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.

翻译：本文研究了VPSPACE（一个被广泛研究的类，Poizat 2008, Koiran和Perifel 2009, Malod 2011, Mahajan和Rao 2013，被认为大于VNP）各种等价定义的自然单调类比。我们观察到这些单调类比并不等价，不同于它们的非单调对应物，并建议将单调VPSPACE（mVPSPACE）定义为Poizat定义的单调类比。通过这一定义，mVPSPACE在指数级别上强于mVNP，并且满足其他类比可能不具备的若干良好封闭性质。我们最初的目标是理解透明多项式的单调复杂性，这一概念由Hrubeš和Yehudayoff（2021）最近引入。在此背景下，我们证明：对于除Poizat定义之外的所有已知VPSPACE定义的单调类比，具有大稀疏度的透明多项式都是困难的。