熵与固定奇偶性信息量的代数表示
Algebraic Representations of Entropy and Fixed-Parity Information Quantities
AI总结：
本文改进 Yeung 的 I-measure，用理想（上集）刻画对数可分解信息量的代数结构，并据此计数、刻画固定奇偶性表达式，证明三变量完全协同系统仅为 XOR。
摘要：
许多信息论量都有相应的集合表示。用于刻画熵的主流带符号测度空间，即 Yeung 的 I-measure，有时无法区分性质上不同的系统。在先前工作中，我们提出了对该带符号测度空间的一种改进，并证明了它能够表示许多量，我们将其称为对数可分解量。在本工作中，我们证明该框架具有自然的代数行为，这种行为可以用理想（在此刻画为上集）来表述；并且我们表明，这种行为使我们能够进行多种计数论证，并刻画许多固定奇偶性的信息量表达式。作为应用，我们给出了一个代数证明：对于三个有限变量 X、Y 和 Z=f(X,Y)，唯一的完全协同系统是 XOR 门。
英文摘要：
Many information-theoretic quantities have corresponding representations in terms of sets. The prevailing signed measure space for characterising entropy, the $I$-measure of Yeung, is occasionally unable to discern between qualitatively distinct systems. In previous work, we presented a refinement of this signed measure space and demonstrated its capability to represent many quantities, which we called logarithmically decomposable quantities. In the present work we demonstrate that this framework has natural algebraic behaviour which can be expressed in terms of ideals (characterised here as upper-sets), and we show that this behaviour allows us to make various counting arguments and characterise many fixed-parity information quantity expressions. As an application, we give an algebraic proof that the only completely synergistic system of three finite variables $X$, $Y$ and $Z = f(X,Y)$ is the XOR gate.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
