Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The framework to unify all complexity dichotomy theorems for Boolean tensor networks

Mingji Xia

TL;DR

A framework is proposed, which observes that for the unresolved $\mathcal{F}$ problems, the binary functions must be a finite group, formed by 2-by-2 matrices over complex numbers, and divides all unsolved problems according to the group categories, into 9 cases.

Abstract

Fixing an arbitrary set $\mathcal{F}$ of complex-valued functions over Boolean variables yields a counting problem $\#\mathcal{F}$. Taking only functions from $\mathcal{F}$ to form a tensor network as the problem's input, the counting problem $\#\mathcal{F}$ asks for the value of the tensor network. These dichotomy or quasi-dichotomy theorems form a partial order according to the inclusion relations of the problem subclasses they characterize. As the number of known dichotomy theorems increases, the number of maximal elements in this partially ordered set first grows, and then shrinks when a new dichotomy theorem unifies several previous maximal ones; currently, there are about five or six. More can be artificially defined. However, it might be the timing to directly study the maximum element in the total partial order, namely, the entire class. This paper proposes such a framework, which observes that for the unresolved $\#\mathcal{F}$ problems, the binary functions must be a finite group, formed by 2-by-2 matrices over complex numbers. The framework, divides all unsolved problems according to the group categories, into 9 cases. This paper: introduces this grand framework; discusses the simplification of matrix forms brought by transposition closure property of the group; discusses the barrier reached by the great realnumrizing method, when a quaternion subgroup is involved; advances the order-1 cyclic group case to a position based on a dichotomy theorem conjecture; and resolves the higher-order cyclic group case.

The framework to unify all complexity dichotomy theorems for Boolean tensor networks

TL;DR

A framework is proposed, which observes that for the unresolved problems, the binary functions must be a finite group, formed by 2-by-2 matrices over complex numbers, and divides all unsolved problems according to the group categories, into 9 cases.

Abstract

Fixing an arbitrary set of complex-valued functions over Boolean variables yields a counting problem . Taking only functions from to form a tensor network as the problem's input, the counting problem asks for the value of the tensor network. These dichotomy or quasi-dichotomy theorems form a partial order according to the inclusion relations of the problem subclasses they characterize. As the number of known dichotomy theorems increases, the number of maximal elements in this partially ordered set first grows, and then shrinks when a new dichotomy theorem unifies several previous maximal ones; currently, there are about five or six. More can be artificially defined. However, it might be the timing to directly study the maximum element in the total partial order, namely, the entire class. This paper proposes such a framework, which observes that for the unresolved problems, the binary functions must be a finite group, formed by 2-by-2 matrices over complex numbers. The framework, divides all unsolved problems according to the group categories, into 9 cases. This paper: introduces this grand framework; discusses the simplification of matrix forms brought by transposition closure property of the group; discusses the barrier reached by the great realnumrizing method, when a quaternion subgroup is involved; advances the order-1 cyclic group case to a position based on a dichotomy theorem conjecture; and resolves the higher-order cyclic group case.
Paper Structure (75 sections, 47 theorems, 72 equations, 4 tables)

This paper contains 75 sections, 47 theorems, 72 equations, 4 tables.

Table of Contents

  1. Recap of Abstract and Chapter Division
  2. Background Knowledge
  3. Introduction to Counting Complexity and Dichotomy Theorems
  4. Definitions Related to Computational Complexity
  5. Tensor Networks
  6. Definition of Counting Problems
  7. Gadget Reduction
  8. Holographic Reduction
  9. Interpolation Reduction
  10. Decomposition Lemma and Reductions
  11. Some Known Dichotomy Theorems
  12. The Binary Function Group of $\#\mathcal{F}$
  13. Pinning Down Matrix Forms Using Transpose Closure
  14. Realification Attempts in the Presence of K4
  15. A Formal Derivation for Full-Rank Arity-4 Functions
  16. ...and 60 more sections

Key Result

Proposition 2.5

For a tensor network $G(V,E,X)$, let $C$ be a cut-set of $G$, such that $G'(V,E-C)$ forms two disconnected parts $G_1(V_1,E_1)$ and $G_2(V_2,E_2)$. (It is not required here that $G_1$ and $G_2$ themselves are connected components.) That is, $G'=G_1 \cup G_2$, $G'=(V_1 \uplus V_2, E_1 \uplus E_2)$, w where $\sigma= \sigma_1 \cup \sigma_2$. This formula can be understood as: using the definition to

Theorems & Definitions (119)

  • Definition 2.1: Graph Primitive/Gadget
  • Definition 2.2: Degree of a Vertex
  • Definition 2.3: Tensor Network
  • Definition 2.4
  • Proposition 2.5: Associativity of Tensor Networks
  • Definition 2.6: $\mathcal{F}$-tensor network problem
  • Definition 2.7: $\mathcal{F}|\mathcal{H}$-tensor network problem
  • Theorem 2.8: Valiant 2004
  • Theorem 2.9: Valiant 2004
  • Definition 2.10: Symmetric Functions and their Shorthand Notation
  • ...and 109 more