models to nonuniform and parameterized complexity classes. The authors develop inductive counting techniques and counter-reduction methods to establish collapses and equivalences, notably showing $2\mathrm{NCT}_4=\mathrm{co}\hbox{-}2\mathrm{NCT}_4$ and $\mathrm{co}\hbox{-}2\mathrm{NPDCT}_k\subseteq 2\mathrm{NPDCT}_{3}$, among others, and demonstrate that polynomial ceilings can eliminate counters, yielding $2\mathrm{NCT}_k/\mathrm{poly}=2\mathrm{N}/\mathrm{poly}$ and related results. They also connect these nonuniform results to parameterized decision problems and LOGCFL/LOGCFL(poly), showing how advice and parameterization bridge nonuniform and classical complexity. The paper further discusses complementation and unambiguity collapses, and extends the analysis to unary alphabets, outlining several open questions and future directions. Overall, the work reveals structural collapses in nonuniform stack-state complexity induced by counters and polynomial ceilings, offering new tools for understanding nonuniform computations and their connections to standard complexity regimes with advice.

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.

Figures (3)

• Figure 1: Containments among nonuniform polynomial (stack-)state complexity classes shown in this work. A blue solid line indicates proper inclusion whereas a black dotted line does simple inclusion (not known to be proper). Remark that the collapse $2\mathrm{U}/\mathrm{poly}=2\mathrm{N}/\mathrm{poly}$ comes from Yam22b and $2\mathrm{U}/\mathrm{unary} = 2\mathrm{N}/\mathrm{unary} = \mathrm{co}\hbox{-}2\mathrm{N}/\mathrm{unary}$ are drawn from GMP07GP11 in Section \ref{['sec:unary-inputs']}.
• Figure 2: A two-way $k$-counter pushdown automaton. Counters are indexed from CT1 to CT$k$. An input-tape head is scanning the $l$th tape cell, which holds an input symbol $\sigma$. The topmost stack cell holds a stack symbol $a$.
• Figure 3: A stack history of a 2npdcta. Rounds $r-1$ and $r$ loosely correspond to stack height. From a conf-interval $(C,0,C',l,r-1)$, we obtain another conf-interval $(C_1,1,C_3,l_2+l_3,r)$ with $l=l_1+l_2+l_3+l_4$ and $l_1=l_4=1$, from which we further obtain two more conf-intervals $(C_1,0,C_2,l_2,r)$ and $(C_2,0,C_3,l_3,r)$ by applying (i) and (ii) in the nondeterministic phase (2) in the proof of Lemma \ref{['pushdown-counter']}.

Theorems & Definitions (14)

• Lemma 3.1
• Lemma 3.3
• Theorem 3.4
• Lemma 3.5
• Theorem 3.6
• Lemma 3.7
• Definition 4.1
• Lemma 4.2
• Definition 4.3
• Theorem 5.1