半環(huán)Markov性質(zhì)的研究

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關(guān)鍵詞：Markov鏈；Markov隨機場;Markov半環(huán);Grobner-Shirshov基;Shirshov算法中圖分類號：029 文獻標(biāo)志碼：A doi：10.12415/j.issn.1671-7872.23107

Research on the Markov Properties of Semirings

NIU Xiaohui，LI Wenxi （School of Microelectronics & Data Science， Anhui University of Technology， Maanshan ）

Abstract：To further simplify complex problems in information theory，the Shirshov algorithm was employed to reduce the specific generation relations， through which simplified algebraic proofs were provided that the reversed chain and subchain of a Markov chain preserve the Markov property.Building upon the semiring-based characterization ofMarkov chains，the algebraic representation of Markov random fields was further explored.The Grobner-Shirshov basis for the generating relations of Markov random fields was computed using the Shirshov algorithm，thereby obtaining the corresponding semiring Markov normal form.Based on this normal form，an algebraiccriterion was establishedfor determining whether random variables forma Markovrandom field，and standard representations were derived for information measures including joint entropy，conditional entropy，and mutual information.Finally，through a concrete example，the Grobner-Shirshov basis and normal form of the generating relations fora Markov random field were computed，and it was proved that the random variables （X1，X2 ， X3 ， X4 ） constitute the given Markov random field if and only if for any p∈K4，yp=θ ， K4={9，10，11}

Keywords：Markov chain; Markov random field; Markov semiring; Grobner-Shirshov basis; Shirshov algorithm

為了尋找解決信息論中困難問題的簡化方法，20世紀(jì)60年代 Hu[1] 開始研究Shannon信息度量的集合論結(jié)構(gòu)，通過符號替換確定每個信息恒等式都對應(yīng)一個集合恒等式。（剩余1307字）