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On the Zero-Error Capacity of Semantic Channels with Input and Output Memories

Qi Cao, Yulin Shao, Shangwei Ge

TL;DR

This letter investigates the zero-error capacity of channels with memory by introducing a new category known as the semantic channel and analyzing the zero-error capacity of the semantic channel using a comprehensive framework that accommodates multiple input and output memories.

Abstract

This paper investigates the zero-error capacity of channels with memory. Motivated by the nuanced requirements of semantic communication that incorporate memory, we advance the classical enlightened dictator channel by introducing a new category known as the semantic channel. We analyze the zero-error capacity of the semantic channel using a comprehensive framework that accommodates multiple input and output memories. Our approach reveals a more sophisticated and detailed model compared to the classical memory channels, highlighting the impact of memory on achieving error-free communication.

On the Zero-Error Capacity of Semantic Channels with Input and Output Memories

TL;DR

This letter investigates the zero-error capacity of channels with memory by introducing a new category known as the semantic channel and analyzing the zero-error capacity of the semantic channel using a comprehensive framework that accommodates multiple input and output memories.

Abstract

This paper investigates the zero-error capacity of channels with memory. Motivated by the nuanced requirements of semantic communication that incorporate memory, we advance the classical enlightened dictator channel by introducing a new category known as the semantic channel. We analyze the zero-error capacity of the semantic channel using a comprehensive framework that accommodates multiple input and output memories. Our approach reveals a more sophisticated and detailed model compared to the classical memory channels, highlighting the impact of memory on achieving error-free communication.
Paper Structure (9 sections, 6 theorems, 33 equations, 4 figures)

This paper contains 9 sections, 6 theorems, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Channel with memory
  4. Zero-error capacity
  5. Main Results
  6. The $K_2=1$ case
  7. The $K_1=1$ case
  8. The $K_1,K_2\geq2$ case
  9. Conclusion

Key Result

Lemma 1

Consider a zero-error code $\mathcal{C}_n$ of length $n$ for a semantic channel. Let $\bm{x}$ be an arbitrary but fixed codeword in $\mathcal{C}_n$. For any sequence $\bm{x}'\in \mathcal{X}^n$ such that $\mathcal{O}_{K_1,K_2}(\bm{x}')\subseteq \mathcal{O}_{K_1,K_2}(\bm{x})$, by replacing $\bm{x}$ by

Figures (4)

  • Figure 1: The output $y_t$ of a memory channel is probabilistically determined by the current input, the previous $K_1-1$ inputs, and the previous $K_2-1$ outputs.
  • Figure 2: Zero-error capacity of the semantic channel when $K_1=1$ or $K_2= 2,3$ or $K_1\ge K_2\ge 4$.
  • Figure 3: An illustration of the upper and lower bounds in \ref{['k2k1_new']}.
  • Figure 4: The hollow points represent the cases where the zero-error capacities are determined. The solid points represent the cases where the zero-error capacities are bounded.

Theorems & Definitions (9)

  • Definition 1: distinguishable sequences
  • Definition 2: zero-error codes
  • Definition 3: zero-error capacity
  • Lemma 1
  • Corollary 1
  • Theorem 1: zero-error capacity of $M_{K_1,1}$ Cai1998
  • Theorem 2
  • Theorem 3: zero-error capacity of $M_{1,K_2}$
  • Theorem 4: zero-error capacity of $M_{K_1,K_2}$, $K_1,K_2\geq2$