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Boolean Circuit Complexity and Two-Dimensional Cover Problems

Bruno P. Cavalar, Igor C. Oliveira

TL;DR

The paper develops a unified discrete-complexity framework that recasts Boolean circuit lower bounds as two-dimensional cover problems, linking the fusion method with graph complexity to enable transfers between circuit and graph models. It introduces core measures $D(A\mid\mathcal{B})$, $D_\cap(A\mid\mathcal{B})$, and $\rho(A,\mathcal{B})$, and proves tight connections and transference results that translate graph lower bounds into circuit lower bounds and vice versa. A central advance is the exact characterization of cover complexity via cyclic discrete complexity, together with a set-theoretic fusion framework that encompasses both acyclic and cyclic constructions; this yields concrete lower bounds in monotone and cyclic circuits, such as for clique-type problems. The work further shows that random graphs have predictable cover complexity, with $\rho(G,\mathcal{G}_{N,N})=\Theta(N)$, and introduces nondeterministic graph complexity to bridge to nondeterministic circuit models. Collectively, these results provide a principled route to deriving and understanding circuit lower bounds through two-dimensional cover problems and their cyclic generalizations, with potential implications for unconditional lower bounds and hardness amplification.

Abstract

We reduce the problem of proving deterministic and nondeterministic Boolean circuit size lower bounds to the analysis of certain two-dimensional combinatorial cover problems. This is obtained by combining results of Razborov (1989), Karchmer (1993), and Wigderson (1993) in the context of the fusion method for circuit lower bounds with the graph complexity framework of Pudlák, Rödl, and Savický (1988). For convenience, we formalize these ideas in the more general setting of "discrete complexity", i.e., the natural set-theoretic formulation of circuit complexity, variants of communication complexity, graph complexity, and other measures. We show that random graphs have linear graph cover complexity, and that explicit super-logarithmic graph cover complexity lower bounds would have significant consequences in circuit complexity. We then use discrete complexity, the fusion method, and a result of Karchmer and Wigderson (1993) to introduce nondeterministic graph complexity. This allows us to establish a connection between graph complexity and nondeterministic circuit complexity. Finally, complementing these results, we describe an exact characterization of the power of the fusion method in discrete complexity. This is obtained via an adaptation of a result of Nakayama and Maruoka (1995) that connects the fusion method to the complexity of "cyclic" Boolean circuits, which generalize the computation of a circuit by allowing cycles in its specification.

Boolean Circuit Complexity and Two-Dimensional Cover Problems

TL;DR

The paper develops a unified discrete-complexity framework that recasts Boolean circuit lower bounds as two-dimensional cover problems, linking the fusion method with graph complexity to enable transfers between circuit and graph models. It introduces core measures , , and , and proves tight connections and transference results that translate graph lower bounds into circuit lower bounds and vice versa. A central advance is the exact characterization of cover complexity via cyclic discrete complexity, together with a set-theoretic fusion framework that encompasses both acyclic and cyclic constructions; this yields concrete lower bounds in monotone and cyclic circuits, such as for clique-type problems. The work further shows that random graphs have predictable cover complexity, with , and introduces nondeterministic graph complexity to bridge to nondeterministic circuit models. Collectively, these results provide a principled route to deriving and understanding circuit lower bounds through two-dimensional cover problems and their cyclic generalizations, with potential implications for unconditional lower bounds and hardness amplification.

Abstract

We reduce the problem of proving deterministic and nondeterministic Boolean circuit size lower bounds to the analysis of certain two-dimensional combinatorial cover problems. This is obtained by combining results of Razborov (1989), Karchmer (1993), and Wigderson (1993) in the context of the fusion method for circuit lower bounds with the graph complexity framework of Pudlák, Rödl, and Savický (1988). For convenience, we formalize these ideas in the more general setting of "discrete complexity", i.e., the natural set-theoretic formulation of circuit complexity, variants of communication complexity, graph complexity, and other measures. We show that random graphs have linear graph cover complexity, and that explicit super-logarithmic graph cover complexity lower bounds would have significant consequences in circuit complexity. We then use discrete complexity, the fusion method, and a result of Karchmer and Wigderson (1993) to introduce nondeterministic graph complexity. This allows us to establish a connection between graph complexity and nondeterministic circuit complexity. Finally, complementing these results, we describe an exact characterization of the power of the fusion method in discrete complexity. This is obtained via an adaptation of a result of Nakayama and Maruoka (1995) that connects the fusion method to the complexity of "cyclic" Boolean circuits, which generalize the computation of a circuit by allowing cycles in its specification.

Paper Structure

This paper contains 27 sections, 26 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Contributions.
  4. Results
  5. Notation.
  6. Organization.
  7. Acknowledgements.
  8. Discrete Complexity
  9. Definitions and notation
  10. Examples
  11. Boolean circuit complexity
  12. Bipartite graph complexity
  13. Higher-dimensional generalizations of graph complexity
  14. Combinatorial rectangles from communication complexity
  15. Basic lemmas and other useful results
  16. ...and 12 more sections

Key Result

Lemma 1

For every non-trivial bipartite graph $G \subseteq [N] \times [N]$ and corresponding Boolean function $f_G \colon \{0,1\}^{n} \times \{0,1\}^n \to \{0,1\}$, we have

Figures (3)

  • Figure 1: A graphical representation of a set $G \subseteq [5] \times [5]$ of intersection complexity $D_\cap(G \mid \mathcal{G}_{5,5}) \leq 2$ via $G = ((R_2 \cup R_4) \cap (C_1 \cup C_3 \cup C_5) ) \cup ((C_2 \cup C_4) \cap (R_1 \cup R_3 \cup R_5) )$.
  • Figure 2: In this example, $\Gamma = [6] \times [22]$, $\mathcal{B} = \mathcal{G}_{6, 22}$ (as in Section \ref{['ss:graph_complexity']}), and the $\{\cdot, \bullet, w\}$-valued matrix above encodes $U = G^c$ (rectangles with $\bullet$), where $G \subseteq \Gamma$ (locations with $\cdot$ and $w$) can be interpreted as a bipartite graph. If a semi-filter $\mathcal{F}$ over $U$ is above $w \in G$ (corresponding to coordinates $(2,15)$), then it must contain the distinguished subsets of $U$ represented in blue $(R_2 \cap U)$ and in orange $(C_{15} \cap U)$, respectively.
  • Figure 3: A graphical representation of $G_\mathsf{NEQ} \subseteq [N] \times [N]$ for $N = 8$. \ref{['prop:NEQ_lb']} shows that for this value of $N$ the intersection complexity is $3$.

Theorems & Definitions (61)

  • Lemma 1: Transference of Lower Bounds
  • Theorem 2: Cover complexity of a random graph
  • Theorem 3: Exact characterization of cover complexity
  • proof
  • Lemma 9: Immediate from DBLP:journals/ipl/Zwick96
  • Proposition 10: Finiteness test
  • proof
  • Lemma 11: Complex sets
  • Lemma 12
  • proof
  • ...and 51 more