Monotone Classes Beyond VNP

TL;DR

This paper explores the monotone analogues of VPSPACE, revealing their non-equivalence, proposing a new definition called mVPSPACE, and demonstrating its strength and properties, especially in relation to transparent polynomials.

Contribution

The paper introduces mVPSPACE as a monotone analogue of VPSPACE, showing it is exponentially stronger than mVNP and analyzing its properties and implications for transparent polynomials.

Findings

mVPSPACE is exponentially stronger than mVNP.

Monotone analogues of VPSPACE are not equivalent.

Transparent polynomials of large sparsity are hard for all monotone VPSPACE analogues except Poizat's.

Abstract

In this work, we study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat 2008, Koiran and Perifel 2009, Malod 2011, Mahajan and Rao 2013) that is believed to be larger than VNP. We observe that these monotone analogues are not equivalent unlike their non-monotone counterparts, and propose monotone VPSPACE (mVPSPACE) to be defined as the monotone analogue of Poizat's definition. With this definition, mVPSPACE turns out to be exponentially stronger than mVNP and also satisfies several desirable closure properties that the other analogues may not. Our initial goal was to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrube\v{s} and Yehudayoff (2021). In that context, we show that transparent polynomials of large sparsity are hard for the monotone analogues of all the known definitions of VPSPACE, except for the one due to Poizat.