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Disjunctive Complexity

Nikita Ivanov, Alexander Rubtsov, Michael Vyalyi

TL;DR

This paper introduces disjunctive complexity $Δ(f)$ as the minimal size of a transition graph representing a Boolean function $f$, and systematically relates it to streaming space and nondeterministic branching programs. It defines restricted DC variants—$Δ_s$, $Δ_{as}$, and $Δ_1$—to capture streaming and write-once models, proving separations such as $Δ_{as}$ vs $Δ_s$ (exponential gap) and, under $NP\nsubseteq P/\!$poly, $Δ$ vs $Δ_1$ (superpolynomial gaps). It also shows that the monotone version of NBP is strictly weaker than DC, and that the space complexity of one-pass streaming algorithms is strictly weaker than DC, while a write-once generalization can capture the full power of DC; the paper identifies uniformly hard functions like clique-graph indicators $C_n$ for which $Δ(C_n)=2^n$ and demonstrates NP-hardness of evaluating $f_G$ via SAT reductions. Overall, the work exposes fundamental separations between DC and classical models, provides new lower-bound techniques via transition graphs, and establishes the existence of uniformly hard DC functions with broad implications for complexity theory and lower-bound methodologies.

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.

Disjunctive Complexity

TL;DR

This paper introduces disjunctive complexity as the minimal size of a transition graph representing a Boolean function , and systematically relates it to streaming space and nondeterministic branching programs. It defines restricted DC variants—, , and —to capture streaming and write-once models, proving separations such as vs (exponential gap) and, under poly, vs (superpolynomial gaps). It also shows that the monotone version of NBP is strictly weaker than DC, and that the space complexity of one-pass streaming algorithms is strictly weaker than DC, while a write-once generalization can capture the full power of DC; the paper identifies uniformly hard functions like clique-graph indicators for which and demonstrates NP-hardness of evaluating via SAT reductions. Overall, the work exposes fundamental separations between DC and classical models, provides new lower-bound techniques via transition graphs, and establishes the existence of uniformly hard DC functions with broad implications for complexity theory and lower-bound methodologies.

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.

Paper Structure

This paper contains 8 sections, 8 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Definitions and Initial Observations
  3. Disjunctive Complexity and Streaming Algorithms
  4. Streaming Disjunctive Complexity
  5. Write-once Disjunctive Complexity
  6. Disjunctive Complexity and Branching Programs
  7. Uniformly Hard Functions
  8. An $\mathrm {NP}$-completeness of an Evaluation a Function Represented by a Transition Graph

Key Result

theorem 1

There exists a sequence of Boolean functions $f_n\colon \{0,1\}^{4n}\to\{0,1\}$ such that $\Delta_{\mathrm{as}}(f_n) = \mathrm {poly}(n)$ and $\Delta_{\mathrm{s}}(f_n) = 2^{\Omega(n)}$.

Theorems & Definitions (24)

  • proof
  • theorem 1
  • proof
  • lemma 1
  • proof
  • theorem 2
  • proof
  • proof
  • proof
  • lemma 2
  • ...and 14 more