Parameterizations of Logarithmic-Space Reductions, Stack-State Complexity of Nonuniform Families of Pushdown Automata, and a Road to the LOGCFL$\subseteq$LOGDCFL/poly Question

TL;DR

This paper explores the stack-state complexity of nonuniform pushdown automata and its connection to the LOGCFL versus LOGDCFL/poly question, providing new insights into automata state complexity and complexity class relationships.

Contribution

It introduces parameterizations of LOGCFL and LOGDCFL and links the open question to the polynomial stack-state complexity of nonuniform pushdown automata.

Findings

Established a connection between LOGCFL LOGDCFL/poly question and pushdown automata complexity.

Analyzed the stack-state complexity of various pushdown automata families.

Discussed the computational complexity of polynomial-size one-way pushdown automata.

Abstract

The complexity class LOGCFL (resp., LOGDCFL) consists of all languages that are many-one reducible to context-free (resp., deterministic context-free) languages using logarithmic space. These complexity classes have been studied over five decades in connection to parallel computation since they are located between Nick's classes $\mathrm{NC}^1$ and $\mathrm{NC}^2$. In contrast, the state complexity of nonuniform finite-automaton families was first discussed in the 1970s and it has been extensively explored lately for various finite-automata families. We extend this old subject to the stack-state complexity (i.e., the total number of inner states plus simultaneously pushable stack symbol series) of nonuniform families of various pushdown automata. We introduce reasonable "parameterizations" of LOGCFL and LOGDCFL and apply them as a technical tool to establish a close connection between the LOGCFL$\subseteq$LOGDCFL/poly question and the polynomial stack-state complexity of nonuniform families of two-way pushdown automata. We also discuss the precise computational complexity of polynomial-size one-way pushdown automata.