Disjunctive Complexity

Nikita Ivanov; Alexander Rubtsov; Michael Vyalyi

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arXiv · 2503.22997

Disjunctive Complexity

Nikita Ivanov·Alexander Rubtsov·Michael Vyalyi

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Abstract

A recently introduced measure of Boolean functions complexity--disjunc\-tive complexity (DC)--is compared with other complexity measures: the space complexity of streaming algorithms and the complexity of nondeterministic branching programs (NBP). We show that DC is incomparable with NBP. Specifically, we present a function that has low NBP but has subexponential DC. Conversely, we provide arguments based on computational complexity conjectures to show that DC can superpolynomially exceed NBP in certain cases. Additionally, we prove that the monotone version of NBP complexity is strictly weaker than DC. We prove that the space complexity of one-pass streaming algorithms is strictly weaker than DC. Furthermore, we introduce a generalization of streaming algorithms that captures the full power of DC. This generalization can be expressed in terms of nondeterministic algorithms that irreversibly write 1s to entries of a Boolean vector (i.e., changes from 1 to 0 are not allowed). Finally, we discuss an unusual phenomenon in disjunctive complexity: the existence of uniformly hard functions. These functions exhibit the property that their disjunctive complexity is maximized, and this property extends to all functions dominated by them.

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Nikita Ivanov, Alexander Rubtsov, Michael Vyalyi. 2025-03-29. Disjunctive Complexity. https://arxiv.org/abs/2503.22997

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