A New Characterization of Streaming Decidability of Boolean CSPs Under Deletions

Abstract

We characterize the single-pass streaming space complexity of Boolean CSP satisfiability in the strict-turnstile model for every fixed constraint language $Γ$ of $k$-ary Boolean relations that is not trivially satisfiable. Constraints can be inserted or deleted with nonnegative multiplicities; satisfiability is decided for those remaining at the end. While Sharma and Velusamy [ESA 2026] characterized insertion-only complexity via non-redundancy (up to logarithmic factors), we show that strict-turnstile complexity is governed by the integer capturing degree. For $R\subseteq\{0,1\}^k$, let $d_R$ be the minimum degree of an integer-coefficient polynomial vanishing exactly on $R$ over the Boolean hypercube, and set $d_Γ=\max_{R\inΓ}d_R$. On polynomial-length streams, we obtain an $O_k(n^{d_Γ}\operatorname{polylog} n)$-space randomized algorithm and an $Ω_k(n^{d_Γ}/\log n)$ lower bound for all algorithms. Moreover, our algorithm outputs a sparse instance preserving every satisfying assignment. This yields, to our knowledge, the first insertion-only versus turnstile separation for streaming CSPs. It also differs from the Fourier-degree characterization of exact optimization by Kol, Paramonov, Saxena, and Yu [ITCS 2023]. For $k\ge3$, $k$-XOR with literals has three distinct space complexities: $\widetildeΘ(n)$ for insertion-only satisfiability, $\widetildeΘ(n^{\lceil k/2\rceil})$ for strict-turnstile satisfiability, and $\widetildeΘ(n^k)$ for exact optimization in either model.