Resolution and Binary Decision Diagrams Cannot Simulate Each Other Polynomially

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运筹学英汉词汇ABC

运筹学英汉词汇(0,1) normalized ――0-1规范化Aactivity ――工序additivity――可加性adjacency matrix――邻接矩阵adjacent――邻接aligned game――结盟对策analytic functional equation――分析函数方程approximation method――近似法arc ――弧artificial constraint technique ――人工约束法artificial variable――人工变量augmenting path――增广路avoid cycle method ――避圈法Bbackward algorithm――后向算法balanced transportation problem――产销平衡运输问题basic feasible solution ――基本可行解basic matrix――基阵basic solution ――基本解basic variable ――基变量basic ――基basis iteration ――换基迭代Bayes decision――贝叶斯决策big M method ――大M 法binary integer programming ――0-1整数规划binary operation――二元运算binary relation――二元关系binary tree――二元树binomial distribution――二项分布bipartite graph――二部图birth and death process――生灭过程Bland rule ――布兰德法则branch node――分支点branch――树枝bridge――桥busy period――忙期Ccapacity of system――系统容量capacity――容量Cartesian product――笛卡儿积chain――链characteristic function――特征函数chord――弦circuit――回路coalition structure――联盟结构coalition――联盟combination me――组合法complement of a graph――补图complement of a set――补集complementary of characteristic function――特征函数的互补性complementary slackness condition ――互补松弛条件complementary slackness property――互补松弛性complete bipartite graph――完全二部图complete graph――完全图completely undeterministic decision――完全不确定型决策complexity――计算复杂性congruence method――同余法connected component――连通分支connected graph――连通图connected graph――连通图constraint condition――约束条件constraint function ――约束函数constraint matrix――约束矩阵constraint method――约束法constraint ――约束continuous game――连续对策convex combination――凸组合convex polyhedron ――凸多面体convex set――凸集core――核心corner-point ――顶点（角点）cost coefficient――费用系数cost function――费用函数cost――费用criterion ; test number――检验数critical activity ――关键工序critical path method ――关键路径法（CMP ）critical path scheduling ――关键路径cross job ――交叉作业curse of dimensionality――维数灾customer resource――顾客源customer――顾客cut magnitude ――截量cut set ――截集cut vertex――割点cutting plane method ――割平面法cycle ――回路cycling ――循环Ddecision fork――决策结点decision maker决――策者decision process of unfixed step number――不定期决策过程decision process――决策过程decision space――决策空间decision variable――决策变量decision决－－策decomposition algorithm――分解算法degenerate basic feasible solution ――退化基本可行解degree――度demand――需求deterministic inventory model――确定贮存模型deterministic type decision――确定型决策diagram method ――图解法dictionary ordered method ――字典序法differential game――微分对策digraph――有向图directed graph――有向图directed tree――有向树disconnected graph――非连通图distance――距离domain――定义域dominate――优超domination of strategies――策略的优超关系domination――优超关系dominion――优超域dual graph――对偶图Dual problem――对偶问题dual simplex algorithm ――对偶单纯形算法dual simplex method――对偶单纯形法dummy activity――虚工序dynamic game――动态对策dynamic programming――动态规划Eearliest finish time――最早可能完工时间earliest start time――最早可能开工时间economic ordering quantity formula――经济定购批量公式edge ――边effective set――有效集efficient solution――有效解efficient variable――有效变量elementary circuit――初级回路elementary path――初级通路elementary ――初等的element――元素empty set――空集entering basic variable ――进基变量equally liability method――等可能性方法equilibrium point――平衡点equipment replacement problem――设备更新问题equipment replacing problem――设备更新问题equivalence relation――等价关系equivalence――等价Erlang distribution――爱尔朗分布Euler circuit――欧拉回路Euler formula――欧拉公式Euler graph――欧拉图Euler path――欧拉通路event――事项expected value criterion――期望值准则expected value of queue length――平均排队长expected value of sojourn time――平均逗留时间expected value of team length――平均队长expected value of waiting time――平均等待时间exponential distribution――指数分布external stability――外部稳定性Ffeasible basis ――可行基feasible flow――可行流feasible point――可行点feasible region ――可行域feasible set in decision space――决策空间上的可行集feasible solution――可行解final fork――结局结点final solution――最终解finite set――有限集合flow――流following activity ――紧后工序forest――森林forward algorithm――前向算法free variable ――自由变量function iterative method――函数迭代法functional basic equation――基本函数方程function――函数fundamental circuit――基本回路fundamental cut-set――基本割集fundamental system of cut-sets――基本割集系统fundamental system of cut-sets――基本回路系统Ggame phenomenon――对策现象game theory――对策论game――对策generator――生成元geometric distribution――几何分布goal programming――目标规划graph theory――图论graph――图HHamilton circuit――哈密顿回路Hamilton graph――哈密顿图Hamilton path――哈密顿通路Hasse diagram――哈斯图hitchock method ――表上作业法hybrid method――混合法Iideal point――理想点idle period――闲期implicit enumeration method――隐枚举法in equilibrium――平衡incidence matrix――关联矩阵incident――关联indegree――入度indifference curve――无差异曲线indifference surface――无差异曲面induced subgraph――导出子图infinite set――无限集合initial basic feasible solution ――初始基本可行解initial basis ――初始基input process――输入过程Integer programming ――整数规划inventory policy―v存贮策略inventory problem―v货物存储问题inverse order method――逆序解法inverse transition method――逆转换法isolated vertex――孤立点isomorphism――同构Kkernel――核knapsack problem ――背包问题Llabeling method ――标号法latest finish time――最迟必须完工时间leaf――树叶least core――最小核心least element――最小元least spanning tree――最小生成树leaving basic variable ――出基变量lexicographic order――字典序lexicographic rule――字典序lexicographically positive――按字典序正linear multiobjective programming――线性多目标规划Linear Programming Model――线性规划模型Linear Programming――线性规划local noninferior solution――局部非劣解loop method――闭回路loop――圈loop――自环（环）loss system――损失制Mmarginal rate of substitution――边际替代率Marquart decision process――马尔可夫决策过程matching problem――匹配问题matching――匹配mathematical programming――数学规划matrix form ――矩阵形式matrix game――矩阵对策maximum element――最大元maximum flow――最大流maximum matching――最大匹配middle square method――平方取中法minimal regret value method――最小后悔值法minimum-cost flow――最小费用流mixed expansion――混合扩充mixed integer programming ――混合整数规划mixed Integer programming――混合整数规划mixed Integer ――混合整数规划mixed situation――混合局势mixed strategy set――混合策略集mixed strategy――混合策略mixed system――混合制most likely estimate――最可能时间multigraph――多重图multiobjective programming――多目标规划multiobjective simplex algorithm――多目标单纯形算法multiple optimal solutions ――多个最优解multistage decision problem――多阶段决策问题multistep decision process――多阶段决策过程Nn- person cooperative game ――n人合作对策n- person noncooperative game――n人非合作对策n probability distribution of customer arrive――顾客到达的n 概率分布natural state――自然状态nature state probability――自然状态概率negative deviational variables――负偏差变量negative exponential distribution――负指数分布network――网络newsboy problem――报童问题no solutions ――无解node――节点non-aligned game――不结盟对策nonbasic variable ――非基变量nondegenerate basic feasible solution――非退化基本可行解nondominated solution――非优超解noninferior set――非劣集noninferior solution――非劣解nonnegative constrains ――非负约束non-zero-sum game――非零和对策normal distribution――正态分布northwest corner method ――西北角法n-person game――多人对策nucleolus――核仁null graph――零图Oobjective function ――目标函数objective( indicator) function――指标函数one estimate approach――三时估计法operational index――运行指标operation――运算optimal basis ――最优基optimal criterion ――最优准则optimal solution ――最优解optimal strategy――最优策略optimal value function――最优值函数optimistic coefficient method――乐观系数法optimistic estimate――最乐观时间optimistic method――乐观法optimum binary tree――最优二元树optimum service rate――最优服务率optional plan――可供选择的方案order method――顺序解法ordered forest――有序森林ordered tree――有序树outdegree――出度outweigh――胜过Ppacking problem ――装箱问题parallel job――平行作业partition problem――分解问题partition――划分path――路path――通路pay-off function――支付函数payoff matrix――支付矩阵payoff――支付pendant edge――悬挂边pendant vertex――悬挂点pessimistic estimate――最悲观时间pessimistic method――悲观法pivot number ――主元plan branch――方案分支plane graph――平面图plant location problem――工厂选址问题player――局中人Poisson distribution――泊松分布Poisson process――泊松流policy――策略polynomial algorithm――多项式算法positive deviational variables――正偏差变量posterior――后验分析potential method ――位势法preceding activity ――紧前工序prediction posterior analysis――预验分析prefix code――前级码price coefficient vector ――价格系数向量primal problem――原问题principal of duality ――对偶原理principle of optimality――最优性原理prior analysis――先验分析prisoner’s dilemma――囚徒困境probability branch――概率分支production scheduling problem――生产计划program evaluation and review technique――计划评审技术(PERT) proof――证明proper noninferior solution――真非劣解pseudo-random number――伪随机数pure integer programming ――纯整数规划pure strategy――纯策略Qqueue discipline――排队规则queue length――排队长queuing theory――排队论Rrandom number――随机数random strategy――随机策略reachability matrix――可达矩阵reachability――可达性regular graph――正则图regular point――正则点regular solution――正则解regular tree――正则树relation――关系replenish――补充resource vector ――资源向量revised simplex method――修正单纯型法risk type decision――风险型决策rooted tree――根树root――树根Ssaddle point――鞍点saturated arc ――饱和弧scheduling (sequencing) problem――排序问题screening method――舍取法sensitivity analysis ――灵敏度分析server――服务台set of admissible decisions(policies) ――允许决策集合set of admissible states――允许状态集合set theory――集合论set――集合shadow price ――影子价格shortest path problem――最短路线问题shortest path――最短路径simple circuit――简单回路simple graph――简单图simple path――简单通路Simplex method of goal programming――目标规划单纯形法Simplex method ――单纯形法Simplex tableau――单纯形表single slack time ――单时差situation――局势situation――局势slack variable ――松弛变量sojourn time――逗留时间spanning graph――支撑子图spanning tree――支撑树spanning tree――生成树stable set――稳定集stage indicator――阶段指标stage variable――阶段变量stage――阶段standard form――标准型state fork――状态结点state of system――系统状态state transition equation――状态转移方程state transition――状态转移state variable――状态变量state――状态static game――静态对策station equilibrium state――统计平衡状态stationary input――平稳输入steady state――稳态stochastic decision process――随机性决策过程stochastic inventory method――随机贮存模型stochastic simulation――随机模拟strategic equivalence――策略等价strategic variable, decision variable ――决策变量strategy (policy) ――策略strategy set――策略集strong duality property ――强对偶性strong ε-core――强ε-核心strongly connected component――强连通分支strongly connected graph――强连通图structure variable ――结构变量subgraph――子图sub-policy――子策略subset――子集subtree――子树surplus variable ――剩余变量surrogate worth trade-off method――代替价值交换法symmetry property ――对称性system reliability problem――系统可靠性问题Tteam length――队长tear cycle method――破圈法technique coefficient vector ――技术系数矩阵test number of cell ――空格检验数the branch-and-bound technique ――分支定界法the fixed-charge problem ――固定费用问题three estimate approach一―时估计法total slack time――总时差traffic intensity――服务强度transportation problem ――运输问题traveling salesman problem――旅行售货员问题tree――树trivial graph――平凡图two person finite zero-sum game二人有限零和对策two-person game――二人对策two-phase simplex method ――两阶段单纯形法Uunbalanced transportation problem ――产销不平衡运输问题unbounded ――无界undirected graph――无向图uniform distribution――均匀分布unilaterally connected component――单向连通分支unilaterally connected graph――单向连通图union of sets――并集utility function――效用函数Vvertex――顶点voting game――投票对策Wwaiting system――等待制waiting time――等待时间weak duality property ――弱对偶性weak noninferior set――弱非劣集weak noninferior solution――弱非劣解weakly connected component――弱连通分支weakly connected graph――弱连通图weighed graph ――赋权图weighted graph――带权图weighting method――加权法win expectation――收益期望值Zzero flow――零流zero-sum game――零和对策zero-sum two person infinite game――二人无限零和对策。

国际自动化与计算杂志.英文版.

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Kannan14.New Delay-dependent Global Asymptotic Stability Condition for Hopfield Neural Networks with Time-varying DelaysGuang-Deng Zong，Jia Liu hHTTp://15.Crosscumulants Based Approaches for the Structure Identification of Volterra ModelsHouda Mathlouthi，Kamel Abederrahim，Faouzi Msahli，Gerard Favier1.Coalition Formation in Weighted Simple-majority Games under Proportional Payoff Allocation RulesZhi-Gang Cao，Xiao-Guang Yang2.Stability Analysis for Recurrent Neural Networks with Time-varying DelayYuan-Yuan Wu，Yu-Qiang Wu3.A New Type of Solution Method for the Generalized Linear Complementarity Problem over a Polyhedral ConeHong-Chun Sun，Yan-Liang Dong4.An Improved Control Algorithm for High-order Nonlinear Systems with Unmodelled DynamicsNa Duan，Fu-Nian Hu，Xin Yu5.Controller Design of High Order Nonholonomic System with Nonlinear DriftsXiu-Yun Zheng，Yu-Qiang Wu6.Directional Filter for SAR Images Based on NonsubsampledContourlet Transform and Immune Clonal SelectionXiao-Hui Yang，Li-Cheng Jiao，Deng-Feng Li7.Text Extraction and Enhancement of Binary Images Using Cellular AutomataG. 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自动验证技术获得图灵奖

Computer Education特别报道自动验证技术获得图灵奖刘瑞挺/文三人分享图灵奖正当我们忙于抗震救灾和准备奥运的时候，2008年6月21日晚，美国计算机协会(ACM)在旧金山召开了2007年度ACM颁奖盛典。

颁布了2007年度图灵奖、ACM Infosys基金会奖以及人工智能、软件系统、计算机理论与实践、计算科学与工程等领域的多个奖项。

2007年度ACM图灵奖由三位研究人员分享，分别是卡内基·梅隆大学的Edmund M. Clarke教授、德克萨斯大学奥斯汀分校的 E. Allen Emerson教授和法国Grenoble大学Verimag实验室的Joseph Sifakis教授，奖金是由英特尔公司和谷歌公司共同提供的25万美元，以表彰他们“在将模型检测发展为硬件和软件业中广泛采用的高效验证技术上的贡献”。

自动验证用途广今天，软件、硬件、网络的规模越来越大，例如一个芯片上就有上亿个晶体管电路，一个嵌入式系统虽小，也五脏俱全。

如果设计出了问题，一旦投产，损失就相当大。

如何在设计时就能证明系统正确，或者如何能及时发现设计中的错误呢?目前，一种成效卓著的自动验证技术(Automatic V erification Technology)已经在硬件和软件工业界得到了广泛的应用，特别是在半导体芯片的设计与生产中得到成功的应用。

正如英特尔研究中心副总裁钱安达(AndrewChien)在评价模型检测技术时所说：“英特尔和整个计算机工业都从他们的贡献中直接获益”。

谷歌的高级工程副总裁Alan Eustace也说：“谷歌像其他同时代的技术公司一样，很大一部分成功都来自于先驱们的研究贡献”。

这些祝贺与评价充满了真诚的感激之情。

模型检测是核心所谓模型检测技术(ModelChecking)，本质上是用严密的数学方法来验证一个设计是否满足预先设定的需求，从而自动地发现设计中的错误。

按照定义，它是一种检查某一给定模型是否满足某一逻辑规则的方法。

两类布尔函数的OBDD阶

i =1 j =1
命题 1 给定 DNF 型布尔函数 f = 的 OBDD 阶为 nm + 2 。
∑∏ xij ( n, m ∈ N + ) ，在变量序 x11 x1m xn1 xnm 下
i =1 j =1
n
n
证明：采用数学归纳法。基础步取 m m ∈ N + 为定值，当 i = 1 时，布尔函数 = f x11 ⋅ x12 x1m 。则 f 在序 x11 x12 x1m 下
x1 x2 xn y1 y2 yn ，f 的 OBDD 阶是 2n +1 。
通过以上示例分析，变量序对 OBDD 的影响是显著的，一个好的变量序可以使布尔函数对应的 OBDD 更紧凑，从而减少空间的占用。而如果不幸选择了坏序，则 OBDD 的阶可能随变量数增多呈指数增长。因此选择一个合适的变量序来构造布尔函数的 OBDD 显得尤为重要。
关键词
OBDD，变量序，DNF，CNF，布尔函数
文章引用: 张媛, 江建国, 胡晓璐. 两类布尔函数的 OBDD 阶[J]. 软件工程与应用, 2018, 7(6): 283-288. DOI: 10.12677/sea.2018.76032
张媛等
Copyright © 2018 by authors and Hans Publishers Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). /licenses/by/4.0/
The OBDD Orders of Two Types of Boolean Functions

02---故障树向二元决策图的转换算法

第44卷第3期原子能科学技术Vo l.44,N o.3 2010年3月Atomic Ener gy Science and T echno logy M ar.2010故障树向二元决策图的转换算法胡文军(中国原子能科学研究院中国实验快堆工程部,北京 102413)摘要:L iv ing PSA (Living pr obability safety analysis)技术已引起核电业主和核安全当局的广泛重视,该技术要求能快速准确地反映核电厂实际运行状态。

因此,基于对新型的可靠性分析算法二元决策图(BD D)的研究和概率安全分析程序NF Risk 的编写经验,介绍了二元决策图的基本概念,故障树向二元决策图转换时的基本事件的排序算法,以及将故障树转换成顶事件的二元决策图的算法。

这些算法均已应用在N FRisk 程序中,并被证明具有良好的计算性能。

关键词:L iv ing PSA ;NF Risk 程序;二元决策图;转换;排序中图分类号:T L 364.5 文献标志码:A 文章编号:1000-6931(2010)03-0289-05Strategy for Fault Tree Conversion to Binary Decision DiagramH U Wen -jun收稿日期:2009-03-01;修回日期:2009-05-08作者简介:胡文军(1978 ),男,湖北荆门人,助理研究员,反应堆工程专业(China I nstitute of A tomic Ener gy ,P.O.Box 275-34,Beij ing 102413,China)Abstract:Living probability safety analysis (Liv ing PSA )technolo gy,w hich is used to get accurate state of nuclear pow er plant,is attractive for the nuclear pow er ow ner and nuclear safety authority.On the basis of the binary decision diag ram (BDD)study and the ex perience in the dev elo pment of NFRisk code,the basic concept of BDD,the so r -ting strateg y o f basic event and the strateg y in fault tree conversion to BDD w ere intro -duced.T hese strateg ies have already used in the platfo rm NFRisk,and the performance is ex cellent.Key words:Liv ing pro bability safety analysis;NFRisk co de;binary decision diagr am ;conversio n;so rting概率安全/风险评价PSA/PRA (probab-i listic safety/risk assessment)自20世纪70年代首次在大型核电站的事故风险分析中使用以来,已引起全世界的普遍关注,并得到广泛应用[1]。

逻辑综合中的大体概念

1. 逻辑综合(Logic Synthesis)EDA工具把数字电路的功能描述(或结构描述)转化为电路的结构描述。

实现上述转换的同时要满足用户给定的约束条件，即速度、功耗、成本等方面的要求。

2. 逻辑电路(Logic Circuit)逻辑电路又称数字电路，在没有特别说明的情况下指的是二值逻辑电路。

其电平在某个阈值之上时看作高电平，在该阈值之下时看作低电平。

通常把高电平看作逻辑值1；把低电平看作逻辑值0。

3. 约束(restriction)设计者给EDA工具提出的附加条件，对逻辑综合而言，约束条件一般包括速度、功耗、成本等方面的要求。

4. 真值表(Truth Table)布尔函数的表格描述形式，描述输入变量每一种组合情况下函数的取值。

输入变量组合以最小项形式表示，函数的取值为真或假(1 或0)。

5. 卡诺图(Karnaugh Map)布尔函数的图形描述形式，图中最小方格和最小项对应，两个相邻的最小方格所对应的最小项只有一个变量的取值不同。

卡诺图适合于用观察法化简布尔函数，但是当变量的个数大于4时，卡诺图的绘制和观察都变得很困难。

6. 单输出函数(Single-output Function)一个布尔函数的单独描述。

7. 多输出函数(Multiple-output Function)输入变量相同的多个布尔函数的统一描述。

8. 最小项(Minterm)设a1,a2,…a i,…a n是n个布尔变量，p为n个因子的乘积。

若是在p中每一变量都以原变量a i或反变量的形式作为因子出现一次且仅出现一次，则称p 为n个变量的一个最小项。

最小项在卡诺图中对应于最小的方格；在立方体表示中对应于极点。

9. 蕴涵项(Implicant)布尔函数f的"与-或"表达式中的每一乘积项都叫作f的蕴涵项。

例如：f=+中的乘积项和都是函数f的蕴涵项。

蕴涵项对应于立方体表示法中的立方体。

10.质蕴涵项(Prime Implicant，PI)设函数f有多个蕴涵项，若某个蕴涵项i所包含的最小项集合不是任何别的蕴涵项所包含的最小项集合的子集的话，则称i为函数f的质蕴涵项。

3_BDD

Introduction to BinaryDecision DiagramProf. Chien-Nan LiuTEL: 03-4227151 ext:4534Email: jimmy@.tw1Outlines•Representing Boolean Functions–Decision graph structure–Reduction to canonical form–Effect of variable ordering–Variants to reduce storage•Algorithms–General framework–Basic operations»Restriction (Cofactor)»If-Then-Else–Derived operations–Computing functional properties3Decision Structures−Vertex represents decision −Follow dashed line for value 0−Follow solid line for value 1−Function value determined by leaf valueX1X2X3f 00000010010001111000101111001111X1X2X2X3X3X3X30110Truth Table1Decision TreeBinary Decision Diagram (BDD)f = x 1x 2+x 3terminal node :•attribute–value(v) = 0–value(v) = 1nonterminal node :•index(v) = i •two children–low(v)–high(v)X1X2X2X3X3X3X31010111: 05BDDA BDD graph which has a vertex v as root corresponds to the function F v :(1) If v is a terminal node :a) if value(v) is 1, then F v = 1b) if value(v) is 0, then F v = 0(2) If F is a nonterminal node (with index(v) = i)F v (x 1, …, x n ) = x i ’F low(v)(x i+1, …, x n ) +x i F high(v)(x i+1, …, x n )Variable Ordering•Assign arbitrary total ordering to variablee.g. X1 < X2 < X3•Variable must appear in ascending order along all paths•Properties–No conflicting variable assignments along path –Simplifies manipulationX1X2X3X1X3X3X2X1X1X1OKNot OK7•Merge equivalent leavesX1X2X2X3X3X3X3010101X1X2X2X3X3X3X301aaaReduction Rule #2•Merge isomorphic nodesX1X2X2X3X3X3X31X1X2X2X3X301X X YZXYZ9•Eliminate Redundant TestsX1X2X2X3X301X1X2X301X YYExample ROBDD•Canonical representation of Boolean function for given variable ordering–Two functions equivalent iff graphs isomorphic»can be tested in linear time–Desirable property : The simplest form is canonicalX1X2X2X3X3X3X30111X1X2X301Initial GraphReduced Graph11Reduce•Visit OBDD bottom up and label each vertex with an identifier •Redundancy–if id( low(v ) ) = id( high(v ) ), then vertex v is redundant ⇒set id(v ) = id( low(v ) )–if id( low(v ) ) = id( low(u ) ) and id( high(v ) ) = id( high(u ) ), then set id(v ) = id(u )• A different identifier is given to each vertex at level i •Terminated when root is reached•An ROBDD is identified by a subset of vertices with different identifiersReducea bbcc101index=1index=2index=3123450101id =1id =1id =2id =1id =2id =3id =3id =4id =3id =51241id =1id =2id =3id =4id =5: 0(a)(b)(c)Construct ROBDD Directly•Using a hash table called unique table–Contain a key for each vertex of an OBDD–Key : (variable, right children, left children)–Constructed bottom up–Each key uniquely identify the specific function–Look up the table can determine if another vertex in thetable implements the same function13The Unique Table•Represent an ROBDD• A strong canonical form•Check equivalence of two Boolean functions by comparing the corresponding identifiers•Can represent multiple-output functions15Multi-Rooted ROBDDabc1id =1id =2id =3id =4id =5f = (a+b) cvariable order (a, b, c)Unique tableIdentifier Variable Right child Left child5 a 3 4 4 b 3 1 3 c 2 1KeyMulti-Rooted ROBDDa bc1id =1id =2id =3id =4id =5did =6f = (a+b) cg = b c df is constructed first and is associated with id =5g : id =6Unique tableIdentifier Variable Right child Left child6 d 4 1 5 a 3 4 4 b 3 1 3 c 2 1Key17The Unique Table•Unique table : hash table mapping (Xi, G, H) into a node in the DAG–before adding a node to the DAG, check to see if it already exists –avoids creating two nodes with the same function–strong canonical form : pointer equality determines function equalityHash Table Mapping (X1, T1, 1 ) --> U1(X1, T2, T1) --> U2(X2, T3, 1 ) --> T1(X2, 0 , T3) --> T2(X3, 0 , 1 ) --> T3X1X2X3U 2X211X11U 1T 1T 2T 3truefalseNon-Shared ROBDDU1 = X1’+ X2’+ X3’U2 = X1’X2’+ X1’X3’X1X2X3U 11110X1X2X3U 2X21119Multi-Rooted (Shared) ROBDD• A DAG node F is represented by a tuple (Xi, G, H)–Xi is called the top variable of F–node (Xi, G, H) represents the function ite(Xi, G, H) = XiG + Xi’H•DAG contains both external and internal functionsX1X2X3U 2X211X11U 1T 1T 2T 3U 1= X 1’+X 2’+X 3’=(X 1, T 1, 1)U 2= X 1’X 2’+X 1’X 3’=(X 1, T 2, T 1)T 1= X 2’+X 3’=(X 2, T 3, 1)T 2= X 2’X 3’=(X 2, 0, T 3)T 3= X 3’=(X 3, 0, 1)0 = (X ∞, 0, 0)1 = (X ∞, 1, 1)External functions User functionsInternal functionsSeparated vs. Shared•Separated–51 nodes for 4-bit adder–12481 nodes for 64-bit adder –Quadratic growth•Shared–31 nodes for 4-bit adder –571 nodes for 64-bit adder –Linear growth21Maintaining Shared ROBDD•Storage Model–Single, multiple-rooted DAG–Function represented by pointer to node in DAG –Maintain Unique (hash) table to keep canonical•Storage Management–User cannot know when storage for node can be freed –Must implement automatic garbage collection•Algorithmic Efficiency–Functions equivalent iff pointer equal»if (p1 == p2) ...–Can test in constant timeOrdering Effects•The size of ROBDD depends on the ordering of variablesex : x 1x 2+ x 3x 4x 1< x 2< x 3< x 4X1X3X21X423Ordering Effects (cont’d)x 1< x 3< x 2< x 4X1X3X31X2X4X2Sample Function Classes•General Experience–Many tasks have reasonable ROBDD representations –Algorithms remain practical for up to 100,000 vertex ROBDD–Heuristic ordering methods generally satisfactoryFunction Class Best WorstOrdering Sensitivity ALU (Add/Sub)Linear Exponential High Symmetric LinearQuadratic None MultipicationExponentialExponentialLow25Symbolic Manipulation•Strategy–Represent data as set of ROBDDs»with identical variable orderings–Express solution method as sequence of symbolic operations –Implement each operation by ROBDD manipulation •Algorithmic Properties–Arguments are ROBDDs with identical variable orderings –Result is ROBDD with same ordering –“Closure Property”•Two Basic Operations–Restriction –If-Then-ElseRestriction Operation•Concept–Effect of setting function argument Xi to constant K(0,1)–Also called Cofactor operation•Implementation–Depth-first traversal–Complexity near-linear in argument graph sizeFF[X i = K]KX i -1X i +1X 1X n….….27Restriction AlgorithmRestrict (F, x, k)Bypass any nodes for variable xChoose Hi child for k = 1Choose Lo child for k = 0Reduce resulte.g. Restrict variable b to 1a bb1c c a bb 01c c a1c c 01cFind nodesBypassReduceFFFFSpecial cases of Restriction•Case 1 : Restrict on root node variablex•Case 2 : Restrict on variable less than root node–e.g. x < yRestrict ( , x, 1 )xRestrict ( , x, 0 )yRestrict ( , x, 1 )y29If-Then-Else Operation•Concept–Basic technique for building ROBDD from network or formula•Argument I (if), T (then), E (else)–Functions over variables X –Represented as ROBDDs •Result–ROBDD representing composite function –IT +I’E •Implementation–combination of depth-first traversal and dynamic programming–Worst case complexity : product of argument graph sizesIf-Then-Else Algorithm•Recursive FormulationITE (I, T, E) = x ITE(I[x=1], T[x=1], E[x=1]) +x’ITE(I[x=0], T[x=0], E[x=0])•General Algorithm–Select top root variable x of I, T and E –Compute restrictions»Guaranteed to be one of special cases –Apply recursively to get results Lo and Hi –Still remain canonical form•Termination Conditions–I = 1 ==> Return T –I = 0 ==> Return E –T = 1, E = 0 ==> Return I31An ITE Example•Given f = ab + bc + ac, g = c under the order a < b < c ITE (f, g, 0)= ITE[ a , ITE( f (a=1), g (a=1), 0), ITE( f (a=0), g (a=0), 0)]= ITE[ a , ITE( b+bc+c, c, 0), ITE( bc, c, 0)]= ITE{ a , ITE[ b , ITE(1, c, 0), ITE(c, c, 0)],ITE[ b , ITE(c, c, 0), ITE(0, c, 0)]}= ITE[ a , ITE(b, c, c), ITE(b, c, 0)]= ITE[ a , c, ITE(b, c, 0)]ab 01c sel1T1E1=sel2T2sel1sel2T1T2E1E2b=1b=0Algorithmic Issues & Derived Operations•Efficiency–Maintain computed table and unique table to increaseefficiency–Worst case complexity product of graph sizes for I, T, E•Derived operations–Express as combination of If-Then-Else and Restrict –Preserve closure property»Result is a ROBDD with the same variable ordering33Detailed ITE AlgorithmITE(f, g, h) {if (terminal case)return (r = trivial result) ;else { /* exploit previous information */if (computed table has entry {(f, g, h), r} )return (r from computed table) ;else {x = top variable of f, g, h ;t = ITE(f x , g x , h x ) ;e = ITE(f x’, g x’, h x’) ;if (t == e) /* children with isomorphic OBDDs */return (t) ;r = find_or_add_unique_table(x, t, e) ; /* add r to unique table if not present */Update computed table with {(f, g, h), r} ;return (r) ;}}}Derived Algebraic Operations•Other common operations can be expressed in terms of If-Then-ElseMUXG F 0XMUXF 1X110F GXF GXAND (F, G)If-Then-Else (F, G, 0)If-Then-Else (F, 1, G)OR (F, G)35ITE OperatorsOperatorEquivalent ite form00f ‧g ite ( f, g, 0 )f ‧g ’ite ( f, g ’, 0 )f ff ’‧g ite ( f, 0, g )g gf ⊕g ite ( f, g ’, g )f + g ite ( f, 1, g )( f + g )’ite ( f, 0, g ’ )( f ⊕ g )’ite ( f, g, g ’ )g ’ite ( g, 0, 1 )f + g ’ite ( f, 1, g ’ )f ’ite ( f, 0, 1 )f ’ + g ite ( f, g, 1 )( f ‧g )’ite ( f, g ’, 1 )11Generating ROBDD from Network•Task : Represent output functions of gate network as ROBDDsa b c bac b cbb a A B C T 1T 2Out A new_var(“a”)B new_var(“b”)C new_var(“c”)T 1AND(A, B)T 2AND(B, C)Out OR(T 1, T 2)abcT1T2OutEvaluationNetwork37Functional Composition•Create new function by composing functions F and G •Useful for composing hierarchical modulesFF[X i = G]X i -1X i +1X 1X n….….GX nX 1…FX i -1X i +1X 1X n……FX i -1X i +1X 1X n……1MUX1GX nX 1…Variable Qualification•Eliminate dependency on some argument through qualificationF∃Xi FX i -1X i +1X 1X n….….∃FX i -1X i +1X 1X n ……FX i -1X i +1X 1X n……139Variants & Optimizations•Concept–Refinements to ROBDD representation –Do not change fundamental properties•Objective–Reduce memory requirement –Improve algorithmic efficiency–Make commonly performed operations faster•Common Optimizations–Share nodes among multiple functions –Negated arcsNegation Arcs•Concept–Dot on arc represents complement operator»Invert function value–Can appear internal or external arc01b a a+ba+b 01b aa+b41Effect of Negation Arcs•StorageSavings–At most 2X reduction in numbers of nodes•Algorithmic Improvement–Can complement function in constant time•Problem–Negation arc allow multiple representations of a function–Modify algorithms with restricted conversions for use of negative arcs1b a a + b 01baabDensity Computation•Definition–p(F) : fraction of variable assignments for which F = 1•Applications–Testability measures –Probability computations•Recursive Formulation–p(F) = [ p( F[x=1] ) + p( F[x=0] ) ] / 2•Computation–Compute bottom-up, starting at leaves –At each node, average density of childrend2d1d1a1a1d0d01a1a0a07/321/43/161/41/81/41/41/41/21/2Let E be a set and A ⊆EThe characteristic function of A is the function X A: E -> { 0, 1 }X A(x) = 1 if x ∈AX A(x) = 0 if x ∉AEx :E = { 1, 2, 3, 4 }A = { 1, 2 }X A(1) = 1X A(3) = 043Characteristic FunctionGiven a Boolean functionf : B n-> B mthe mapping relation denoted as F ⊆B n×B m isdefined asF(x, y) = { (x, y) ∈B n×B m| y = f(x) }The characteristic function of a function f is definedfor (x, y) s.t. X f(x, y) = 1 iff(x, y) ∈F45Ex : y = f(x1, x2) = x1 + x2x1 x2 y 0 0 00 1 11 0 11 1 1Fy(x1, x2, y) =x1 x2 y F 0 0 0 10 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1Summary•ROBDD–Reduced graph representation of Boolean Function –Canonical for given variable ordering –Size sensitive to variable ordering•Algorithmic Principles–Operations maintain closure property»Result ROBDD with same ordering as arguments »Can perform further operations on results –Limited set of basic operations to implement»Restrict, If-Then-Else»Other operations defined in terms of basic operations。

计算机系统形式化验证中的模型检测方法综述

计算机系统形式化验证中的模型检测方法综述形式化方法是用数学和规律的方法来描述和验证系统设计是否满意需求。

它将系统属性和系统行为定义在抽象层次上，以形式化的标准语言去描述系统。

形式化的描述语言有多种，如一阶规律，Z 语言，时序规律等。

采纳形式化方法可以有效提高系统的平安性、全都性和正确性，关心分析冗杂系统并且及早觉察错误。

形式化验证是保证系统正确性的重要方法，主要包括以数学、规律推理为根底的演绎验证(deductive verification)和以穷举状态为根底的模型检测(model checking)。

演绎验证是基于人工数学来证明系统模型的正确性。

它利用规律公式来描述系统，通过定理或证明规章来证明系统的某些性质。

演绎验证既可以处理有限状态系统，又可以解决无限状态问题。

但是演绎验证的过程一般为定理证明器帮助，人工参加，无法做到完全自动化，推导过程冗杂，工作量大，效率低，不能适用于大型的冗杂系统，因此适用范围较窄。

常见的演绎验证工具有HOL，ACL2，PVS和TLV等。

模型检测主要应用于验证并发的状态转换系统，通过遍历系统的状态空间，对有限状态系统进展全自动验证，快速高效地验证出系统是否满意其设计期望。

下面将主要介绍模型检测方法的进展历史和讨论现状，以及当前面临的挑战和将来进展方向等问题。

2 模型检测及相关技术模型检测方法最初由Clarke，Emerson等人于1981年提出，因其自动化高效等特点，在过去的几十年里被广泛用于实时系统、概率系统和量子等多个领域。

模型检测根本要素有系统模型和系统需满意的属性，其中属性被描述成时态规律公式Φ。

检测系统模型是否满意时态规律公式Φ，假设满意那么返回“是”，不满意那么返回“否”及其错误路径或反例。

时态规律主要有线性时态规律LTL(Linear TemporalLogic)和计算树规律CTL(Computation Tree Logic)。

2.1 线性时态规律对一个系统进展检测，重要的是对系统状态正确性要求的形式化，其中一个根本维度是时间，同时需要知道检验结果与时间维度的关系。

数字IC的设计流程及验证方法介绍

数字IC的设计流程，如下图所示：形式验证（Formal VerificaTIon）是一种IC设计的验证方法，它的主要思想是通过使用数学证明的方式来验证一个设计的功能是否正确。

形式验证可以分为三大类：等价性检查（Equivalence Checking）形式模型检查（Formal Model Checking）（也被称作特性检查）定理证明（Theory Prover）为什么要做形式验证？电路不也是工具综合出来的吗？为什么不能保证一致性？因为工具也是人做出来的，也有可能会出错，所以要确认。

我们平时做的最多的模拟仿真，就是给各种case的输入，穷尽各种组合，总是希望100%的验证到所有的情况。

但是有些情况下，你不太可能达到这一个目的。

假如有一个32位的比较器：比较产生等于、大于、大于的结果。

假设采用一个快速模拟器，每微秒运行一个向量，则用模拟器模拟完全部模拟向量需要的时间为：264 （all input patterns）X 10^-6—————————————————3600 （seconds）X 24 （hours）X 365 （days）≈584，942 years显然这是一个不切实际的验证时间。

而形式验证使用严格的数学推理来证明待测试设计的正确性，由于其静态、数学的特性，避免了对所有可能测试向量的枚举，而且能够达到100%无死角的检测。

定理证明是形式验证技术中最高大上的，它需要设计行为的形式化描述，通过严格的数学证明，比较HDL描述的设计和系统的形式化描述在所有可能输入下是否一致。

这种验证方法需要非常深厚的数学功底，而且不能完全自动化，所以应用案例较少。

当然还是有一些例子，例如HOL系统、PVS系统和ACL2系统等，并且都有成功应用案例。

Moore等人验证了AMD5K86芯片的除法算法的微码，Brock等验证了Motorola的CAP处理器，Clark等验证了SRT除法算法。

模型检验是一种检测设计是否具有所需属性的方法，如安全性、活性和公平性。

2007年图灵奖获得者

2007年图灵奖获得者：Edmund M. Clarke，Allen Emerson和Joseph Sifakis美国计算机协会2008年2月4日宣布了2007年图灵奖获得者：Edmund M. Clarke(艾德蒙德·克拉克)，Allen Emerson(艾伦·埃莫森)和Joseph Sifakis(约瑟夫·西法基斯)三位科学家，表彰他们开发模型检测技术，并使之成为一个广泛应用在硬件和软件工业中非常有效的算法验证技术所做的奠基性贡献。

背景知识：模型检查及其历史模型检测（Model-Checking，也译为模型检验，仿真术语里称为模型校验）是一类“验证”，分析设计背后的逻辑，就像数学家用证明来判断一个定理是否正确。

其本质上是用严密的数学方法来验证设计是否满足预设的需求，从而自动化地发现设计中的错误。

按Wikipedia的定义，它是一种检查某一给定模型是否满足某一逻辑规则的方法。

其中一种重要的方法，就是通过算法来验证形式化系统，具体方法是验证由硬件或者软件设计导出的模型是否满足通常用模态逻辑规则表示的形式化规范。

在硬件业，包括半导体业和嵌入式系统中，模型检查已经成为一项非常关键的主流技术。

要知道，在硬件行业，如果设计有问题，一旦投产，损失就太大了。

正因为这样，图灵奖赞助方之一Intel对三位获奖者的祝贺可以说是充满了感激之情。

此外，在通信协议、安全算法的设计方面，模型检查也发挥了关键作用。

但是，软件业对模型检查的重视似乎很不够。

一线的软件开发人员可能都对它比较陌生，感觉比较学院化。

当然，由于存在可计算性导致的缺陷，以及软件本身的复杂性，模型检查是不可能完全解决软件设计中的bug的。

但是，软件业对这种方法的忽视，是否也是软件总体质量不如硬件，或者说低级错误更多的一个原因呢？总之，模型检查在工业检测方面有诸多应用：如芯片检测、通信协议、外部设备主控软件、嵌入式系统（如在飞机、火车、火箭、卫星或移动电话）以及安全算法等。

材料科学专业英语词汇（B）

材料科学专业英语词汇(B)b-h curveb-h 曲线(同磁滞曲线)b.n.f. jet testb.n.f 喷射试验(测量电镀层厚度) babbitt metal1 巴比合金2巴氏合金back and design 後端设计back annotation 背面注解back rack 背後接线架back roll 背压轧辊back sand 背砂back side damage 背侧损伤back side reference method 背侧基准法back spattering 反电泼(离子蚀刻)back stamp 背款back surface luster 背面光泽度back-reflection focusing came 背面反射聚焦照像机back-reflection pinhole camera 背面反射针孔照像机back-reflection x-ray exami 背面反射x 光检验backing 背里backing pad 衬垫backing plate 支撑板backlash 齿隙backside rinse 背面冲洗backspill 逆溢料backstop tongs 後挡钳backstreaming 回流backwall 後墙backward extrusion 逆向挤制backward spinning 逆向旋弯成型backward welding 後退熔接bacterial corrosion 细菌腐蚀bacterial oxidation 细菌氧化baddeleyite 二氧化锆矿badging 落款；标记baffle 挡板baffle mark 挡板痕bag filter 袋滤器bag wall 盾墙bagging cement 袋装水泥bahnmetal 道金属(含0.7%ca,0/6%na,0.2%al 0.05%si, li 或ni 之铅基轴承合金)bainite chin 弯勒铁颚baked permeability 烘乾透气性baked strength 烘乾强度bakelite 电木baking core 烘乾砂心baking furnace 烘焙炉baking temperature uniformity 烘烤温度均质性baking unit oven 烘烤炉balance 天平；秤；均衡balanced blast cupola 均衡鼓风化铁炉balanced steel 平衡钢(一种半静钢)balbach electrolytic process 巴巴赫电解法balbach-thum cell 巴森电解槽ball bearing 滚珠轴承ball bearing stee 滚珠轴承钢ball bonding 球形接合，球形压接ball clay 球[状][黏]土ball grid array 球状栅极阵列封装体ball mill1 粉机2球磨机ball shape 球形，球状ball shear strength 球部抗切强度ball shear tester 球状压接端切变强度测试机ball size 球头尺寸，球形大小ballast 道碴balling 球状化bamboo leaves pattern 竹叶状(铁水)花纹band blade 条带刀片band saw 带锯band steel 钢带band theory 能带论band-run gravel 河岸砾石banded structure 带状组织banding 描边；边圈bank controller 触排控制器bank kiln 坡窑bank sand 河砂banka tin 彭克锡(印尼彭克岛生产纯度99.7%之锡)banks 斜床bannering 修边banox process 斑诺克斯法(以磷酸处理钢线，使拉制时产生润滑作用)bar1 棒2杆3条bar drawing 条杆拉制bar iron 铁条bar mill 棒料轧机bar solder 软焊条barba's law 巴巴定律bare electrode 裸熔接条bare metal 裸金属barffing 发黑处理(形成黑色磁铁层之蒸汽处理)barite 重晶石barium (ba, 56)钡barium ferrite 钡铁氧磁体barium titanate 钛酸钡bark hausen effect 巴好生效应(磁化)bark hausen jumps 巴好生跳跃(磁化时之一种现象)barn (1barn=10-24 cm2)邦(原子核截面积单位)barr-bardgett creep test 巴尔-巴德格潜变试验barrel asher 圆筒型灰化机barrel finishing 打磨barrel type plasma etching system 圆筒型等离子体蚀刻系统barreling 装桶barret effect 巴瑞德效应(材料磁化体积膨胀现象)barrier 障壁barronia 巴隆尼尔(防锈黄铜，用於冷凝管,82-85%cu, 4%sn, 1%fe, 其余为zn)basal plane 基面basalt 玄武岩basalt ware 玄武陶base bullion 含银粗铅base coat 底涂base exchange 硷交换base line 基线base metal1 卑金属2母材base metal thermocouple 卑金属热电偶base permeability 原砂透气性base-centered monoclinic 底心单斜base-centered orthor hombic 底心斜方basic bessemer converter 硷性柏思麦转炉basic bessemer process 硷性柏思麦法basic bessemer steel 硷性柏思麦钢basic brick 硷性砖basic electric arc furnace 硷性电弧炉basic electric furnace 硷性电炉basic lining1 硷性衬里 2硷性炉衬basic material 硷性材料basic open hearth furnace 硷性平炉basic open-hearth furnace 硷性平炉basic oxygen converter 硷性氧气转炉basic oxygen process 硷性氧气炼钢法basic pig iron 硷性生铁basic refractory 硷性耐火材料basic slag 硷性渣basic steel 硷性钢basic water content 基本含水量basicity 硷度basin 槽，池，盆basis metal 心材(包层产品之中心金属) basket ware 蓝纹陶bat 垫饼；泥饼；半截砖batch 批[料]batch meter 批量计batch processing 分批处理batch wafer retrieval 晶圆片回批取出batch weights 批重batch [type] furnace 批式炉batch-to-batch dose uniformity 批次间之注入均质性batch-type annealing furnace 批式退火炉batch-type mixer 批拌机batch-type mixer 分批混合机batching equipment 配料设备batching plant 配料工厂bath 浴bath metal 脆铜(廉价装饰用具，其成份为55%cu, 其余为zn) bath stabilizer 镀浴安定剂bath to bath transport time 槽间输送时间bath with filter for circulation solution 循环过滤洗条槽battering tool 鎚击工具batterium alloy 含镍铝青铜(9%al, 1%ni, 其余为cu)batting out machine 平椪机baume hydrometer 波美比重计baume's scale 波美度(比重单位)bauschinger effect 鲍辛格效应bauxite 水矾土，铝矾土bauxite brick 矾土砖bauxite cement 铝矾土水泥bayer process 拜耳法bead1 熔珠；焊珠2珠bead weld 联珠泰接beaded rolls 起脊辊子beading 起脊beading enamel 逸缘釉beam 梁beam current 波束电流beam diameter 光束直径beam energy 波束能量beam filter 波束滤波器beam focusing system 波束聚焦系统beam positioner 光束定位器beam positioning accuracy 光束定位精确度beam stability 波束安定性beam, hh 型梁beam, ii 型梁beam, wide-flange 宽缘梁bearing bronze 轴承青铜bearing corrosion 轴承腐蚀bearing metals 轴承金属beck aluminium recovery process 贝克铝品收法bed 床面bed cok 底焦bedding 垫底bedding system 层集系统bedding, stratification 层理beehive coke 蜂巢炉焦炭beehive oven 蜂房炼焦炉behavioral description language 性能记述语言behavioral schematic editor 性能简图编辑器behavioral simulator 性能模拟器behavioral synthesis/behavioral synthesizer 性能合成/性能合成器beidellite 贝德石(从矿)beilby layer 比耳拜层(金属研磨样品表面形成之30-50%a 厚之非晶体层)belite 矽酸二钙bell and hoppe 钟斗形盖\(鼓风炉)bell bronze 钟青铜，响铜bell damper 钟形风门bell metal 钟用合金bell-type furnace 钟式炉belljar 钟罩型反应器bellows pump 风箱泵belly 炉腹belt conveyor 带运机belt drop hammer 皮带落鎚belt grinding[ 连续]带研磨belt hammer 皮带鎚belt kiln 带式窑beltless transfer system 无带式输送系统bench mold 台制砂模bench molde 台制砂模工bend test 弯曲试验bendability 弯曲性bender1 弯曲机2 弯曲模bender, free-flow 开放式弯曲模bender, trapped-stock 关闭式弯曲模bending 弯曲bending roll 卷板机bending, air 空中弯曲bending, compression 压缩弯曲bending, roll 辊筒弯曲bending, stretch 拉伸弯曲benedict metal 班耐迪克金属(10%ni, 20%zn, 10%pb, 2%sn，其余为铜)benefaction 初选(矿)beneficiation 提选bengough-stuart process 班哥-可徙尔法(轻金属在3%铬酸中之阳极处理法)bentonite1 膨土2土般土3火山黏土berkelium (bk, 97)(金北)berlin porcelain 柏林瓷bernoulli chuck 伯努利吸盘berrelius process 柏里法beryl 绿宝石beryllia 敛气(气化铍)berylliosis 铍毒效应beryllium (be, 4)铍beryllium bronze 铍青铜beryllium-copper alloy 铍铜合金bessemer cast steel 柏思麦铸钢bessemer converter 柏思麦转炉bessemer iron 柏思麦生铁bessemer ore 柏思麦铁矿bessemer pig iron 柏思麦生铁bessemer process 柏思麦法bessemer steel 柏思麦钢best fit plan reference 最妥适平面基准best fit plane reference 最妥适平面基准beta particleβ 粒子beta structureβ 组织beta treatmentβ 处理beta (β)brassβ 黄铜beta (β)graphiteβ 石墨beta (β)ironβ 铁betts electrolytic lead process 柏兹电解铅法betts electrolytic process 柏兹电解法bevel 斜角；截成斜角bevel brick 斜砖bevel cut 斜角切割bias 偏压bias sputtering system 偏压溅镀系统biaxial stress 双轴应力biaxiality 双轴应力比billet 小[纲]坯billet mill 小坯轧机bimetallic corrosion 双金属腐蚀bimetals 双金属bin 测试结果之分门类别bin-dicator 料面指示器binary alloy 二元合金binary decision diagram 双择判定图binary scale pattern recognition 二值标度图案识别binary scan 二进扫描binary search 二次搜索binary system 二元系binder 黏结剂binding material 结合剂binding, draw 拉延弯曲biological corrosion 生物腐蚀biotite 黑云母biquartz 双石英片birefringent 双折射birmabright 柏玛勃莱合金(含1-7%mg 之一组铝合金) birmetal 柏尔合金(含0-4.5%cu, 5-6%zn, 0.5-3%mg, 0.3%cr 之铝合金)birmingham platinum (或platima)伯明罕白金(20-45%cu, 其余为zn)biscuit 素烧坯biscuit firing 素烧biscuit ware 无釉器；素烧呣bismanol 铋化锰(一种磁性材料)bismuth (bi, 83)铋bismuth luster 铋光料bismuth telluride 石帝化铋(一种热电材料)bisque 素烧坯bit defect test 点缺陷试验bitstone 碎矽石bitter pattern 毕德氏磁区图bituminous cement 沥青胶泥bituminous coal 烟煤bituminous coating 沥青涂面black annealing 黑色退火black ash 黑灰black body 黑体black edging 黑边black heart malleable cast ir 黑心可锻铸铁，黑心展性铸铁black iron plate 黑铁板black lead 石墨black lignite 黑褐煤black oxide finish 黑色氧化处理black sand 旧砂black sheet iron 黑铁皮black skin 黑皮，毛坯面black speck 黑斑black wash 砂模黑浆blacking 碳质材料(如石墨等)blacking holes 黑洞(铸疵)blade 刀片blade breakage detection device 刀片破损检测装置blade deflection 刀片偏差blade deflection sensor 刀片挠曲敢测器blade exposure 刀刃曝露量blade height calibration 刀片高度校准blade retension 刀片再加张力blade tension 刀片张力blade tensioning frame 刀片张力框架blade wear compensation 刀片摩损补偿blaes 布拉土岩(从矿)blank 坯料blank carburizing 空白渗碳blank holder1 压料板2坯料夹blank nitriding 空白氮化blanket feed 毡式加料blanking 切坯blast 鼓风blast cleaning 喷砂处理blast furnace 鼓风炉，高炉blast furnace coke 鼓风炉焦碳，高炉焦碳blast furnace refractory 高炉(鼓风炉)用耐火材料blast furnace slag 鼓风炉渣blast furnace slag cement 鼓风炉渣水泥，高炉水泥blast main 送风主管blast pipe 送风管blast pressure 鼓风压力blast regulator 风量调节器blast roasting 鼓风焙烧blast volume meter 风量计blasting 喷击净面法bleaching 漂白bleaching clay 漂白黏土bleb 气泡bleeding 渗出bleeding of concrete 混凝土渗水(泌水) blended sand 调合砂blending 搀合；拌合bleu persan 波斯蓝底白花彩blind core 暗砂心blind feeder 暗补给口blind hole 盲孔blind riser 暗冒口blinding 堵塞blister1 气泡2起泡blister copper 泡铜(粗铜)blister steel 泡面钢bloating 胀大block 砖[块]block cut 方块切割block diagram 区块图；方块图block handle 实心柄block rake 刮痕blocker 成坯型(锻)blocker-type forging 坯锻blocking 锻坯blocking impression 成坯型blood 钢面红锈bloom[ 金属]块料bloomer 块料炉blooming mill 块料轧机blooming test 图像浮散试验blotch 残迹；輚痕blotter press 压滤机blow and blow process 重吹[法]blow hole 气孔blow mould 吹模blower 鼓风机blowing1 喷出溶体(钢锭题耳结壳後喷出之溶体)2 送风blowing iron 吹管blown casting 含气泡铸件blown metal 吹津金属(转炉中杂质已氧化除去之金属) blue annealing 显蓝退火(温度较blueing 高)blue brick 青砖blue brittleness 青脆性，蓝脆性blue heating 发蓝加热blue shortness 青脆性，蓝脆性blueing 青化法blueing (bluing)发蓝blunger 拌和器blurring-highlight test 耐击试验(釉)blushing 抹红board drop hammer 板落鎚board hammer 板鎚board insert check function 接线板插入检查功能board level simulation 基板位阶模拟boart (or bort)黑钻石(工业用)boash 炉腹(鼓风炉)boat 晶舟boat elevator 晶舟升降机boat handler 晶舟处理机boat lift travel 晶舟移动行程boat loader 晶舟搭载机boat transfer 晶舟输送器boding parameter 焊接参数body 坯[胎]body-centered cubic (b.c.c.)体心立方body-centered cubic lattice 体心立方格子body-centered orthorhombic 体心斜方body-centered tetragonal 体心正方body-core 主体砂心boehmite 柏买石(从矿)bogie hearth furnace 通心式炉，台车式炉bohr magnetron 波尔磁子boil 气泡boiler corrosion 锅炉腐蚀boiler embrittlement 锅炉脆化boiler plate 锅炉钢板boiler scale 锅垢boiler water 锅炉用水boiling 沸腾bolster 承板boltzmann constant 波子曼常数bomb reduction process 高压罐还原法bond 黏结；叠砌法；键bond energy1 束缚能2结合能bond strength 黏结强度bonded abrasive 黏合研磨剂bonder 接头砖bonderizing 磷酸盐[防锈]处理bonding accuracy 焊接精确度bonding force 结合力，焊线强度bonding head 压接头bonding length 接合长度bonding pad 焊垫，接合垫bonding silicon on insulator wafer 矽绝缘体（soi）接合晶圆bonding speed 焊接速度，接合速度bonding wire 焊接线，压接线bondur alloy 邦杜合金(锻造用合金之一，类似於含4%铜之杜拉铝)bone ash 骨灰bone china 骨灰瓷bone-dry body 乾透坯book mould 书型模boost melting 辅熔法(玻)boot 醮料筒boot leg[ 矿]孔片[冶]初缩borate flint 含硼火玻璃borax 硼砂borazon 氮化硼borcher' furnace 波其尔炉boric acid 硼酸boric oxide 氧化硼borides 硼化物boriding 渗硼处理borings 搪屑boron (b, 5)硼boron carbide 碳化硼(b4c)boron fiber 硼纤维boron nitride 氮化硼(bn)boron steel 硼钢boronising 渗硼法borosiliconizing 渗硼矽处理boroxal 波拉索尔(类似於波拉尔材料，由b2o2 与al 制成) bort 圆粒金刚石bosh 炉腹(从机)botting 堵塞botting clay 封塞土bottle brick 瓶形砖bottle kiln 瓶形窑bottle oven 瓶形炉bottom 底部bottom pouring 底浇bottom pouring ladle 底浇盛桶bottom-up design 由下而上之设计boulder clay 漂砾土boundary crystal 晶浇声界boundary displacement 界壁位移boundary domain 磁域界壁boundary energy 声院能(晶粒间)boundary layer 界面层，边界层boundary scan test 边界绍]扫描测试boundary scattering 声界散射boundary tension 界面张力bow 弯曲bowing trench 沟壁内凹bowl rinse 碗盘冲洗bowl temperature and humidity control 碗盘温度与湿度控制bowor-barf process 包尔-巴夫法(钢铁防锈处理法之一)box annealing 箱式退火box carburizing 箱式渗碳法box feeder 箱饲机box funace 箱式炉bracken glass 蕨灰玻璃brackish water 淡盐水(盐分界於河海水之间)bragg reflection 布勤格反射bragg's law 布勒格定律brake lining 勒衬；煞车衬brale indenter[ 钻石]压痕器brass 黄铜braunite 布劳奈铁(层状氧化铁)bravia lattice 布拉菲格子braze welding 硬焊泰接brazing 硬焊brazing alloy 硬焊合金brazing solder 硬焊料breakaway corrosion 剥离蚀breaking load 断裂负荷breaking point 断裂点breaking quenching 辉面淬火breaking strength 断裂强度breaking stress 断裂应力breast 炉胸breast wall 胸墙breeder reactor 滋生反应器，增殖反应器breeding 滋生，增殖breeze 焦碳粉breeze coke 成型焦碳bremsstrahlung 制动辐射(braking-radiation)breunnerite 铁菱镁矿brick 砖brick clay 砖土brick earth 砖土brick lining 砖衬bridging1 题面结壳(坩埚内炉料上部固结现像)2桥接(直炉风口上部炉料固结现像)3拱陷bridging oxygen 架桥氧bridgman [crystal growing] technique 布里吉曼[晶体生长]技术bright annealing 辉桶退火bright coal 辉煤bright dip 辉面浸渍bright etching 光亮蚀刻bright field 明视野bright goldliquid gold bright hardening 辉面硬化bright heat treatment 辉面热处理brightener 上光剂brightray alloys 布莱崔合金(高铬之镍合金)brillouin zone 布利洛区brine leaching 盐水浸滤，盐水浸洗brinell hardness 勃氏硬度brinell hardness test 勃氏硬度试验bring etching 光亮蚀刻briquet (briquette)1媒砖2方块briquette 试锭bristol glaze 布里斯脱釉britannia metal 不列颠金属(一种锡锑合金又名pewter)brittle boundary technique 边界脆化术brittle material 脆性材料brittle rupture 脆性破坏brittle-ring test 脆环试验brittleness 脆性broad line (转)宽(绕射)线(x- 光) broeggerite 金士铀矿bromine (br, 35)溴bromo-gelatine 溴明胶bronze 青铜bronze welding 青铜焊brookite 板钛矿brown and bharp wire gauge 线号规brown coal 褐煤brownies 棕斑brownmillerite 铁铝酸四钙brownsdon-bannister jet test 布-邦式气击试验brtittle fracture 脆性断裂brucite 氢氧镁石brush marks 刷痕brush plating 刷覆电镀法brush scrubber 刷子擦洗机brushing 刷洗brushing glaze 刷釉brushing machine 刷子清除机bruttin furnace 布劳顿炉bubble cap 泡罩bubble leak tester 漏泄气泡测试器bubble nucleation 气泡成核bubble-pressure method 加压孔隙测定法buck stay 撑柱buckling1 扭曲，翘曲2翘度(核子化工)buff 擦光轮buffer1 缓冲剂2缓冲器buffing 擦光轮buffing compound 擦光剂buffing paste 擦光膏bug 缺陷build-up 堆积building stone 建筑石材built-in self test 内建自我测试bulge forming 胀大成形bulging 胀大bulk 表体；积体bulk cement 散装水泥bulk defects 表体缺陷bulk delivery car 散装车bulk density 总体密度bulk modulus of elasticity 体积密度bulk specific gravity 体比重bulk volume 容体积bulk weld process 体积弹性系数bulking 胀大bull's eye structure 金银堆bullard-dunn process 布粉电弧熔接法bulldozer 布拉-杜恩去锈镀锡法bullet envelope 推土机bullion 弹头外壳(85%cu, 15%zn，即薄皮金属) bum 胶bump 隆起物，凸块bumper 牛眼组织bundle 振动机(造砂模用)bundy tubing 压块bung 匣柱bur-in controller 老化测试控制器burden 卷焊钢管burden ratio 进炉料burdening 配料比bureau of standards 标准局burger's vector 柏格向量burgers circuit 柏格环路burmister flow trough 布氏流槽burn-in board 老化测试基板burn-in board checker 老化测试检验器burn-in board ejector 老化测试基板拔除器burn-in board inserter 老化测试板插入器burn-in cable tray 老化测试电缆架burn-in chamber 老化测试恒温槽burn-in rack 老化测试架burn-in stress function 老化测试应力施加功能burn-in system 老化测试系统burn-in timer 老化测试计时器burn-in tracking 老化测试追踪burn-on 脱腊burn-up 燃耗率(核反应器)burned lime 烧石灰burned sand 铸砂烧结burning 燃烧burning on 浇补burnishing 压光burnt brass 烧毁黄铜burnt deposit 无光泽析出物(电镀时电流密度过大所成不正常) burnt dolomite 烧结白云石burnt metal 烧毁金属burnt steel 焚钢burr 毛边burr/flash/bleed 毛头/(塑模)溢料/残渣burring1 凸出成形2除毛头burst test 胀裂试验(水压)bursting 膨裂bursting expansion 瓶裂膨胀bursting off 瓶裂(从机)bushing 衬套bustle pipe 炉腹风管butane atmospheres 丁烷炉气butler finish 中度光面butt brass 白特黄铜butt joint 对接butt strap 搭板butt welding 对头熔接butterfly valve 蝶形阀buttering 熔接缝预镀层butters vacuum filter 百塔尔真空过滤机button 金属钮button test 熔流性试验by-pass capacitor relay driver 旁路电容器中继驱动器by-product 副产品by-product coke 副产焦碳。

专业英语词典-B

back annotation 逆向注解back channel 反向通道，反向信道back electromotive force 反电动势back filling 倒填⼊back light 背景照明back panel 背板back ripple current 逆向脉动电流back, loop 回环back-annotate 后端注释back-end 后端back-off, bus 总线退出back-scattering 反向散射back-to-back testing 背对背测试backbone cabling 主⼲线布线，主⼲布线backbone network 中枢络backbone network architecture (BNA) 中枢络结构background 背景background acquisition 背景采集backplane 底板；背板；基架backplane bus 基架总线backplane transceiver logic (BTL) 基架收发器逻辑backscatter 反向散射backtrace 回索backup 备存backup gap 备存间隙backup protection 备存保护backward diode 逆向⼆极管backward estimation 后向估计backward recovery 向后恢复backward-compatible 逆向相容baffle 隔板balance force 平衡⼒balance, analytic 分析天平balance, charge 电荷平衡balanced 平衡的balanced amplifier 平衡放⼤器balanced capacitance 平衡电容balanced circuit 平衡电路balanced conditions 平衡条件balanced line 平衡线路balanced mixer 平衡混合器balanced, link access protocol (LAP-B) 链路存取协定平衡balancing network 平衡络ball bearing 滚珠轴承ball, solder 焊锡球ball-bearing fan 滚珠轴承风扇ballast 镇流管ballistic galvanometer 冲击电流计balun 平衡－不平衡变换器banana plug ⾹蕉插头band 频带band spectrum 频带谱band spreading 频带散布band, carrier 载波频带band, citizen (CB) 民⽤波段band, effective 有效频带band, guard 防护频带band, pass 通频带band, side 侧频带band, single-side (SSB) 单⾯频带band, voice 声⾳频带band-edge 频带边沿band-elimination filter 带阻滤波器bandgap 频带间隙banding, rubber 弹性连接bandpass 通频带bandpass filter 带通滤波器bandpass limiter 带通限制器，带通限幅器bandwidth 频宽bandwidth on demand 额定带宽bandwidth, communications 通讯频宽bandwidth, frequency 频率频宽bandwidth, frequency selective 频率选择频宽bandwidth, luminance 亮度频宽bandwidth, read 读出频宽bandwidth, resolution (RBW) 解析度频宽bandwidth, sampling frequency 取样频率频宽bandwidth-limited operation 频宽限制操作bank 组bank switching 存储器组转换bank winding 存储器组缠绕bank, multi-register 多寄存器组bank, register 寄存器组bar 巴bar code 条码bar generator 条状信号产⽣器bar graph ⽓压记录仪bar magnet 条形磁铁bar pattern 条形图案bar-code reader 条码阅读器bar-code scanner 条码扫描器bar-code system 条码系统bar-type current transformer 条形电流变换器bare board *板bare conductor *线bare die *⽚bare-board testing *板测试bargraph 柱状图表barium (Ba) 钡barometer ⽓压计barometer, aneroid ⽆液⽓压计barrel connector 圆柱式连接器barrel distortion 桶形畸变barrel shifter 柱式位移器barrier 障碍；阻挡barrier layer 阻挡层base 基本；基底base active power 基本有功功率base address 基底地址base bus 基底总线base drive 基极驱动base electrode 基极base frequency 基本频率base impedance 天线座端阻抗base load 基本负载base management information, (MIB) 管理信息基本原则base page address 基页地址base port address 基本端⼝地址base pressure 基本压⼒base region 基极区base register 基本寄存器；基极寄存器base resistivity 基极电阻base station 基站base station controller (BSC) 基站控制器base station subsystem (BSS) 基站⼦系统base transceiver station (BTS) 基站base transistor 基极晶体管base unit 基本单位base, common 共基极base, time 时基baseband 基频带baseband LAN 基带局域baseband coaxial cable 基频带同轴电缆baseband modem 基带调制解调器baseband network 基频带络baseband signal 基频带信号baseband-multiplexed 基频带多⼯baseline delay 基线延迟baseline offset 基线位移量baseline overshoot 基线过冲baseline privacy 基线私密baseline wander (BLW) 基线漂移baseplate 底板baseplate temperature 底板温度basic HTML (HyperText Markup Language) 超⽂本标识语⾔basic access 基本存取basic access method 基本存取⽅法basic access model 基本存取模式basic audio processor (BAP) 基本声频处理器basic cell 基本单元basic communications services 基本通讯服务basic control element 基本控制元素basic direct access method 基本直接存取⽅法basic encoding rule (BER) 基本编码法则basic input/output system (BIOS) 基本输⼊／输出系统basic input/output system data area 基本输⼊／输出系统数据区域basic media access controller (BMAC) 基本媒介存取控制器basic part 基本零件basic rate 基本速率basic rate access (BRA) 基本速率存取basic rate interface (BRI) 基本速率接⼝basic repetition rate 基本重复率basic telecommunications access method (BTAM) 基本远程通讯传取法basic transmission unit (BTU) 基本传输单位bass boost 低⾳提升batch 批次batch mode 批[处理]⽅式batch processing 批次处理作业bath voltage 槽电压bath, solder 焊槽bath, water ⽔浴槽bathtub curve 澡盆曲线battery 电池battery back-up 备⽤电池battery boost 电池升压battery charger 电池充电器battery eliminator 等效电池battery pack 电池包；电池组件battery, alkaline 硷性电池battery, lithium 锂电池battery, nickel cadmium (NiCd) 镍镉电池battery, nickel metal hydride (NiMH) 镍⾦属氢电池battery, primary 原电池battery, secondary 次级电池battery, solar 太阳能电池baud 波特baud rate 波特率baud-rate generator 波特率产⽣器beacon 标⽰beacon frame 指⽰信息段bead, solder 焊球beam 束；光束beam crossover loss 射束交迭损耗beam finder 寻束器beam forming 射束形成，聚束，成束beam splitter 分光镜beam steering 射束[⽅向]控制beam, electron 电⼦束beam, focused ion (FIB) 聚离⼦束beam, gallium ion 镓离⼦光束beam, ion 离⼦束beam, laser 激光束beam, molecular 分⼦束beam, photolithographic 光刻线beam-deflection tube 射束偏转管bearer service 承载业务bearing 轴承bearing accuracy ⽅位准确度bearing, ball 滚珠轴承bearing, hydrostatic 静液压轴承bearing, roller 滚桶轴承bearing, spindle 转轴轴承beat 拍beat frequency 拍频beat note 拍⾳bed-of-nails 针床behavior, transient 瞬变⾏为behavioral ⾏为的；性能的behavioral data 性能数据behavioral definition 性能定义behavioral description 性能描述behavioral simulation 性能摸拟behavioral simulation model 性能摸拟模型bellow 摺箱belt, conveyer 传送带benchmark 基准点benchmarking program 基准点测试程序bend loss [ 光纤的]弯曲损耗bend radius 弯曲半径benzene 苯beryllium (Be) 铍beta circuit У缏?beta testing Р馐?beveled 斜削的bezel 挡板；遮光板bi-directional 双向的bi-directional bus 双向总线bi-directional pin 双向接脚bi-phase coding ⼆相编码，双相编码bias 偏置；偏压bias circuit 偏压电路bias current 偏置电流bias distortion 偏置失真bias power 偏置功率bias slope 偏置斜率bias, fixed 固定偏压bias, high-temperature reverse- (HTRB) ⾼温反向偏压bias, negative 负偏压biased amplifier 偏置放⼤器biconcave lens 双凹⾯透镜biconical antenna 双锥形天线biconvex lens 双凸⾯透镜bifocal lens 双焦点透镜bilateral network 双向络bilateral roaming agreement 双向漫游协定，双向漫游协议bilateral transducer 双向传感器bill of materials (BOM) 材料单bimetallic strip 双层⾦属条binary ⼆元的；⼆进制binary arithmetic ⼆进制算术binary code ⼆进制码binary compatible ⼆进制兼容binary countdown ⼆进制倒数binary counter ⼆进制计数器binary data ⼆进制数据binary decision diagram (BDD) ⼆元判定图binary encoding ⼆进制编码binary frequency-shift keying (BFSK) ⼆进制频率变换调制binary mask ⼆元掩膜binary notation ⼆进制记数法binary number system ⼆进制数制binary phase-shift keying (BPSK) ⼆进制相位变换调制binary search ⼆元搜寻binary synchronous communications (BSC) ⼆进制同步通讯binary synchronous control (BSC) ⼆进制同步控制binary synchronous interface (BSI) ⼆进制同步接⼝binary synchronous protocol (BISYNC) ⼆进制同步协定binary tree search ⼆叉树搜寻binary unit ⼆进制单元binary with eight zero substitution (B8ZS) ⼋位⼆进制零替换binary-coded decimal (BCD) ⼆进码⼗进数binary-tree search ⼆元树状搜寻binaural 双⽿的，双声道的，⽴体声的binder 黏合剂binding 联编，汇集，邦定binding energy 结合能binhex ⼆ -- ⼗六进制biodegradable 可⽣物降解biplexer ⾼速通道bipolar 双极bipolar analog process 双极模拟过程bipolar coding 双极性编码bipolar coding with zero suppression 消零双极性编码bipolar complementary metal oxide semiconductor (BiCMOS) 双极互补⾦属氧化半导体bipolar logic 双极逻辑bipolar metal oxide semiconductor (BiMOS) 双极⾦属氧化半导体bipolar power transistor 双极功率晶体管bipolar signal 双极信号bipolar transistor 双极晶体管biquinary ⼆-五进制bismuth compound 铋化合物bistability 双稳定性bistable 双稳态bit 位bit cell 位单元bit error rate (BER) 位错误率bit error rate (ber) ⽐特差错率，误码率bit error rate testing (BERT) 位错误率测试bit field 位资料栏bit map 位映射图bit plane 位平⾯bit problem 位问题bit rate 位速率，⽐特率bit serial 位串⾏bit shift 位移位bit stream 位流bit stuffing 位填充，⽐特填充bit synchronous 位同步bit, accessed 存取位bit, asynchronous 异步位bit, framing 帧划分位bit, horizontal parity ⽔平奇偶位bit, interleaved 交错位bit, least significant (LSB) 最⼩有效位bit, lower 低位bit, missing 缺失位bit, parity 奇偶位bit, significant 有效位bit, start 起始位bit, status 状态位bit, stop 终⽌位bit, storage 储存位bit, upper ⾼位bit, vertical parity 垂直奇偶位bit-aligned block transfer 对位式区块传输bit-block transfer (BitBLT) 位区块传输bit-cell polysilicon 位单元多晶硅bit-error rate testing (BERT) 位错误率测试bit-error rates 误码率bit-error ratio (BER) 位错误率bit-mapped graphics 位映射图像bit-mapped raster image 位映射光栅影像bit-oriented protocol ⾯向⽐特的协议bit-oriented protocol (BOP) 位取向协定bit-parallel 位并⾏bit-per-second (bps) 每秒位数bit-retention time 位保存时间bit-reversal permutation 位反向置换bit-serial 位串⾏bit-sliced 位式⽚black and white ⿊⽩的black body ⿊体black box ⿊盒black compression ⿊区信号压缩black level ⿊⾊电平black light ⿊光；不可见光black peak ⿊⾊峰值black-body radiation ⿊体幅射black-oxide process ⿊氧化处理black-surface reflector (BSR) ⿊⾯反射器blackening, relative 相对⿊度blade 叶⽚；⼑刃blade antenna 叶⽚形天线blade tip 叶梢blank 空⽩；清除blank character 空⽩字符blanked picture signal 消隐画⾯信号blanking 消隐blanking interval 消隐间隔blanking level 消隐信号电平bleed 渗出，泻漏；⾊料扩散；[图⽂]超出版⾯blind hole 闭孔blind spot 盲点blind via 盲通孔blink 闪烁blister ⽓泡block 区块block allocation 区块配置block code 分组码block command 字块命令block diagram ⽅块图block erase 区段式擦除block error rate (BLER) 资料段错误率block mode 区块模式；资料段模式block parity 区块奇偶校验block sequence number (BSN) 块序列号block transfer 区块传输block, building 汇编区块block, configurable logic (CLB) 可配置逻辑区块block, data 数据资料段block, expanded memory (EMB) 扩充存储器区块block, file control (FCB) 档案控制区块block, functional 功能区块block, generic logic (GLB) 通⽤逻辑区块block, logic array (LAB) 逻辑阵列区块block, upper memory (UMB) 上存储器区块block-mode transfer 资料段模式传输blockcheck character (BCC) 区块检测符blocking capacitor 阻塞电容器blocking diode 阻塞⼆极管blocking probability 阻塞概率blocking voltage 阻塞电压blow hole 熔固孔blower 吹风机blowout coil 减弧线圈board layer 电路板层board library 电路板程序库board test lab 电路板测试实验室，插件板测试实验室board warpage 电路板扭曲board, bare *板board, bread 实验⽤电路板board, bulletin 电⼦布告板board, burn-in (BIB) ⽼化测试板board, bus master 总线控制器板board, chip-on- (COB) 板上式芯⽚board, daughter ⼦板board, double-sided 双⾯电路板board, field programmable circuit (FPCB) 可现场编程电路板board, functional system (FSB) 功能系统板board, main 主机板board, multilayer 多层电路板board, plated-through 镀通电路板board, printed circuit (PCB) 印刷电路板board, printed wiring (PWB) 印刷电路板board, single-sided 单⾯电路板board, through-hole 穿孔式电路板board, wiring terminal 路由终端板bobbin 线轴bobble [ 装饰]软球body capacitance 体电容body, electrified 带电体body, illuminated 受照体body, luminous 发光体body, non-luminous ⾮发光体body, opaque 不透明体body, perfectly elastic 完全弹性体body, rigid 刚体body, translucent 半透明体boiling 沸腾boiling point 沸点bond 键；黏接bond pad 接合焊盘，结合区bond wire 接合线bond, covalent 共价键bond, ionic 离⼦键bond, metallic ⾦属键bonded wafer 已粘接晶圆bonding 粘结，压焊，结合，搭接bonding alloy 粘结合⾦bonding pads 结合⽚，结合区，连接填料bonding, metal electrode face (MELF) ⾦属电极表⾯黏合bonding, tape automated (TAB) 卷带⾃动接合技术bonding, transferred-bump tape automated (T-BTAB) 传递碰撞式卷带式⾃动接合技术book-to-bill ratio 订购运销值⽐boolean algebra 布尔代数，逻辑代数boolean equation 布尔⽅程boolean expansion 布尔延伸式boolean logic 布尔逻辑boolean operator 布尔运算⼦boolean reduction 布尔约简boost 升压boost regulator 升压调节器boost, buck- 冲跳升压booster 升压器；放⼤器booster charge 再充电boot 启动boot infector 启动感染程序boot sector 启动磁区boot, cold 冷启动boot, warm 暖启动boot-up 启动booting 引导，引导装⼊bootstrap 启动程序bootstrap circuit 启动电路bootstrap loader 启动程序载⼊器bootstrap supply ⾃举电源border 边界border gateway 边界关border gateway protocol (BGP) 边界关协议bottom centering jaw 下定中⽖bounce 跳动bounce pad 回弹插⼊,回弹垫⽚bounce, ground 接地跳动bound charge 束缚电荷bound electron 束缚电⼦boundary condition 边界条件boundary controller 边界控制器boundary functional test (BFT) 边界功能测试boundary in-circuit test (BICT) 边界内电路测试boundary representation (B-rep) 边界表⽰法boundary scan 边界扫描boundary tag method (BTM) 边界标志法boundary value 边界值boundary, framing 信息段范围boundary, kernel 核⼼边缘boundary-element method (BEM) 临界元素法boundary-scan architecture 边界扫描结构boundary-scan description language (BSDL) 边界扫描描述语⾔braid 包线brain drain ⼈才外流brake anti-lock, 防震刹车braking resistor 制动电阻器branch 分⽀；转位branch network 分⽀络branch target cache (BTC) 分⽀式⽬标⾼速缓冲存储器braze 铜锌合⾦焊，硬钎焊breadboard 实验⽤电路板break detect 中断检测break indication 中断指⽰breakdown 中断点breakdown current 击穿电流breakdown impedance 击穿阻抗breakdown rating 击穿额定值breakdown region 击穿区域breakdown voltage 击穿电压breakdown, collector-base 集极－基极击穿breakdown, secondary 次级击穿breaker 断路；开关breakout box 断接盒breakpoint 断点breakpoint trigger 触断点breakthrough 贯穿bridge 电桥；桥接bridge amplifier 桥式放⼤器bridge circuit 桥接电路bridge current 桥接电流bridge rectifier 桥接整流器bridge, Wheatstone 惠斯登电桥bridge, Wien 惠恩电桥bridge, solder 焊桥bridged-T network 桥接Ｔ形四端络bridging 桥接，跨接；分路，分流bridle wire 跳线；绝缘跨接线brightness 光度brightness, cross 互串光度brine 盐⽔broadband 宽频带broadband Internet 宽带因特broadband coaxial cable 宽带同轴电缆broadband detection 宽频带检测broadband directional coupler 宽频带定向耦合器broadband power divider 宽频带功率分配器broadband telecommunications 宽频带远程通讯broadband wireless 宽带⽆线[通信]broadcast ⼴播broadcast address ⼴播地址broadcast control channel (BCCH) ⼴播控制信道broadcast encryption ⼴播加密broadcast management entity (BME) ⼴播管理机构broadcast network ⼴播broadside array 垂射天线bronze conductor 青铜传导体brouter 桥由器，桥式路由器brownout 电压起伏及闪烁brownout 降压，节电browse 浏览browser 浏览器。

数字电子技术基础英语词汇

数字电子技术基础英语词汇1．二进制：Binary2.数：number3.逻辑：logic;4.下降时间：fall time;5.计数器：counter;6.加法器：adder;7.分辨率：resolution;8.存储器：memory;9.时钟：clock;10.触发：trigger;11.字：Word;12.译码器：decoder;13.反相器：i nverter;14.电平：level;15.门：gate;16.符号：symbol;17.函数：function;18.常数：constant;19.编码：coding;20.二极管：diode;21.真值表：True table;22.量化：quantification;23.总线：bus;24.复位：reset;25.定时器：timer;26.脉冲：pulse;27.状态：state;28.图：diagram;29.动态：dynamic; 30.线：line;31.门限电压Threshold voltage; ：32.阵列：array;33.建立时间：setup time;34.可编程逻辑阵列：PLD;35.数据选择器：multiplexer;36.表达式：expression;37.寄存器：register;38.锁存器：latch;39.时序图：timer diagram40.位：bit41.参考电压：reference voltage42.拉电流：draw off current43.函数发生器：function generator44.权：weight45.多谐振荡器：astable multibrator46.半：half47.全：full48.分频：frequency division49.静态：static50.传输特性：transfer characteristics51.算术电路：arithmetic circuit52.现：present53.下降沿：fall edge54.上升沿：rise edge55.三态：three state56.地址：address57.十进制：decimal58.同步：synchronous59.上升时间：rise time60.无关项：don’t care terms61.施密特触发器：schmitt trigger62.奇偶校验：odd even check63.扇入：fan in64.扇出：fan out65.保持时间：hold time66.恢复时间：recovery time67.读写控制：read write control68.数据分配器：demuliplexer69.噪声容限：noise margin70.编码器：encoder71.数字电路：digital circuit72.模拟开关：analog switch73.双积分：dual slope74.算术逻辑单元：ALU75.触发器：flip-flop76.数字显示：digital display77.双向：bidirectional78.纹波计数器：ripple counter79.传输门：TG80.刷新：refresh81.随机访问：random access82.只读：read only83.布尔代数：boolean algebra84.双稳态：bistable85.次：next86.闪烁存储器：flash memory87.存储时间：storage time88.或门：OR89.非门：NOT90.与门：AND 91.或非门：NOR92.与非门：NAND93.集电极开路：open collector94.上拉电阻：pull up resistor95.驱动方程：driving equation96.卡诺图：karnaugh map97.三极管-三极管逻辑电路：TTL98.金属氧化物互补对称电路：CMOS99.减法器：subtractor100.移位寄存器：shift register。

逻辑综合理论

关于RM逻辑介绍逻辑综合与优化是一类用逻辑门实现电路功能或描述的完整过程，而逻辑优化的关键内容之一是电路表达式或函数的化简。

这是由于电路的面积，功耗，速度和可验证性与电路结构直接相关，而具体的电路结构可由表达式或函数的繁简程度反映。

因此，函数表达式的化简是很有必要的，IC 设计者可根据需求对电路表达式进行改善，以实现理想的面积、速度和功耗等性能。

对于运算电路、通信电路、奇偶检测电路等特定电路，使用 RM 逻辑往往能够实现更好的面积、速度和功耗等性能RM 逻辑电路主要包括 XOR/AND 和XNOR/OR 这两种表示形式，依据极性分为固定极性 Reed-Muller(FPRM)表达式、混合极性 Reed-Muller(MPRM)表达式； fixed polarity固定；mixedpolarity 混合；XOR——异或门，符号标志为“⊕”；XNOR——同或门，数学符号为“⊙”；Boolean 逻辑仍是当前电路设计的主流逻辑形式，为了更好的使用 RM 逻辑并进行相关优化，首先就需要实现从 Boolean 逻辑函数到 RM 逻辑函数的转换。

极性转换方法提供了 Boolean 逻辑到RM 逻辑以及 RM 逻辑中极性间的转换。

FPRM 电路相较于MPRM 电路实现更简单，其极性转换方法更简便适用；FPRM 电路相关的极性转换方法较多，主要有：列表法、系数矩阵法、不相交乘积项等；MPRM 电路的极性转换方法主要有：图形变换法、OKFDDs(Ordered Kronecker Functional Decision Diagrams)法。

逻辑综合概述认识逻辑综合用Verilog之类的程序设计语言将硬件的高级描述转换成一个优化的数字电路网表，一个由相互连接的布尔逻辑门组成的网络，从而实现该功能。

逻辑综合设计流程大型数字电路设计流程如下：EDA是用来完成芯片的功能设计、综合、验证、物理设计等流程的设计方式，其中，逻辑级自动综合与优化属于EDA前端设计技术；逻辑综合完成就进入后端设计阶段；布局：就是将综合后的门级电路网表的每个工艺单元合理的摆放到芯片的各个位置；布局的任务是确定每个单元的位置，尽可能减小布线的开销。

synthesis复习总结

Synthesis(综合):synthesis is the transformation of an ideainto a manufacturable device to carry out an intended function.Hold time(保持时间):the length of the time that data must remain stable at the input pin after the active clock transition.Wire load model(线载模型):it is an estimate of a net's RC parasitics based on the net's fan-out.Constraint(约束):the informations about the timing, area,and the environmental attributes for the design.critical path(关键路径): The timing path which has the largest delay.clock skew(时钟偏斜):to account for varying delays betweenthe clock network branches.Jitter(时钟抖动):b ecause some uncertain factors, which leads to the clock happen drift.STA(静态时序分析)：static timing analysis ;a method for determining of a circuit meets timing constraint without dynamic simulation.Setup time(建立时间):the length of the time that data must stabilize before the clock transition.TCL(tool commend language):BDD(binary decision diagram):SOLD( Synopsys on-line documentation):timing path:design time breaks designs into sets of signal paths ,each has a start point and an endpoint. Startpoint :input ports, clock pin of sequential devices; endpoint: output port ,data input pin of the sequential devices.PVT(process, voltage, temperature):operating condition: STA scale each cell and net delay basedon process ,voltage ,and temperature (PVT) variations.2:synthesis = translation + optimization + mappingdesign object:clock: A timing reference object in DC memory which describes a waveform for timing analysisport: The input or output of a design.cell: An instance of a design or library primitive within a designnet: The wire that connects ports to pin and/or pins to each otherdesign: A circuit that performs one or more logical functionspin: The input or the output of a designreference: The name of the original design that a cell instance “points to”Levels of circuit abstraction:idea, function, behavioral, register transfer, gate-level, physical device .synopsys_dc.setup:Define the path of target library, symbol library, link library, search path and other parameters for the logic synthesis.Library in the synopsys_dc.setupTarget library: the ASIC technology that the design is mapped to.Symbol library: used during schematic generationLink library: the library used for interpreting input descriptionSearch path: the path to search for unsoveled referencelibrary or design3 to 8 decoder：Write verilog code.HDL Synthesis process:——YONG。

对电路的等价性检验方法的探讨

对电路的等价性检验方法的探讨摘要：等价性检验是目前电路设计验证中应用最为广泛的形式化方法。

为了提高验证的效率,通常使用组合验证的方法来验证大型时序电路。

大多数组合等价性检验方法都以二叉判决图(bdd)为主要推理引擎,可能导致内存爆炸题。

基于增量的方法是利用两个电路内部的结构相似性,把要验证的问题分解为多个子任务、增量地完成验证。

本文则在此基础上对电路的等价性检验方法作出一番探讨。

关键词：等价性检验电路验证1、引言一般来说,形式化验证方法可以分为等价性检验(equivalence checking)、模型检验(model checking)和定理证明(theorem proving)方法。

而等价性检验被广泛地应用到设计的各个阶段。

它的基本原理是建立被比较的两个模型之间的关系。

检验的依据是数学的定理和公理,以及设计实现所利用的标准的单元库的精确描述。

等价性检验程序自动确定被比较的两个设计的关系,而不需要用户的输入,它的优点是使用简单,容易集成到设计流程中。

等价性检验方法又包括基于符号和基于增量两种方法。

基于符号的检验方法依赖于基于bdd(binary decision diagram)遍历有限状态机(fsm)来实现等价性检验的。

在基于增量的方法中,利用被验证的两个电路的结构相似性来检验所刻画的系统是否与实现一致,它被进一步划分为:基于替换的方法、基于学习的方法和基于变换的方法。

2、等价性检验模型传统的组合电路功能等价性验证是通过构造两个电路的规范表示形式,如真值表或二叉判定图bdds,当且仅当它们的规范形式同构时,这两个电路功能等价。

为了验证两个时序电路的等价性,通常需要把它们当成有限状态机,并构造这两者的积自动机。

brand将这种计算模型称为miter。

它是通过把两个状态机相应的每一对原始输入联接到一起,同时把相应的每一对原始输出联接到一个异或门,而这些异或门就构成了积自动机的输出。

如果对于每一个输入序列,积自动机的每个原始输出恒为0,那么这两个时序电路就是等价的。

高性能电脑形式化验证技术研究

高性能电脑形式化验证技术研究第一章：引言随着计算机科学的快速发展和普及，计算机系统已经成为现代社会的重要基础设施之一。

在一个计算机系统中，无数的软件程序和硬件组件相互作用，形成了复杂的系统。

为了保证这些系统的可靠性和正确性，必须对其进行全面的测试和验证。

而在计算机系统中，形式化验证技术是一种非常有效的方法。

本文将介绍高性能电脑形式化验证技术，本文将从形式化验证的基本概念和原理入手，介绍现有高性能电脑形式化验证技术的发展和应用，最后探讨高性能电脑形式化验证技术的未来发展方向。

第二章：形式化验证基本概念和原理形式化验证是指使用数学逻辑和自动推理技术来验证计算机系统是否符合一定的规范和要求。

形式化验证的方法是建立一个数学模型来表示系统的行为和性质，然后通过模型检查等技术来验证系统的正确性。

形式化验证的基本原理是：通过证明某个系统符合某种规范来保证其正确性，而不是简单地依靠测试来验证。

测试只能覆盖系统的一部分可能的行为和情况，因此不能保证系统的完全正确性。

而形式化验证则可以通过数学推理来检查系统的全部行为和情况，可以保证其正确性。

第三章：高性能电脑形式化验证技术的发展和应用高性能电脑形式化验证技术是在传统形式化验证技术的基础上发展起来的，其特点是可以处理更加复杂、规模更大的系统。

现有的高性能电脑形式化验证技术主要包括SAT求解器、BDD、SMT求解器等。

SAT求解器是目前应用最广泛的高性能电脑形式化验证技术之一，它可以用来解决布尔公式可满足性问题。

由于布尔公式是许多形式化验证问题的基础，因此SAT求解器技术在形式化验证领域应用非常广泛。

BDD（Binary Decision Diagram）是另一种非常常用的高性能电脑形式化验证技术。

它是一种用于表示布尔函数的数据结构，在各种硬件和软件验证中都经常使用。

SMT求解器则是一种可以解决一般逻辑公式可满足性问题的高性能电脑形式化验证技术。

与SAT求解器、BDD相比，SMT求解器可以处理更加复杂的逻辑问题，包括一般谓词逻辑和量词逻辑等。

一类新型抽象数据类型:有序二叉决策图

一类新型抽象数据类型：有序二叉决策图古天龙【摘要】有序二叉决策图OBDD(Ordered Binary Decision Diagram)是布尔函数的一种规范表达形式、一种的新的数据结构.基于OBDD能够完成布尔函数的有效表述和操作运算,可以看作为一类新的抽象数据类型.OBDD在VLSI逻辑综合和验证的成功应用结果引起了学术界和工业应用界的极大关注.迄今为止,OBDD技术及其工业应用已有了长足的发展、产生了不少的研究结果.本文对OBDD相关技术问题、OBDD扩展形式、OBDD应用等方面的研究现状进行了综述和讨论.【期刊名称】《桂林电子科技大学学报》【年(卷),期】2010(030)005【总页数】15页(P374-388)【关键词】有序二叉决策图;抽象数据类型;数据结构;符号技术;布尔函数【作者】古天龙【作者单位】桂林电子科技大学,计算机科学与工程学院,广西,桂林,541004【正文语种】中文【中图分类】TP311.12布尔代数是计算机科学和逻辑系统设计的重要基石[1]。

VLSI系统CAD、软件测试和验证、AI规划与调度等领域中的诸多问题都归结为逻辑函数及其序列的操作和运算。

毫无疑问,布尔函数的表述及其操作对这些领域中问题的合理有效解决起着尤为关键的作用。

不幸的是,布尔函数的可满足性和等价性等问题的NP完备性所导致的状态组合爆炸问题严重地制约了大规模、甚至工业小规模问题的解决[2]。

然而,在实际问题的处理中,对布尔函数采取恰当的描述、并建立相应描述下的操作算法,可以达到有效地避免和减缓问题处理过程中的状态组合复杂性。

有序二叉决策图OBDD则是该方面的有益探索和结果。

OBDD是一种新的数据结构、一种新型抽象数据类型,是迄今为止布尔函数表述和操作中最为有效的技术之一[3-6]。

OBDD的发展起源于Lee和Akers的布尔函数的二叉决策图BDD(Binary Decision Diagram)表述[7-8]。

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Resolution and Binary Decision Diagramscannot simulate each other polynomiallyJan Friso Groote1,2Hans Zantema1,3JanFriso.Groote@cwi.nl hansz@cs.uu.nl1:CWI,P.O.Box94.079,1090GB Amsterdam,The Netherlands2:Department of Mathematics and Computing Science,Eindhoven University of TechnologyP.O.Box513,5600MB Eindhoven,The Netherlands3:Department of Computer Science,Utrecht University,P.O.Box80.089,3508TB Utrecht,The NetherlandsAbstractThere are many diﬀerent ways of proving formulas in proposition logic.Many of these caneasily be characterized as forms of resolution(e.g.[12]and[9]).Others use so-called binarydecision diagrams(BDDs)[2,10].Experimental evidence suggests that BDDs and resolutionbased techniques are fundamentally diﬀerent,in the sense that their performance can diﬀer verymuch on benchmarks[14].In this paper we conﬁrm theseﬁndings by mathematical proof.Weprovide examples that are easy for BDDs and exponentially hard for any form of resolution,andvice versa,examples that are easy for resolution and exponentially hard for BDDs.1IntroductionWe consider formulas in proposition logic:formulas consisting of proposition letters from some set P,constants t(true)and f(false)and connectives∨,∧,¬,→and↔.There are diﬀerent ways of proving the correctness of these formulas,i.e.,proving that a given formula is a tautology.In the automatic reasoning community resolution is a popular proof technique, underlying the vast majority of all proof search techniques in this area,including for instance the well known branch-and-bound based technique named after Davis-Putnam-Loveland[5] or the remarkably eﬀective methods by St˚almarck[12]and the GRASP prover[9].In the VLSI and the process analysis communities binary decision diagrams(BDDs)are popular[2,10].BDDs have caused a considerable increase of the scale of systems that can be veriﬁed,far beyond anything a resolution based method has achieved.On the other hand there are many examples where resolution based techniques out-perform BDDs with a major factor,for instance in proving safety of railway interlockings([7]).Out-performance in both directions has been described in[14].However,benchmark studies only provide an impression,saying very little about the real relation of resolution and BDDs.The results may be inﬂuenced by badly chosen variable orderings in BDDs or non optimal proof search strategies in resolution.Actually,givensuch benchmarks it can not be excluded that there exist a resolution based technique that always out-performs BDDs,provided a proper proof search strategy would be chosen.So,a mathematical comparison between the techniques is called for.This is not straightforward, as resolution and BDDs look very diﬀerent.BDDs work on arbitrary formulas,whereas resolution is strictly linked to formulas in conjunctive normal form.And the resolution rule and the BDD construction algorithms appear of a totally dissimilar nature.Moreover,classical(polynomial)complexity bounds cannot be used,as the problem we are dealing with is(co-)NP-complete.Fortunately,polynomial simulations provide an elegant way of dealing with this(see e.g.[16]).We say that proof system A polynomially simulates proof system B if for every formulaφthe size of the proof ofφin system A is smaller than a polynomial applied to the size of the proof ofφin system B.Of course,if the polynomial is more than linear,proofs in system A may still be substantial longer than proofs in system B, but at least the proofs in A are never exponentially longer.It is self evident that for practical applications it is important that the order of the polynomial is low.If it can be shown that for some formulas in B the proofs are exponentially longer than those in A we consider A as a strictly better proof system than B.It has for instance been shown that‘extended resolution’is strictly better than resolution[8],being strictly better than Davis-Putnam resolution[6]; for an extended overview of comparisons of systems based on resolution,Frege systems and Gentzen systems we refer to[16].We explicitly construct a sequence of biconditional formulas that are easy for BDDs,but exponentially hard for resolution.The proof that they are indeed hard for resolution is based on results from[15,1].The reverse is easier,namely showing that there is a class of formulas easy for any‘reasonable’form of resolution and exponentially hard for BDDs.Here the formulas are related to the pigeon hole formulas,for which we prove that the BDD approach is exponentially hard,which is of interest in itself.It was proven before in[8]that for the same formulas resolution is exponentially hard for every strategy.We start with preliminaries on OBDDs in Section2.In Section3we prove that OBDD proofs are exponential for pigeon hole formulas and related formulas.In Section4we prove that OBDD proofs are polynomial for biconditional formulas.In Section5we present our results on resolution.In Section6we present the our main results in comparing resolution and OBDDs.Finally,in Section7we describe some points of further research. Acknowledgment.Special thanks go to Oliver Kullmann and Alasdair Urquhart for their help with lower bounds for resolution.2Binary Decision DiagramsThe kind of Binary Decision Diagrams that we use presupposes a total ordering<on P,and therefore are also called Ordered Binary Decision Diagrams(OBDDs).First we present some basic deﬁnitions and properties as they are found in e.g.[2,10].An OBDD is a Directed Acyclic Graph(DAG)where each node is labeled by a proposition letter from P,except for nodes that are labeled by0and1.From every node labeled by a proposition letter,there are two outgoing edges,labeled‘left’and‘right’,to nodes labeled by0or1,or a proposition letter strictly higher in the ordering>.The nodes labeled by0and1do not have outgoing edges.An OBDD compactly represents which valuations are valid,and which are not.Given a valuationσand an OBDD B,theσwalk of B is determined by starting at the root of theDAG,and iteratively following the left edge ifσvalidates the label of the current node,and otherwise taking the right edge.If0is reached by aσ-walk then B makesσinvalid,and if1 is reached then B makesσvalid.We say that an OBDD represents a formula if the formula and the OBDD validate exactly the same valuations.An OBDD is called reduced if the following two requirements are satisﬁed.1.For no node its left and right edge go to the same node.It is straightforward to seethat a node with such a property can be removed.We call this the eliminate operation.2.There are no two nodes with the same label of which the left edges go to the same node,and the right edges go to the same node.If this is the case these nodes can be taken together,which we call the merge operation.Applying the merge and the eliminate operator to obtain a reduced OBDD can be done in linear time.Reduced OBDDs have the following very nice property.Lemma2.1.For aﬁxed order<on P,every propositional formulaφis uniquely represented by a reduced OBDD B(φ,<),andφand psi are equivalent if and only if B(φ,<)=B(ψ,<). As a consequence,a propositional formulaφis a contradiction if and only if B(φ,<)=0, and it is a tautology if and only if B(φ,<)=1.Hence by computing B(φ,<)for any suitable order<we can establish whetherφis a contradiction,orφis a tautology,orφis satisﬁable. If the order<isﬁxed we shortly write B(φ)instead of B(φ,<).We write#(B(φ))for the number of internal nodes in B(φ).The main ingredient for the computation of B(φ)is the apply-operation:given the reduced OBBDs B(φ)and B(ψ)for formulasφandψand a binary connective⋄∈{∨,∧,→,↔}as parameters,the apply-operation computes B(φ⋄ψ).For the usual implementation of apply as described in[2,10]both time and space complexity are O(#(B(φ))∗#(B(ψ))).If B(φ)is known then B(¬φ)is computed in linear time simply by replacing every0by1and vice versa; this computation is considered as a particular case of an apply-operation.Now for everyφits reduced OBDD B(φ)can be computed by recursively calling the apply-operation.As the basis of this recursion we need the reduced OBDDs for the single proposition letters.These are simple:the reduced OBDD for p consists of a node labeled by p,having a left outgoing edge to0and a right outgoing edge to1.By maintaining a hash-table for all sub-formulas it can be avoided that for multiple occurrences of sub-formulas the reduced OBDD is computed more than once.By the OBDD proof of a formulaφwe mean the recursive computation of B(φ)using the apply-operation as described above.Ifφconsists of n boolean connectives then this proof consists of exactly n calls of the apply-operation.However,by the expansion of sizes of the arguments of apply this computation can be of exponential complexity,even if it ends in B(φ)=0.As the satisﬁability problem is NP-complete,this is expected to be unavoidable for every way to compute B(φ).We give an explicit construction of formulas for which we prove that the OBDD proofs are of exponential size,independently of the order<on P.In[3] it was proved that representing the middle bits of a binary multiplier requires an exponential OBDD;this function is easily represented by a small circuit,but not by a small formula,and hence does not serve for our goal of having a small formula with an exponential OBDD proof.3Pigeon hole formulasIn this section we prove lower bounds for OBDD proofs for pigeon hole formulas and related formulas.Deﬁnition 3.1.Let m,n be positive integers and let p ij be distinct variables for i =1,...,m and j =1,...,n .LetC m,n =m i =1(n j =1p ij ),R m,n =n j =1(m i =1p ij ),R m,n = j =1,...,n,1≤i<k ≤m (¬p ij ∨¬p kj ),CR m,n =C m,n ∧R m,n PF m,n =C m,n ∧R m,n .In order to understand these formulas put the variables in a matrix according to the indexes.The formula C m,n states that in every of the m columns at least one variable is true,the formula R m,n states that in every of the n rows at least one variable is true,and the formula R m,n states that in every of the n rows at most one variable is true.Hence if C m,n holds then at least m of the variables p ij are true and if R m,n holds then at most n of the variables p ij are true.Hence if m >n then PF m,n is a contradiction.Since this reasoning describes the well-known pigeon hole principle,the formulas PF m,n are called pigeon hole formulas.Note that PF m,n is in conjunctive normal form.In [8]it has been proved that for every resolution proof for PF n +1,n the length is at least exponential in n .Here we prove a similar exponential lower bound for OBDD proofs,which is of interest in itself since pigeon hole formulas are widely considered as benchmark formulas.For the main result of the paper however we get better results by using similar lower bounds for CR m,n instead since the size of CR n,n is quadratic in n while pigeon hole formulas have cubic sizes.The contradictory formula in the main result is p ∧(¬p ∧CR n,n ).Our proof of these lower bounds has been inspired by the proof from [14]that every OBDD for CR n,n has a size that is exponential in √n ,which we improve to a size that is exponential in n .First we need two lemmas.Lemma 3.2.Let φbe a formula over variables in any ﬁnite set P .Let <be a total order on P .Let k <#P .Write I B ={0,1}.Let f φ:I B #P →I B the function representing φ,in such a way that the smallest k elements of P with respect to <correspond to the ﬁrst k arguments of f φ.Let A ⊆{1,...,k }.Let z ∈I B k .Assume that for every distinct x , x ′∈I B k satisfying x i =x ′i =z i for all i ∈A there existsy ∈I B #P−k such that f φ( x , y )=f φ( x ′, y ).Then #B (φ,<)≥2#A .Proof.There are 2#A diﬀerent ways to choose x ∈I B k satisfying x i =z i for all i ∈A .Now from the assumption it is clear that by ﬁxing the ﬁrst k arguments of f φ,at least 2#A diﬀerent functions in the remaining #P −k arguments are obtained.All of these functions correspond to diﬀerent nodes in the reduced OBDD B (φ,<),proving the lemma.2Lemma 3.3.Let m,n ≥1.Consider a matrix of n rows and m columns.Let the matrix entries be colored equally white and black,i.e.,the diﬀerence between the number of whiteentries and the number of black entries is at most one.Then at least (m −1)√22columns or at least (n −1)√22rows contain both a black and a white entry.Proof.If all rows contain both a black and a white entry we are done,so we may assume that at least one row consists of entries of the same color.By symmetry we may assume all entries of this row are white.If also a row exists with only black entries,then all columns contain both a black and a white entry and we are done.Since there is a full white row,we conclude that no full black column exists.Let r be the number of full white rows and c be the number of full white columns.The number of entries in these full white rows and columns together is mr +cn −cr ,and the total number of white entries is at most mn +12,hencemn +12≥mr +cn −cr =mn −(m −c )(n −r ).Assume the lemma does not hold.Then m −c <(m −1)√22and n −r <(n −1)√22,andmn +12≥mn −(m −c )(n −r )>mn −(m −1)√22∗(n −1)√22=mn −(m −1)(n −1)2from which we conclude m +n <2,contradiction.2Theorem 3.4.For m ≥n ≥1and for every total order <on P ={p ij |i =1,...,m,j =1,...,n }both time and space complexity of the OBDD proofs of both CR m,n and PF m,n is Ω(1.63n ).Proof.The last step in the OBDD proof of CR m,n is the application of apply on B (C m,n ,<)and B (R m,n ,<);the last step in the OBDD proof of PF m,n is the application of apply on B (C m,n ,<)and B (R m,n ,<).We prove that at either the OBDD B (C m,n ,<)has size at least 2(m −1)√22or both the OBDDs B (R m,n ,<)and B (R m,n ,<)have size at least 2(n −1)√22.Since m ≥n and 2√22>1.63,then the theorem immediately follows.Let P <⊂P consist of the ⌊nm 2⌋smallest elements of P with respect to <,and let P >=P \P <.hence elements of P >are greater than elements of P <.We say that row j ={p ij |i =1,...,m }is mixed if i,i ′exist such that p ij ∈P <and p i ′j ∈P >;we say that column i ={p ij |j =1,...,n }is mixed if j,j ′exist such that p ij ∈P <and p ij ′∈P >.From Lemma 3.3we conclude that either at least (n −1)√22rows are mixed or at least (m −1)√22columns are mixed.For both cases we will apply Lemma 3.2for k =⌊nm 2⌋.We number the elements of P from 1to mn such that the numbers 1,...,k correspond to the elements of P <.Assume that at least (m −1)√22columns are mixed.For every mixed column ﬁx one element of P <in that column;collect the numbers of these elements in the set A .For i =1,...,k deﬁne z i =1for i corresponding to matrix elements in non-mixed columns and z i =0for i corresponding to matrix elements in mixed columns.Choose x , x ′∈I B k satisfying x = x ′and x i =x ′i =z i for all i ∈A .Then there exists i ∈A such that x i =x ′i .Now let y =(y k +1,...,y mn )be the vector deﬁned by y j =0if j ∈P >corresponds to a matrix element in the same column as i ,and y j =1otherwise.Interpret the concatenation of x and y as an assignment to {0,1}on the matrix entries.Non-mixed columns contain only the value 1,and every mixed columns contains at least one value 1,except for one column which consists purely of zeros if and only if x i =0.Hence f C m,n ( x , y )=x i ,and similarlyf Cm,n( x′, y)=x′i.Since x i=x′i we obtain f C m,n( x, y)=f C m,n( x′, y).Now by Lemma3.2weconclude that#B(C m,n,<)≥2#A≥2(m−1)√22.For the remaining case assume that at least(n−1)√22rows are mixed.The required boundfor#B(R m,n,<)follows exactly as above by symmetry.It remains to prove the bound for #B(R m,n,<).For every mixed rowﬁx one element of P<in that row;collect all these elements in the set A.Deﬁne z i=0for all i=1,...,k.Choose x, x′∈I B k satisfying x= x′and x i=x′i=z i=0for all i∈A.Then there exists i∈A such that x i=x′i.Now deﬁney=(y k+1,...,y mn)by choosing y j=0for all but one j,and y j=1for one single j for which i and j correspond to matrix elements in the same row.This is possible because i corresponds to an entry in a mixed row.Since in every other row at most one value is set to 1all corresponding clauses in R m,n are true.The only clause in R m,n that is possibly falseis the one corresponding to i and j.We obtain f Rm,n ( x, y)=¬x i and f Rm,n( x′, y)=¬x′i.Since x i=x′i we have f m,n( x, y)=f m,n( x′, y).Now by Lemma3.2we conclude that #B(R m,n,<)≥2#A≥2(n−1)√22.2 Note that we proved that either C m,n or both R m,n and R m,n must have OBDDs of exponential size.However,for each of these formulas seperately a properly chosen order may lead to small OBDDs.Indeed,ifp ij<p i′j′⇐⇒(i<i′)∨(i=i′∧j<j′)then#B(C m,n,<)=mn and ifp ij<p i′j′⇐⇒(j<j′)∨(j=j′∧i<i′)then#B(R m,n,<)=mn and#B(R m,n,<)=2(m−1)n,all being linear in the number of variables.4Biconditional formulasAn interesting class of formulas are biconditional formulas consisting of proposition letters, biconditionals(↔)and negations(¬).Biconditionals have very nice properties:they are associative,φ↔(ψ↔χ)≡(φ↔ψ)↔χ,commutative,φ↔ψ≡ψ↔φ,idempotent,φ↔φ≡t and negation distributes over the biconditionalφ↔¬ψ≡¬(φ↔ψ).Using these properties it is easy to show that there exists for every biconditional formulaφa biconditional normal formψin which there is at most one negation,and each proposition letter occurs at most once,such thatφ≡ψ.For a string S=p1,p2,p3,...,p n of proposition letters,where letters are allowed to occur more than once,we write[S]=p1↔(p2↔(p3···(p n−1↔p n))···).It is not diﬃcult to see that[S]is a tautology if and only if all letters occur an even number of times in S.The BDD technique turns out to be very eﬀective for biconditional formulas.We show that for any biconditional formulaφits OBDD proof has a polynomial complexity.For anybiconditional formula φ,we write |φ|for the size of φ,α(φ)for the number of variables occurring in φand αodd (φ)for the number of variables that occur an odd number of times in φ.It is useful to speak about the OBDD of n formulas,φ1,...,φn .This OBDD is a single DAG with up to n root nodes.The notion reduced carries over to these OBDDs.In particular,if φi and φj are equivalent,then the i th and j th root node are the same.Again the size of a DAG is deﬁned to be the number of its internal nodes.We have the following lemma,showing that each reduced OBDD for a biconditional formula is small.Lemma 4.1.Let φbe a biconditional formula.Any reduced OBDD for φand ¬φhas size 2αodd (φ).Proof.First ﬁx an arbitrary ordering <on the proposition letters.Note that there is a biconditional normal form ψthat is equivalent to φ.As by Lemma 2.1the reduced OBDD of φand ψare the same,we can as well construct the OBDD of ψ.Moreover,αodd (φ)=αodd (ψ).We prove this theorem by induction on αodd (ψ).•αodd (ψ)=0.As ψis a biconditional normal form,it does not contain any proposition letter,and hence is either equivalent to true or false.So,the reduced OBDD of φand ¬φdoes not contain internal nodes at all,and has size 0.•α(ψ)odd =n +1.Consider the ﬁrst letter in the ordering <that occurs in ψand let it be p .The OBDDs for ψand ¬ψlook like:•p B ψ ©d d d •t t t B ψ[1/p ]•t t t B ψ[0/p ]•pB ¬ψ ©d d d •t t t B ¬ψ[1/p ]•t t t B ¬ψ[0/p ]Here ψ[v/p ]is the formula ψwhere v has been substituted for p .Clearly,as p occurs an odd time in ψ,ψ[0/p ]≡¬ψ[1/p ]and ψ[1/p ]≡¬ψ[0/p ].So,the reduced OBDD of ψ[0/p ],¬ψ[1/p ],ψ[1/p ]and ¬ψ[0/p ]is the same as the OBDD of ψ[0/p ]and ¬ψ[0/p ].Using the induction hypothesis,the size of this OBDD must be 2n .The reduced OBDD for ψand ¬ψadds two new nodes.So,the size of the reduced OBDD of ψand ¬ψis 2n +2.This equals 2αodd (ψ)+2,ﬁnishing the proof.2Theorem 4.2.Let <be an ordering on the proposition letters.•The complexity of the corresponding OBDD proof for any biconditional formula φis O (|φ|3).•The complexity of the corresponding OBDD proof for [S ]or ¬[S ]for any string S of proposition letters is O (|S |2).Proof.The OBDD proof forφconsists of O(|φ|)applications of apply applied on reduced OBDDs of sub-formulas ofφ.By Lemma4.1each of these reduced OBDDs has size O(|φ|).Since the complexity of apply(↔,B,B′)is O(#B∗#B′)and the complexity of apply(¬,B) is O(#B)for every apply operation the complexity is O(|φ|2),yielding O(|φ|3)for the fullOBDD proof forφ.For the OBDD proof for[S]or¬[S]only applications of apply(↔,B,B′)occur with #B=1,giving the complexity O(#B′),yielding O(|S|2)for the full OBDD proof.25ResolutionResolution is a very common technique to prove formulas.Contrary to the BDD technique, it is applied to formulas in conjunctive normal form(CNF),i.e.formulas of the formi∈I j∈J il ijwhere I and J i areﬁnite index sets and l ij is a literal,i.e.a formula of the form p or¬p for a proposition letter p.Each sub-formula j∈J i l ij is called a clause.As∧and∨are associative, commutative and idempotent it is allowed and convenient to view clauses as sets of literals and CNFs as sets of clauses.The resolution rule can be formulated by:{p,l1,...,l n}{¬p,l′1,...,l′n′}{l1,...,l n,l′1,...,l′n′}where p is a proposition letter and l i,l′j are literals.A proof of a set of clauses F is a sequence of clauses where the last clause is empty and each clause in the sequence is either taken from F,or matches the conclusion of the resolution rule,where both premises occur earlier in the sequence.In case one of the clauses involved is a single literal l,by this resolution rule all occurrences of the negation of l in all other clauses may be removed.Moreover,all other clauses containing l then may be ignored.Eliminating all occurrences of l and its negation in this way is called unit resolution.We call a resolution proof search system reasonable if it starts with doing unit resolution as long as there is a clause consisting of a single literal.All practical resolution proof search systems are reasonable.In order to apply resolution on arbitrary formulas,these formulas mustﬁrst be translated to CNF.This can be done in linear time maintaining satisﬁability using the Tseitin transfor-mation[13].A disadvantage of this transformation is the introduction of new variables,but it is well-known that a transformation to CNF without the introduction of new variables is necessarily exponential.For instance,it is not diﬃcult to prove that for(···((p1↔p2)↔p3)······↔p n)every clause in a CNF contains either p i or¬p i for every i.Since one such clause of n literals causes only one zero in the truth table of the formula,the full CNF contains2n−1 of these clauses to obtain all2n−1zeros in its truth table.Hence without the introduction of new variables every CNF of this formula is of exponential size.More general for every biconditional formulaφwithout the introduction of new variables every CNF consists of at least2αodd(φ)−1clauses each consisting of at leastαodd(φ)literals.The Tseitin transformation works as follows.Given a formulaφ.Every sub-formulaψof φnot being a proposition letter is assigned a new letter pψ.Now the Tseitin transformation ofφconsists of•the single literal pφ;•the conjunctive normal form of pψ↔(pψ1⋄pψ2)for every subtermψofφof the shape ψ=ψ1⋄ψ2for a binary operator⋄;•the conjunctive normal form of pψ↔¬pψ1for every subtermψofφof the shape ψ=¬ψ1.It is easy to see that this set of clauses is satisﬁable if and only ifφis satisﬁable.Moreover, every clause consists of at most three literals,and the number of clauses is linear in the size of the original formulaφ.It is not diﬃcult to see that after applying the Tseitin transformation to a CNF,by a number of resolution steps linear in the size of the CNF,the original CNF can be re-obtained. By a resolution proof for an arbitrary formula we mean a resolution proof after the Tseitin transformation has been applied.We now give a construction of strings S n in which all letters occur exactly twice by which ¬[S n]is a contradiction,and for which we prove that every resolution proof of¬[S n]is very long.Although the construction is somewhat involved,we think that simpler constructions do not suﬃce.In[16]for instance it was proved that¬[p1,p2,...,p n,p1,p2,...,p n]admits a resolution proof that is quadratic in n.For a string S and a label i we write lab(S,i)for the string obtained from S by replacing every symbol p by a fresh symbol p i.For a string S of length n∗2n we write ins(n,S)for the string obtained from S by inserting the symbol i after the(i∗n)-th symbol for i=1,2,...,n. We deﬁneS1=1,1,andS n+1=ins(n,lab(S n,0)),ins(n,lab(S n,1)),for n>0.For instance,we haveS1=1,1 ,S2=10,1,10,2 ,11,1 ,11,2 ,S3=100,10,1,101,21,2,111,11,3.,111,21,4,101,11,1,100,20,2,110,10,3,110,20,4Clearly S n is a string of length n∗2n over n∗2n−1symbols each occurring exactly twice.The string S n can be considered to consist of2n consecutive groups of n symbols,called n-groups. In the examples S1,S2and S3above the n-groups are under-braced.Write g n,k to be the k-th n-group in S n,for n>1and1≤k≤2n.Lemma5.1.Let A⊆{1,2,...,2n}for any n>0.Then there are at least min(#A,2n−#A) pairs(k,k′)such that k,k′∈{1,2,...,2n},k∈A,k′∈A and g n,k and g n,k′have a common symbol.Proof.We apply induction on n;for n=1the lemma clearly holds.Let m0=#{k∈A|k≤2n−1}and m1=#{k∈A|k>2n−1}.Say that(k,k′)is a matching pair if k∈A, k′∈A and g n,k and g n,k′have a common symbol.If k,k′≤2n−1then by construction g n,k and g n,k′have a common symbol if g n−1,k and g n−1,k′have a common symbol.If k,k′>2n−1 then by construction g n,k and g n,k′have a common symbol if g n−1,k−2n−1and g n−1,k′−2n−1 have a common symbol.Hence by induction hypothesis there are at least min(m0,2n−1−m0) matching pairs(k,k′)with k,k′≤2n−1and at least min(m1,2n−1−m1)matching pairs(k,k′) with k,k′>2n−1.Since by construction g n,k and g n,k+2n−1have a common symbol for every k=1,2,...,2n−1,there are at least|m0−m1|matching pairs(k,k′)with|k−k′|=2n−1. Hence the total number of matching pairs is at least|m0−m1|+min(m0,2n−1−m0)+min(m1,2n−1−m1).A simple case analysis shows that this is at least min(m0+m1,2n−m0−m1)=min(#A,2n−#A).2 Essentially this lemma states the well-known fact that for any set A of vertices of an n-dimensional cube there are at least min(#A,2n−#A)edges for which one end is in A and the other is not.It is applied in the next lemma stating a lower bound on connections between separate elements of S n rather than connections between n-groups.Lemma5.2.Let n>0and let B⊆{1,2,...,n∗2n}.Let X⊆{1,2,...,n∗2n}2consist of the pairs(i,j)for which i∈B and j∈B and for which either|i−j|=1or the i-th element of S n is equal to the j-th element of S n.Then#X≥min(#B,n∗2n−#B)2n.Proof.Assume that#B≤n∗2n−1,otherwise replace B by its complement.Let A be the set of numbers k∈{1,...,2n}for which all elements of the corresponding n-group g n,k correspond to elements of B,i.e.,{(k−1)∗n+1,...,k∗n}⊆B.Let m1=#A.Let m2 be the number of n-groups for which none of the elements correspond to elements of B,i.e., m2=#{k∈{1,...,2n}|{(k−1)∗n+1,...,k∗n}∩B=∅}.Let m3be the number of remaining n-groups,i.e.,n-groups containing elements corresponding to both elements of B and outside B.Clearly n∗m1≤#B≤n∗(m1+m3).Each of the m3remaining groups gives rise to a pair(i,j)∈X for which|i−j|=1.Hence#X≥m3.Now assume that m1>m3.Since n∗m1≤#B≤n∗2n−1we have m1=#A≤2n−1. By Lemma5.1we obtain at least m1pairs(k,k′)such that k∈A,k′∈A and g n,k and g n,k′have a common symbol.For at least m1−m3of the corresponding n-groups g n,k′none of the elements correspond to elements of B.Since g n,k and g n,k′have a common symbol for every corresponding pair(k,k′)this gives rise to at least m1−m3pairs(i,j)∈X for which the i-th element of S n is equal to the j-th element of S n.Hence in case m1>m3we conclude #X≥m3+(m1−m3)=m1.We conclude#X≥max(m3,m1)≥m1+m32≥#B2n.2Theorem5.3.Every resolution proof of¬[S n]contains2Ω(2n/n)resolution steps.。