On Weak-Space Complexity over Complex Numbers

Abstract

Defining a feasible notion of space over the Blum-Shub-Smale (BSS) model of algebraic computation is a long standing open problem. In an attempt to define a right notion of space complexity for the BSS model, Naurois [CiE 2007] introduced the notion of weak-space. We investigate the weak-space bounded computations and their plausible relationship with the classical space bounded computations. For weak-space bounded, division-free computations over BSS machines over complex numbers with $\mathop {=}\limits ^{?}0$ tests, we show the following:

• 1. The Boolean part of the weak log-space class is contained in deterministic log-space, i.e., $\mathsf{BP}(\mathsf{LOGSPACE_W}) \subseteq \mathsf{DLOG}$;

• 2. There is a set $L\in$ $\mathsf {NC}^{1}_{\mathbb {C}}$ that cannot be decided by any deterministic BSS machine whose weak-space is bounded above by a polynomial in the input length, i.e., ${\mathsf {NC}}^1_{\mathbb {C}} \nsubseteq \mathsf{PSPACE_W}$.

The second result above resolves the first part of Conjecture 1 stated in [6] over complex numbers and exhibits a limitation of weak-space. The proof is based on the structural properties of the semi-algebraic sets contained in $\mathsf{PSPACE_W}$ and the result that any polynomial divisible by a degree-$\omega (1)$ elementary symmetric polynomial cannot be sparse. The lower bound on the sparsity is proved via an argument involving Newton polytopes of polynomials and bounds on number of vertices of these polytopes, which might be of an independent interest.