發(fā)布時(shí)間：2018-06-17 20:17

  本文選題：布爾網(wǎng)絡(luò) + 不動(dòng)點(diǎn)　； 參考：《河北師范大學(xué)》2017年碩士論文

【摘要】：布爾網(wǎng)絡(luò)是自然和人造的非線性動(dòng)態(tài)網(wǎng)絡(luò)的一種緊湊模型,近年來(lái),對(duì)于有關(guān)布爾網(wǎng)絡(luò)的研究受到了人們的廣泛關(guān)注.布爾網(wǎng)絡(luò)是一個(gè)有向圖,但是與一般圖的的區(qū)別是:布爾網(wǎng)絡(luò)中節(jié)點(diǎn)的狀態(tài)變量取值只能是“0”或“1”,關(guān)于布爾網(wǎng)絡(luò)的相關(guān)計(jì)算都是在二元布爾代數(shù)上進(jìn)行的.需要注意的是這里的“0”和“1”不再是單純的數(shù)字,而是符號(hào).形式邏輯中命題的“真”和“假”,開(kāi)關(guān)網(wǎng)絡(luò)中出現(xiàn)的有電與無(wú)電,高壓與低壓,導(dǎo)通與截止等,都可以視為二元布爾代數(shù)中兩個(gè)元素“1”和“0”的現(xiàn)實(shí)模型.由于現(xiàn)實(shí)生活中的網(wǎng)絡(luò)模型并不總是強(qiáng)連通的,當(dāng)這些網(wǎng)絡(luò)不特殊的時(shí)候,我們希望用更少的條件來(lái)找到它們的不動(dòng)點(diǎn).本文主要研究了一類(lèi)布爾網(wǎng)絡(luò),首先根據(jù)有向圖中節(jié)點(diǎn)的等價(jià)關(guān)系,把布爾網(wǎng)絡(luò)分成了若干個(gè)極大強(qiáng)連通子網(wǎng),并為子網(wǎng)進(jìn)行編號(hào);基于子網(wǎng)的反饋節(jié)點(diǎn)集,給出了為每個(gè)子網(wǎng)構(gòu)造對(duì)應(yīng)新子網(wǎng)的算法,這些新子網(wǎng)也合成了新網(wǎng)絡(luò);由此,證明了原布爾網(wǎng)絡(luò)的不動(dòng)點(diǎn)與新布爾網(wǎng)絡(luò)的不動(dòng)點(diǎn)相同.然后,通過(guò)新增加的反饋節(jié)點(diǎn)集中節(jié)點(diǎn)的樹(shù)形函數(shù),給出了求解布爾網(wǎng)絡(luò)不動(dòng)點(diǎn)的一個(gè)充分必要條件,即:在新構(gòu)造網(wǎng)絡(luò)的基礎(chǔ)上,通過(guò)移動(dòng)網(wǎng)絡(luò)中的弧,使其變?yōu)榉茄h(huán)網(wǎng)絡(luò),按照節(jié)點(diǎn)的拓?fù)渑判?給出了為新增加的反饋節(jié)點(diǎn)集中節(jié)點(diǎn)構(gòu)造樹(shù)形函數(shù)的算法;在此基礎(chǔ)上,利用狀態(tài)變量和節(jié)點(diǎn)的樹(shù)形函數(shù)所滿足的方程,給出了確定不動(dòng)點(diǎn)的必要條件以及條件加強(qiáng)下的求解不動(dòng)點(diǎn)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(wǎng)絡(luò)不動(dòng)點(diǎn)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(xué)位授予單位】：河北師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O157.5

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本文編號(hào)：2032318