Nondeterministic Auxiliary Depth-Bounded Storage Automata and Semi-Unbounded Fan-in Cascading Circuits

TL;DR

This paper introduces a nondeterministic variant of auxiliary depth-bounded storage automata and a new semi-unbounded fan-in circuit model, providing complexity characterizations and bounds for these automata classes.

Contribution

It proposes a new circuit model with cascading blocks and characterizes nondeterministic auxiliary automata using this model, establishing complexity bounds and relationships.

Findings

Characterizes nondeterministic auxiliary automata with new circuit model.

Establishes upper bounds on automata complexity classes.

Relates cascading circuits to polynomial-time computation.

Abstract

We discuss a nondeterministic variant of the recently introduced machine model of deterministic auxiliary depth-$k$ storage automata (or aux-$k$-sda's) by Yamakami. It was proven that all languages recognized by polynomial-time logarithmic-space aux-$k$-sda's are located between $\mathrm{LOGDCFL}$ and $\mathrm{SC}^k$ (the $k$th level of Steve's class SC). We further propose a new and simple computational model of semi-unbounded fan-in Boolean circuits composed partly of cascading blocks, in which the first few AND gates of unbounded fan-out (called AND$_{(\omega)}$ gates) at each layer from the left (where all gates at each layer are indexed from left to right) are linked in a "cascading" manner to their right neighbors though specific AND and OR gates. We use this new circuit model to characterize a nondeterministic variant of the aux-$2k$-sda's (called aux-$2k$-sna's) that run in polynomial time using logarithmic work space. By relaxing the requirement for cascading circuits, we also demonstrate how such cascading circuit families characterize the complexity class $\mathrm{P}$. This yields an upper bound on the computational complexity of $\mathrm{LOG}k\mathrm{SNA}$ by $\mathrm{P}$.