本文選題：布爾網絡 + 不動點��； 參考：《河北師范大學》2017年碩士論文

【摘要】：布爾網絡是自然和人造的非線性動態(tài)網絡的一種緊湊模型,近年來,對于有關布爾網絡的研究受到了人們的廣泛關注.布爾網絡是一個有向圖,但是與一般圖的的區(qū)別是:布爾網絡中節(jié)點的狀態(tài)變量取值只能是“0”或“1”,關于布爾網絡的相關計算都是在二元布爾代數上進行的.需要注意的是這里的“0”和“1”不再是單純的數字,而是符號.形式邏輯中命題的“真”和“假”,開關網絡中出現的有電與無電,高壓與低壓,導通與截止等,都可以視為二元布爾代數中兩個元素“1”和“0”的現實模型.由于現實生活中的網絡模型并不總是強連通的,當這些網絡不特殊的時候,我們希望用更少的條件來找到它們的不動點.本文主要研究了一類布爾網絡,首先根據有向圖中節(jié)點的等價關系,把布爾網絡分成了若干個極大強連通子網,并為子網進行編號;基于子網的反饋節(jié)點集,給出了為每個子網構造對應新子網的算法,這些新子網也合成了新網絡;由此,證明了原布爾網絡的不動點與新布爾網絡的不動點相同.然后,通過新增加的反饋節(jié)點集中節(jié)點的樹形函數,給出了求解布爾網絡不動點的一個充分必要條件,即:在新構造網絡的基礎上,通過移動網絡中的弧,使其變?yōu)榉茄h(huán)網絡,按照節(jié)點的拓撲排序,給出了為新增加的反饋節(jié)點集中節(jié)點構造樹形函數的算法;在此基礎上,利用狀態(tài)變量和節(jié)點的樹形函數所滿足的方程,給出了確定不動點的必要條件以及條件加強下的求解不動點的充分條件.從而用更少的方程和更少的條件給出了求解布爾網絡不動點的充要條件.
[Abstract]:Boolean network is a compact model of natural and artificial nonlinear dynamic networks. In recent years, the research on Boolean networks has been paid more and more attention. Boolean network is a directed graph, but the difference between Boolean network and general graph is that the state variable of node in Boolean network can only be "0" or "1", and the calculation of Boolean network is carried out on binary Boolean algebra. It is important to note that the "0" and "1" here are no longer simple numbers, but symbols. The "truth" and "falsehood" of propositions in formal logic, the existence of electricity and no electricity in switching networks, high voltage and low voltage, conduction and cutoff can all be regarded as the realistic models of two elements "1" and "0" in binary Boolean algebra. Because network models in real life are not always strongly connected, when these networks are not special, we hope to find their fixed points with fewer conditions. In this paper, we mainly study a class of Boolean networks. Firstly, according to the equivalent relations of nodes in directed graphs, Boolean networks are divided into several maximal strongly connected subnets and numbered for subnets. An algorithm for constructing new subnets for each subnet is presented, and the new subnets are also synthesized, and it is proved that the fixed points of the original Boolean networks are the same as the fixed points of the new Boolean networks. Then, a necessary and sufficient condition for solving fixed points of Boolean network is given by using the tree function of the nodes in the new feedback node set, that is, on the basis of the new construction of the network, through the arc in the mobile network, a necessary and sufficient condition is given for solving the fixed point of the Boolean network. The algorithm of constructing the tree function for the newly added nodes with feedback nodes is given according to the topological sort of nodes, on the basis of which, the equations satisfied by the state variables and the tree functions of nodes are used. The necessary conditions for determining fixed points and the sufficient conditions for solving fixed points under strengthened conditions are given. The necessary and sufficient conditions for solving fixed points of Boolean networks are given by using fewer equations and less conditions.
【學位授予單位】：河北師范大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O157.5

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本文編號：2032318