Core Concepts
The author presents a combinatorial view for Holant problems on higher domains, focusing on generalized Fibonacci gates.
Abstract
Introduction and Background
Holant problems encompass a broad class of counting problems.
Symmetric constraint functions are equivalent to edge-coloring models.
Fibonacci Gates on a Domain of Size 3
Tractable ternary symmetric signatures have specific forms.
Parameters satisfy linear relationships in medium-sized triangles.
Fibonacci Gates on Domain 4
Ternary domain 4 symmetric signatures have tractable forms.
Parameters satisfy cubic and quadratic relationships.
Theorems
Theorem 1: Generalized Fibonacci gates on size 3 lead to computable Holant problems.
Theorem 2: Generalized Fibonacci gates on size 4 also result in computable Holant problems.
Proofs
Detailed proofs provided for the theorems using symmetry and linear relationships.
Stats
On the Boolean domain, it is over all {0, 1}-edge assignments.
On the domain of size 3, it is over all {R, G, B}-edge assignments.
On the domain of size 4, it is over all {R, G, B, W}-edge assignments.