Core Concepts
Quantum mechanics and finite automata provide insights into graph capacity quantization.
Abstract
The article explores zero-error capacity in communication channels, introducing the zero-error unitary capacity. It discusses the relationship between Shannon capacity and independent sets in graphs. The difficulty of controlling the behavior of Shannon capacity is highlighted, along with upper bounds based on conic programming. The paper proposes a new upper bound on Shannon capacity using a Sum of Squares hierarchy. It delves into regular languages, growth rates, connectedness properties of DFAs, and correctness checking algorithms. The concept of reversible DFAs is introduced, leading to discussions on reversible capacity and its relationship to Shannon capacity. The article concludes by discussing unitary capacities in quantum finite automata.
Stats
Θ(C5) ≥ √5 was open for 24 years until Lovasz provided a matching upper bound.
Bi & Tang showed that upper bounds based on conic programming cannot beat Lovasz's bound.
Regular languages have many nice properties making it easy to reason about their growth rate.
There is a polynomial-time algorithm to determine if a regular language is a graph language for G.
For a regular language L with k symbols with a d-state DFA, there is an algorithm to find the growth rate of L.
Quotes
"The behavior of the Shannon capacity is difficult to control."
"Regular languages have many nice properties."
"There is a polynomial-time algorithm to determine if a regular language is a graph language for G."