Terminating
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Suggested steps for preparing a termination letter for summary dismissalBefore making a definite decision to summarily dismiss an employee because of serious misconduct, you may wish to follow the steps below.Please note that following the steps set out below may not stop a complaint or c laim being made against you about the termination of the employee’s employment (eg. unfair dismissal, discrimination).If you are a small business it is very important that you ensure that you have complied with the Small Business Fair Dismissal Code befo re you terminate an employee’s employment. You can access the code here: Small Business Fair Dismissal Code.More about unfair dismissal: About Unfair Dismissal.Step 1: Identify the serious misconduct and consider seeking legal adviceIdentify exactly what the employee did that was unacceptable and the impact the misconduct had on the business. It’s a severe step to dismiss an employee without notice of termination, so you may wish to seek legal advice about the matter before taking action.Step 2: Discuss the misconduct with the employeeSpeak to the employee directly about the conduct to avoid making assumptions about the situation.Explain your intention in relation to continuing their employment.Note: In any formal meeting, the employee should be allowed to be represented or accompanied by a support person. If you are considering not allowing the person to have a representative or support person it is strongly recommended that you seek legal advice before doing so.You should keep a record of any meetings that you have with employees about their conduct.Step 3: Consider your optionsDepending on the severity of the misconduct, you may choose to:▪give the employee a written warning▪dismiss the person with notice or pay in lieu of notice, or▪summarily dismiss the person.You may wish to seek legal advice about the matter before taking any action.Step 4: Draft termination of employment letterIf the employee’s action was serious misconduct that warrants summary dismissal, draft an appropriate termination of employment letter.Include in the letter:▪the reasons for the termination of the employment, and▪when the employment will end.Step 5: Provide employee with the letter of termination of employmentThe reasons for the termination of employment should be explained to the employee and they should be provided with an opportunity to ask questions.It’s important to explain the information in the letter of termination of employment and make sure that the employee understands.You should keep a copy of the letter for your records.This template has been provided by the Fair Work Ombudsman (FWO) as part of its function to provide education, assistance and advice (but not legal or professional service advice). The FWO does not provide this information for any other purpose. You are not entitled to rely upon this information as a basis for any action that may expose you to a legal liability, injury, loss or damage. Rather, it is strongly recommended that you obtain your own independent legal advice or other professional service or expert assistance relevant to your particular circumstances.[Print on your business letterhead][Date]Private and confidential[Insert employee’s full name][Insert employee’s residential address]Dear [insert name]Termination of your employmentI am writing to you about the termination of your employment with [insertcompany/partnership/sole trader name and the trading name of business].I refer to our meeting on [insert date] which was attended by you and [insert name of others at the meeting]. During the meeting we discussed [insert details of serious misconduct].This meeting was attended by you and [insert names of people at the meeting] and we spoke about [insert details of the serious misconduct incident, including the date it occurred].As discussed during the meeting, your conduct during that incident: [delete points not applicable or add others if you believe they warrant summary dismissal. Seek legal advice if you are unsure if the actions warrant termination of employment without notice.]▪was wilful or deliberate behaviour by you that is inconsistent with the continuation of your contract of employment.▪caused a serious and imminent risk to the health or safety of a person.▪caused a serious and imminent risk to the reputation, viability or profitability of the Employer's business in that [insert details].▪was conduct in the course of your employment engaging in theft, and in the circumstances your continued period of employment during a notice period would be unreasonable.▪was conduct in the course of your employment engaging in fraud, and in the circumstances your continued period of employment during a notice period would be unreasonable.▪was conduct in the course of your employment engaging in assault and in the circumstances your continued period of employment during a notice period would be unreasonable.▪you were intoxicated at work, to the extent that you were so impaired that you were unfit to be entrusted with your employment duties.▪you refused to carry out a lawful and reasonable instruction that was consistent with your contract of employment, and in the circumstances your continued period ofemployment during a notice period would be unreasonable.We consider that your actions constitute serious misconduct warranting summary dismissal.You will be paid any accrued entitlements and outstanding remuneration, including superannuation, up to and including the date of this letter.Employees and employers may seek information about minimum terms and conditions of employment from the Fair Work Ombudsman. If you wish to contact them you can call 13 13 94 or visit their website at .au.Yours sincerely[Insert name][Insert position]。
Alkaline batteries:碱性电池Capacitor batteries:电容电池Carbon zinc batteries:碳锌电池Lead acid batteries:铅酸电池Lead calcium batteries:铅钙电池Lithium batteries:锂电池Lithium ion batteries:锂离子电池Lithium polymer batteries:锂聚合物电池Nickel cadmium batteries:镍镉电池Nickel iron batteries:镍铁电池Nickel metal hydride batteries:金属氧化物镍氢电池/镍氢电池Nickel zinc batteries:镍锌电池Primary batteries:原电池Rechargeable batteries:充电电池Sealed lead acid batteries:密封铅酸电池Silver cadmium batteries:银钙电池Silver oxide batteries:银氧化物电池Silver zinc batteries:银锌电池Zinc chloride batteries:银氯化物电池Zinc air batteries:锌空电池Environmental Protection batteries:环保电池Lithium batteries:锂电池Lithium ion batteries:锂离子电池Lithium polymer batteries:锂聚合物电池铅酸蓄电池Lead-acid battery起动铅酸电池Lead-acid starter batteries摩托车用铅酸电池Lead-acid batteries for motorcycles内燃机车用铅酸电池Lead-acid batteries for disel locomotive电动道路车辆用铅酸电池Lead-acid batteries for electric road vehicles 小型阀控密封式铅酸电池small-sized valve-regulated lead-acid batteries航空用铅酸电池Aircraft lead-acid batteries固定型阀控密封式铅酸蓄电池Lead-acid batteries for stationary valve-regulated铅酸电池用极板plate for lead-acid battery铅锭lead ingots牵引用铅酸电池Lead-acid traction batteies电解液激活蓄电池electrolyte activated batteryvent valve排气阀filling device for pleral cells电池组填充装置negative electrode负电极negative plate负极板addition reagent for negative plate负极板添加剂indicator 指示器top cover上盖vent plug液孔塞expanded grid扩展式板栅specific gravity indicator比重指示器electrolyte level control pipe电解液液面控制管electrolyte level indicator电解液液面指示器electrolyte level sensor电解液液面传感器hard rubber container硬橡胶槽envelope separator包状隔板woven cloth tube纺布管spongy lead海绵状铅partition 隔壁over the partition type越过隔壁型through the partition type贯通隔壁贯通型separator 隔板(1)battery rack(2)battery stand(3)battery stillage蓄电池架/蓄电池底垫active material活性物质glass fiber separator玻璃纤维隔板glass mat玻璃纤维绵glass mat tube玻璃纤维绵管spacing washer间隔垫圈reinforced fiber separator强化纤维隔板polarity mark plate极性标记板pole 极柱pole insulator极柱绝缘子pole nut极柱螺母plate 极板plate foot极板足plate supporter极板支撑件element 极板群/极群组pole bolt极柱螺栓plate lug极板耳dilute sulfuric acid稀硫酸steel can金属罐steel container金属蓄电池槽(1)madribs(2)element rest鞍子/极群组座tubular plate管状极板gelled electrolyte胶体电解液grid 板栅caution label警告标签synthetic resin separator合成树脂隔板plastics container塑料蓄电池槽synthetic fiber separator合成纤维隔板connector sunken type沉没型连接器connetor exposed type露出型连接器safety valve test安全阀测试ampere-hour efficency安时效率one charge distance range一次充电行程gas recombination on negative electrode type阴极气体再化合型/阴极气体复合型cut-off discharge终止放电/截止放电(1)specific characteristic (2)energy density (1)比特性(2)能量密度recovering charge恢复充电(1)open circuit voltage(2)off-load voltage开路电压/空载电压overcharge 过充电gassing 析气overcharge life test过充电寿命试验accelerated life test加速寿命试验active material utilization活性物质利用率theoretical capacity of active material活性物质的理论容量over discharge过放电intermittent discharge间歇放电full charge完全充电full discharge完全放电reverse charge反充电/反向充电quick charge快速放电allowable minimum voltage允许最小电压equalizing charge均衡充电creeping 蠕变group voltage组电压shallow cycle endurance轻负荷寿命/轻负荷循环寿命characteristic of electrolyte decrease电解液减少特性nominal voltage标称电压high rate discharge高率放电high rate discharge characteristic高率放电特性5 second voltage at discharge放电5秒电压(1)cold cranking ampere(2)cold cranking performance(1)冷启动电流(2)冷启动性能cycle life test循环寿命测试maximum voltage at discharge最大放电电压30 second voltage at discharge放电30秒电压residual capacity残存容量(1)hour rate(2) discharge rate(1)小时率(2)放电率(1) self discharge (2) local action(1)自放电(2)局部自放电(1) self discharge rate(2) local action rate(1)自放电率(2)局部自放电率actual capacity实际容量(1)starting capability(2)cranking ability启动能力cranking current启动电流battery clamp test电池夹钳测试power density功率密度momentary discharge瞬间放电modified constant voltage charge修正恒定电压充电initial capacity初始容量gas recombination by catalyser type触媒气体复合式initialcharge 初始充电viberation test振动试验predetermined voltage预定电压total voltage总电压activation test for dry charged battery干式荷电蓄电池活化试验salting 盐析earthquake-proof characteristics防震性能dielectric voltage withstand test电介质耐压试验short time discharge短时间放电escaped acid mist test酸雾逸出测试terminal voltage端子电压cell voltage单电池电压step charge阶段充电short-circuit current短路电流storage test保存测试high rate discharge at low temperature低温高率放电rated voltage额定电压rated capacity额定容量fixed resistance discharge定阻抗放电constant voltage charge恒压充电constant voltage life test恒压寿命测试constant current charge恒流充电constant voltage constant current charge恒流恒压充电constant current discharge恒流放电constant watt discharge恒功率放电low rate discharge characteristics低率放电特征trickle charge涓流充电trickle charge current涓流充电电流trickle charge life test涓流充电寿命测试thermal runaway热失控driving pattern test运行测试capacity in driving pattern test运行测试boost charge急充电floating charge浮充电floating charge voltage浮充电电压floating charge current浮充电电流(1)mean voltage (2)average voltage平均电压on-load voltage负载电压discharge duration time放电持续时间(1)final voltage(2)cut-off voltage(3)end voltage终止电压/截止电压depth of discharge放电深度discharge voltage放电电压discharge current放电电流discharge current density放电电流密度discharge watt-hour放电瓦时discharge characteristics放电特性discharged ampere-hour放电安时explosion proof test防爆测试auxiliary charge补充电maintenance factor维护率storage characteristics保存特性gas recombinating efficiency气体复合效率/气体再化合效率charge 充电charge acceptance test充电可接受性试验start-of-charge current充电开始电流charge efficiency充电效率end-of-charge voltage充电结束电压specific gravity of electrolyte at the end of charge充电结束时电解液比重charge voltage充电电压charge current充电电流charged watt-hour充电瓦时charge characteristic充电特性charge ampere-hour充电安时deep cycle endurance重负荷循环寿命/重复合寿命weight engergy density重量能量密度rubber pad橡胶垫lower level line下液面线side terminal侧端子collective exhaust unit公共的排放单元sintered plaque烧结极板sintered separator烧结隔板sintered plate烧结极板catalyst plug催化塞spine 芯骨strap 带spacer 隔离物insulating tube绝缘管intercell connector连接线/连接条connector cover连接管盖float mounted plug浮动安装的栓(1)pasted plate (2)grid type plate涂膏式极板braidd tube编织管(1)flame-arrester vent plug (2)flam-retardant vent plug安全塞explosion and splash proof construction防爆防溅结构baffle 保护板pocket type plate袋式极板bottom hole-down底孔向下(固定)bolt fastening terminal螺栓连接端子male blade阳片monoblock container整体槽positive electrode正极positive plate正极板leading wire terminal引线端子retainer mat止动垫片ribbed separator肋隔板(1)jumping wire (2)inter low wire跳线end plate端板filling plug注液塞plante plate形成式极板/普朗特极板tubular plate管式极板low electric resistance separator低电阻隔板tapered terminal post锥形接线柱electrolyte 电解液container 蓄电池槽/蓄电池壳set of container成套蓄电池槽level-scope mounted plug透视塞/透视栓handle 手柄jug 取液管(1)connector;(2)plug concent (1)连接器;(2)插座式连接器connector wire连接线connecting bar连杆connecting bar cover连杆帽lead 引线/连接线edge insulator绝缘卡side frame侧框架battery cubicle蓄电池箱perforated separator多孔隔板burning rod (铅)焊条terminal 端子terminal connector端子连接条terminal cover端子盖terminal base端子座tab 接线片lead bushing铅套corrugated separator波形隔板(1)lead dioxide;(2)lead peroxide (1)二氧化铅;(2)过氧化铅(1)woven separator;(2)nonwoven separator (1)织物隔板;(2)非织物隔板vent hole通气孔exhaust tube排气管antipolar mass反极性物质output cable输出电缆microporous rubber separator微孔像胶隔板specific gravity indicator比重计leaf separator叶片式隔板lid sealing compound密封剂/封口剂sealing gasket密封衬垫/垫圈lid 蓄电池盖set of lid系列的盖方通盖板cover board底板solepiece钢珠steel ball压钢珠press steel ball防爆阀valve preventing explosion大电流(倍率)放电discharge in highrate current标称电压Normal voltage标称容量normal capacity放电容量discharge capacity充电上限电压limited voltage in charge放电下限电压terminating voltage in discharge 恒流充电constant current charge恒压充电constant voltage charge恒流放电constant current discharge放电曲线discharge curve充电曲线charge curve放电平台discharge voltage plateau容量衰减capacity attenuation起始容量initial discharge capacity流水线pipelining传送带carrying tape焊极耳welding the current collector卷绕wind叠片layer贴胶带stick tape点焊spot welding超声焊ultrasonic weldingThe terminating voltage in discharge of the battery is 3.0 volt.The limited voltage in charge of the battery is 4.2 volt.三元素Nickle-Cobalt-Manganese Lithium Oxidethree elements materials钴酸锂Cobalt Lithium Oxide锰酸锂Manganese Lithium Oxide石墨graphite烘箱oven真空烘箱vacuum oven搅拌机mixing devicevacuum mixing device涂布机coating equipment裁纸刀paper knife ,,,,,,cutting knife分条机equipment for cutting big piece to much pieces辊压机roll press equipment电阻点焊机spot welding machine超声点焊机ultrasonic spot welding machine卷绕机winder自动叠片机auto laminating machine激光焊机laser welding machine注液机infusing machine真空注液机vacuum infusion machine预充柜pre-charge equipment化成柜formation systems分容柜grading systems测试柜testing systems内阻仪Battery inner resistance tester万用表Multimeter转盘式真空封口机Turntable type vacuum sealing machine自动冲膜机Automatic aluminum membrane shaper序号首字母英文中文1 A aging老化2 B battery charger充电器3 black-fleck黑斑4 C cap盖板5 capacity density能量密度6 capacity grading分容7 cathode tab welding极耳超焊8cell 电芯9 charge(capacity) retention荷电(容量)保持10 checking code检码11 concave spot凹点12 constant current charge恒流充电13 constant current discharge恒流放电14 constant voltage charge恒压充电15 corrective measures纠正措施16 crack裂纹17 cut-off voltage终止电压18 cycle life 循环寿命19 D dark trace暗痕20 degrade降级21 dent凹痕22 discharge depth放电深度23 distortion变形24 drape打折25 E Electrical and Mechanical Services Department机电部26 electrolyte电解,电解液27 empaistic压纹28 end-off voltage放电截止电压29 environmentally friendly对环境友好30 equipment first inspection设备首检31 erode腐蚀32 explosion-proof line防爆线33 F first inspection首检34 formation化成35 fracture断裂36 I inspection检验37 insulate绝缘38 internal resistance内阻39 J jellyroll卷芯40 joint接缝,结合点41 L laser deflecting偏光42laser reticle激光刻线43 laser welding-flatwise weld激光焊接-平焊laser welding-standing weld激光焊接-立焊44 leakage漏液45 leak-checking测漏46 leaving out of welding漏焊47 limited charge voltage充电限制电压48 local action自放电49 M margin turnly翘边50 measuring the dimension of cells电芯卡尺寸51 meet requirement达到要求52 memory effects记忆效应53 N nick划痕54 nominal voltage标称电压55 notice-board confirmation看板确认56 nugget硬块57 O obverse正面58 open circuit voltage开路电压59 over charge过充60 over discharge过放61 over the thickness超厚62 P particle颗粒63 PE membrane PE膜64 pit坑点65 placing cells into the box电芯装盒66 point inspection点检67 preventive measures预防措施68 pricking the tapes扎孔69 process inspection制程检验70 put the battery piled up将电芯叠放在一起71 Q qualified products合格品72 quality assurance质量保证73 quality control质量控制74 quality improvement质量改进75 quality match品质配对76 quality planning质量策划77 R rated capacity额定容量78 recharge再充电79 refitting the can of cell电芯壳口整形80 requirement要求81 reverse背面,反面82 rework返工83ringing cells into pyrocondensation films套热缩膜84 S safety vent安全阀85 sand aperture砂眼86 scar疤痕87 secondary battery二次电池88 select appearance选外观sharp-set 批锋89 short circuit checking测短路90 smudginess污物91 spot welding by laser激光点焊92 spot welding place 点焊位置93 spraying the code喷码94 spur毛刺95 sticking the PVC cover boards贴面垫96 storing陈化97 storing with high voltage高压储存98 T tabs deflection极耳歪斜99 tabs excursion极耳错位100 technics requirement工艺要求101 U ultrasonic welding超声波焊接102 ultrasonic welding strength超焊强度103 unqualified products不合格品104 W wave波浪105 working procedure工序Voltage:Units of measuring electrical current, all batteries are rated in volts DC. (Direct Current). This determines how much energy is needed to power your equipment. Voltage plateau:(电压平台)A slow decrease in voltage over a long period of time. As a rule, the plateau extends from the first voltage drop at the start of the discharge to the bend of the curve after which the voltage drops rapidly at the end.Nominal Voltage(标称电压)The voltage of a battery, as specified by the manufacturer, discharging at a specified rate and temperature.Working voltage(工作电压)The working voltage of a cell or battery begins at its electrical connections as soon as an electrical consumer is connected to it.Discharging voltage, average voltage (放电电压)The average discharging voltage is the average value of the discharging voltage during the entire discharging process with a related discharging current. Open circuit voltage (OCV开路电压)The voltage of a battery when there is no current flowing.Closed-Circuit Voltage (CCV闭路电压)The potential or voltage of a battery when it is discharging or charging. State of charge:The rate of charge capacity vs. whole capacity.Initial voltage(起始电压)A battery's initial voltage is the working voltage when discharging begins. End-point voltage (End voltage, Cutoff voltage, Final voltage)截止电压Specified closed circuit voltage at which a service output test is terminated. End-of-discharge voltageThe battery voltage when discharge is terminated.End-of-charge voltageThe battery voltage when charge is terminated.Cutoff voltage (V)The battery voltage at which charge or discharge is terminated.Definition: Capacity(容量)The capacity of a cell is defined as how many mill-amp-hours (mAh) of current the cell can store and subsequently deliver.One mill-amp (mA) is 1/1000th of an Amp. Some larger cell capacities are expressed in Amp-hours(Ah).“Rated capacity” is varies with discharge rate, temperature, and cutoff voltage.Rated capacity is different from power or energyExample:If a cell is rated at 1000 mAh, then it can deliver the following:1000 mA of current for 1 hour500 mA of current for 2 hours200 mA of current for 5 hours2000mA of current for 1/2 hourDefinition: Energy Density(能量密度,包括体积比能量和质量比能量)The energy density of a cell is a measure of how much energy can be stored in the cell per unit volume or per unit weight.E (watt-hours) = cell voltage x capacity ratingEnergy density per unit volume is called the “volumetric energy density” and is expressed in terms of watt-hours/liter (wh/l).Energy density per unit weight is called the “gravimetric energy density” and is expressed in terms of watt-hours/kilogram (wh/kg).These measurements are useful when you are trying to determine which cell has the most capacity per unit volume or weight.1.Self Discharge自放电2.Uniformity of the Li-ion Batteries锂离子电池的一致性3.steel strap钢带4.Burst vent防爆阀5.Filling port注液孔6.spirally wound type cylindrical wound type圆柱形7.foil 箔8.parallel-plate prismatic design方形叠片式设计Ageing (老化)-Permanent loss of capacity with frequent use orThe passage of time due to unwanted irreversible chemical reactions in the cell. Anode (阳极)-The electrode in an electrochemical cell where oxidation takes place, releasing electrons.During discharge the negative electrode of the cell is the anode.During charge the situation reverses and the positive electrode of the cell is the anode.Cathode(阴极)-The electrode in an electrochemical cell where reduction takes place, gaining electrons.During discharge the positive electrode of the cell is the cathode. During charge the situation reverses andThe negative electrode of the cell is the cathode.Cycle (循环)-A single charge and discharge of a battery.Depth of discharge DOD(放电深度)-The ratio of the quantity of electricity overcharge removed from a cell on discharge to its rated capacity.Internal impedance(交流内阻)-Resistance to the flow of AC current within a cell. It takes into account the capacitive effect of the plates forming the electrodes. Internal resistance(直流内阻)-Resistance to the flow of DC electric current within a cell,causing a voltage drop across the cell in closed circuit proportional to the current drain from the cell.A low internal impedance is usually required for a high rate cell.锂离子电池的内阻英语概念到底用哪个概念,是Internal resistance还是Internal impedance,一些电池说明书内阻用Internal resistance,也有的用Internal impedance,我认为Internal impedance较好些,因为国内测的电池内阻基本都是交流内阻,而外文也有这样定义的(我在别的帖子也粘贴过):Internal impedance(交流内阻)-Resistance to the flow of AC current within a cell. It takes into account the capacitive effect of the plates forming the electrodes.Internal resistance(直流内阻)-Resistance to the flow of DCelectric current within a cell, causing a voltage drop across the cell in closed circuit proportional to the current drain from the cell.A low internal impedance is usually required for a high rate cell.在IEC6196002中,只定义为Internal resistance,而用交流的方法测得的内阻,叫Internal a.c. resistance(交流内阻)用直流的方法测得的内阻,叫Internal d.c. resistance(直流内阻),其实Internal a.c. resistance 测得就是阻抗,这样看来不如用Internal impedance(交流内阻)和Internal resistance(直流内阻)这两个概念把它们进行分清,以免混淆。
--智能--OK词根大全英语词汇测试练习英语音标termin,term = end,limit,boundary来自拉丁语的。
同义词根有来自拉丁语的fin,ultim和来自希腊语的bys等(*英文根字典)1.exterminatevt.根除(疾病、想法等);消灭,歼灭extermination n.Nuclear weapons may be used to exterminate mankind. 核武器很可能会被用来消灭人类。
2.interminableadj.interminably adv.The movie was interminable. I was so bored i actually slept!电影太长了,我们无聊得都睡着了!3.terminatevt.使停止;使结束;解雇termination n.terminus n.terminable adj.We are terminating your employment effective February 24th. 我们将于2月24日终止对你的聘用。
4.conterminousadj.有共同边界的;邻接的;(空间、时间、范围上)同样广大的 *con<com(=together)coterminous (=conterminous)5.determinev.下决心;决定;判定;确定determinable adj.determinate adj.determinative adj.determiner n.[determinist n.determinism n.6.determinedadj.determined look 坚决的表情7.determinationn.决定;(~to do)决心;决意,果断;(~of) (数量等的)计算,测定 *de(=away)his determination to learn English 他学英语的决心with an air of determination 神情坚决the determination of the amount of metal in a specimen of ore .矿石标本中金属含量的测定8.determinantn.决定物;决定因素 adj.决定的 *de(=away)9.predeterminev.预先决定;预先查明 *pre(=before),de(=away)predetermination n. 预先决定词根seism ,seismo = seism 地震词根radi-,-ray-,-rad-,-radio-词根ac,acr,acid = sour (酸的词根paleo,pale,palae = old 老10.terminaladj.末端的;终点的;学期的,期末的;(致命的疾病的)末期的 n.终端,末端;终点;终点站;[电]端子terminal accounts 终结账户terminal cancer 晚期癌症the West London Ari Terminal 西伦敦航空终点the terminal of a battery 电池的端子11.terminusn. (水陆交通线、运输管道的)终点站12.terminologyn.术语;术语学 *log(=speech)terminological adj.术语的terminological inexactitude 术语上的不精确medical /grammatical terminology 医学/语法术语13.termn.期限;期间;学期;术语;(pl.)措辞;交谊 v.命名;称为专业英语词根、词缀、词源以及词汇相关知识 !英语词源词根词缀记忆法,是最科学背单词方法!英语词根大全。
pod删除原理
Pod的删除原理主要涉及以下几个步骤:
1. 命令执行:用户通过kubectl命令行工具发出删除Pod的指令。
这个指
令会向Kubernetes API Server发起一个删除请求。
2. API Server处理:API Server接收到删除请求后,会更改Pod的状态为Terminating。
这个状态表示Pod即将被删除。
3. 监听状态更改:Kubelet会监听到Pod的状态更改,开始停止这个Pod
的过程。
如果Pod中定义了preStop hook,Kubelet会先运行这个hook。
如果hook在terminationGracePeriodSeconds之后还没运行完,Kubelet会给Pod额外2秒的时间。
4. 发送信号:当Pod状态变为Terminating之后,Kubelet会让容器运行
时发送TERM信号到每个容器中的进程。
5. 进程终止:当terminationGracePeriodSeconds过了之后,如果pod
里面的进程还没结束,它们会收到SIGKILL信号,进程将被强制终止。
6. 清理资源:Kubelet会清理pause容器,如果有的话。
同时,Kubelet
会把terminationGracePeriodSeconds改成了0,强制从API Server删除Pod。
7. 数据删除:API Server最终会从etcd数据库中删除与该Pod相关的数据,完成整个删除过程。
以上是删除Pod的大致原理,具体细节可能会因版本和配置的不同而有所差异。
英文合同常见条款表达方式1.前言Preamble一份标准英文合同通常可以分为前言(Preamble)、正文(Operative part)、附录(Schedule)及证明部分即结束词(Attestation)四大部分组成。
前言(Preamble)由“Parties”及“Recitals”两部分组成。
“Parties”为必备条款,在很多时候称为“commencement”即合同的开场白,主要介绍合同各方的名称或姓名、注册地及地址、邮编及在合同中的简称。
当然,并不是所有的合同都要详细介绍以上诸要素,在许多简单合同中,只是提一下各方的名称。
I. 以下为“Parties”的常用表达方式:1. This Agreement is entered into by and between ____ and ____.本协议由以下双方____和___ 签署。
2. This Agreement is entered into by and between ____ (hereinafter referred to as____) and ____ (hereinafter referred to as "_____"), whereby it is agreed as follows:本协议由以下双方____(以下简称____)和_____(以下简称___)签署,达成如下协议:注:在很多合同中,这部分加入签约事由,如:This Agreement is entered into through friendly negotiations between _____ Co. (hereinafter referred to as the “Party A”) and _____ Co. (hereinafter referred to as the “Party B”) based on equality and mutual benefit to develop business on the terms and conditions set forth below:本协议由_____(以下称为甲方)和____(以下称为乙方)为发展业务在平等互利的基础上签订,其条款如下:This Agreement is entered into between _____ (hereinafter referred to as "Company"), and ______, (hereinafter referred to as "Employee") pursuant to paragraph VIII(2) of the Employee Handbook, whereby it is agreed as follows:本“协议”由_____(以下简称“公司”)与_____(以下简称“雇员”)根据“雇员手册”第VIII(2)款签署,“协议”内容如下:II. 以下为标准的“Parties”条款:3. This Agreement is made and entered into this _____th day of _____ in the year of ____ by and between ______, a company duly organized and existing under and by virtue of the laws of ______, with its principal place of business at _____ (hereinafter referred to as “_____”), and ______, a company duly organized and existing under and by virtue of the laws of _____, with its principal place of business at _____ (hereinafter referred to as “_____”), whereby it is agreed as follows:本合约由______,在_____法律之下并凭该等法律正式组织并存在的公司,其主要营业地点在______(下称_____),与_____,在_____法律之下并凭该等法律正式组织并存在的公司,其主要营业地点______(下称代理人),于_____日签订和缔结,协议如下:III. “Recitals”由数个以"Whereas"字样开头的句子所组合而成(这些句子俗称为“Whereas Clauses”),表示当事人乃是在基于对这些事实(例如订约的目的、背景来由等)的共同认识,订立此合约。
2020职称英语词汇T开头:terminate是什么意思及用法动词[I,T]【正式】(使)结束,(使)终止词形变化:时态:terminated,terminated,terminating,terminates。
同义词:end;dismiss,fire,force out,give notice,give the axe,give the sack,can,sack,send away;end;end,finish,cease,stop。
反义词:employ,engage,hire;commence,commence,get,get down,begin,begin,lead off,set about,set out,start,start,start out;begin,start。
单词分析:这些动词均含“结束,完成”之意。
complete:侧重指完成预定的任务或使某事完善,补足缺少的部分等。
finish:与complete基本同义,着重圆满地结束或完成已着手的事。
end:最普通用词,着重事情的完成。
也指某种活动因达到目的而自然结束或因为某种原因而突然中止。
close:普通用词,着重行为的终止或结束,不强调其目的。
conclude:正式用词,多指以某事或活动达到预期目的而告终。
terminate:强调有一个空间和时间的限度,届时必须终止。
书面语用词。
accomplish:正式用词,强调一个过程的完成;也可指依靠努力达到一定目的,或取得一定的结果。
英语句子Is it the bus terminal?它是终点站吗?International terminal国际候机楼a terminable annuity.一笔有期限的年金IK TerminatorIK终结terminal airport终点航空站英语解释be the end of; be the last or concluding part ofterminate the employment ofbring to an end or halthave an end, in a temporal, spatial, or quantitative sense; either spatial or metaphorical相似短语terminate cable 终接电缆terminate contract 解除合同terminate load 终端负载terminate resistance 终端电阻terminate a contract 【法】终止契约相似单词terminates [ terminate ]的第三人称单数;[ terminating ]的第三人称单数terminate v. [I,T]【正式】(使)结束,(使)终止terminater n.转录终止信号,终结者。
第1篇Case Overview:In this legal English case, John Doe, a former employee of XYZ Corporation, filed a lawsuit against the company for wrongful termination and violation of the Employment Contract. The case revolves around the termination of John Doe's employment and the subsequent dispute between the parties.Background:John Doe was employed by XYZ Corporation as a Sales Manager. He was hired on a fixed-term employment contract, which was due to expire on December 31, 2020. According to the contract, XYZ Corporation was required to provide John Doe with a written notice of termination at least 30 days before the expiration of the contract.On November 15, 2020, XYZ Corporation terminated John Doe's employment without providing the required notice. John Doe claims that the termination was wrongful and in violation of the Employment Contract. He further alleges that the termination was due to discrimination based on his age and that he was replaced by a younger employee.Facts:1. John Doe was hired by XYZ Corporation on February 1, 2015, as a Sales Manager.2. He was employed under a fixed-term employment contract, which was due to expire on December 31, 2020.3. The Employment Contract explicitly states that XYZ Corporation must provide John Doe with a written notice of termination at least 30 days before the expiration of the contract.4. On November 15, 2020, XYZ Corporation terminated John Doe's employment without providing the required notice.5. John Doe claims that he was replaced by a younger employee who was offered the same position and salary.6. John Doe filed a complaint with the Equal Employment Opportunity Commission (EEOC) and received a right-to-sue letter.Issues:1. Did XYZ Corporation violate the Employment Contract by terminating John Doe's employment without providing the required notice?2. Did XYZ Corporation discriminate against John Doe based on his age in terminating his employment?3. Is John Doe entitled to damages for wrongful termination and discrimination?Arguments:1. Violation of the Employment Contract:John Doe argues that XYZ Corporation violated the Employment Contract by terminating his employment without providing the required notice. He claims that the contract clearly stipulates that XYZ Corporation must provide him with a written notice of termination at least 30 days before the expiration of the contract. By terminating his employment without notice, XYZ Corporation breached the contract.2. Age Discrimination:John Doe further argues that XYZ Corporation discriminated against him based on his age in terminating his employment. He claims that he was replaced by a younger employee who was offered the same position and salary. John Doe argues that this action demonstrates age discrimination and violates anti-discrimination laws.3. Entitlement to Damages:John Doe asserts that he is entitled to damages for wrongful termination and discrimination. He seeks compensation for lost wages, emotional distress, and any other damages resulting from the termination and discrimination.Defenses:1. Just Cause for Termination:XYZ Corporation argues that there was just cause for terminating John Doe's employment. They claim that John Doe engaged in misconduct and violated company policies, which warranted immediate termination.2. No Age Discrimination:XYZ Corporation denies the claim of age discrimination. They argue that the decision to replace John Doe was based on performance and business needs, not age.3. Lack of Notice Requirement:XYZ Corporation argues that the Employment Contract does not require them to provide notice of termination before the expiration of the contract. They claim that the contract only requires notice for termination during the contract term.Judgment:The court found that XYZ Corporation violated the Employment Contract by terminating John Doe's employment without providing the required notice. The court held that the contract clearly stipulated the notice requirement and XYZ Corporation breached its obligations under the contract.Regarding the claim of age discrimination, the court found that there was insufficient evidence to prove that XYZ Corporation discriminated against John Doe based on his age. The court noted that the replacement of John Doe with a younger employee was based on performance and business needs, not age.As for the entitlement to damages, the court awarded John Doe lost wages for the period between the termination and the expiration of the contract. The court denied the claim for emotional distress and other damages, finding that John Doe failed to provide sufficient evidence to support such claims.Conclusion:In this legal English case, the court held that XYZ Corporation violated the Employment Contract by terminating John Doe's employment without providing the required notice. However, the court found insufficient evidence to support the claim of age discrimination. John Doe was awarded lost wages as damages, while the claim for emotional distressand other damages was denied. This case highlights the importance of adhering to employment contracts and the potential consequences of wrongful termination and discrimination.第2篇Case No.: 12345 ABCDCourt: Superior Court of the State of California, County of Los AngelesDate: January 15, 2023Facts:In the summer of 2021, John Smith, a resident of Los Angeles, purchased a new car from Johnson Motors, a local dealership. The car, a 2021 model, was advertised as being in "like new" condition and had a warranty of 36 months or 36,000 miles, whichever came first. Smith drove the car for approximately 6 months before he noticed that the engine was emitting excessive smoke and making unusual noises. Upon investigation, it was discovered that the engine had been severely overheated and had suffered irreparable damage.Smith took the car back to Johnson Motors, presenting the warranty and seeking a repair or replacement. Johnson Motors denied the claim,stating that the damage was due to misuse and not covered under the warranty. Smith, believing that the car was sold with false representations, filed a lawsuit against Johnson Motors for breach of warranty and fraudulent misrepresentation.Issues:1. Whether the warranty terms are enforceable.2. Whether Johnson Motors breached the warranty by denying Smith's claim.3. Whether Johnson Motors engaged in fraudulent misrepresentation.Arguments:1. Enforceability of the Warranty Terms:Smith argues that the warranty terms are enforceable and that Johnson Motors is bound by the terms of the warranty. He points to the fact that the warranty was clearly stated in the sales contract and that he relied on these terms when purchasing the car. Johnson Motors counters that the warranty terms were ambiguous and that the damage to the engine was not covered by the warranty.2. Breach of Warranty:Smith claims that Johnson Motors breached the warranty by denying his claim for repair or replacement. He argues that the warranty explicitly covers engine issues and that the damage to his car falls within the scope of the warranty. Johnson Motors argues that the damage was caused by Smith's misuse of the vehicle and that the warranty does not cover such damages.3. Fraudulent Misrepresentation:Smith further claims that Johnson Motors engaged in fraudulent misrepresentation by advertising the car as being in "like new" condition when it was, in fact, defective. He argues that this false representation induced him to purchase the car and that he suffered damages as a result.Ruling:The court, after hearing the arguments and examining the evidence, rendered the following decision:1. Enforceability of the Warranty Terms:The court held that the warranty terms are enforceable and that Johnson Motors is bound by the terms of the warranty. The court found that the warranty was clear and unambiguous and that Smith relied on these terms when purchasing the car.2. Breach of Warranty:The court found that Johnson Motors breached the warranty by denying Smith's claim. The court noted that the warranty explicitly covers engine issues and that the damage to Smith's car fell within the scope of the warranty. The court also found that the evidence did not support Johnson Motors' claim that the damage was due to misuse.3. Fraudulent Misrepresentation:The court found that Johnson Motors did engage in fraudulent misrepresentation. The court held that the advertisement of the car as being in "like new" condition was false and that it induced Smith to purchase the car. The court found that Smith suffered damages as a result of this misrepresentation.Remedies:The court awarded Smith the following remedies:1. The cost of repairing or replacing the car, as covered under the warranty.2. Compensatory damages for the emotional distress and inconvenience caused by the car's malfunction.3. Punitive damages against Johnson Motors for engaging in fraudulent misrepresentation.Conclusion:This case serves as a reminder to consumers to carefully read and understand warranty terms before making a purchase. It also underscores the importance of honesty in advertising and the legal consequences of fraudulent misrepresentation. The court's decision in favor of Smith sends a strong message to businesses that they must honor their warranties and provide accurate information to their customers.Legal Principles Involved:1. Warranty Law: The court applied principles of warranty law to determine whether Johnson Motors breached the warranty terms.2. Fraudulent Misrepresentation: The court applied principles of fraudulent misrepresentation to find that Johnson Motors made false statements that induced Smith to purchase the car.3. Damages: The court awarded damages to Smith for the breach of warranty and fraudulent misrepresentation.End of Case.第3篇Court: Superior Court of the State of California, County of Los AngelesCitation: 123 Cal. Rptr. 3d 456 (2010)Facts:In the year 2005, John Smith ("Plaintiff") purchased a house in Los Angeles, California, from James Johnson ("Defendant"). The property was advertised as a "fixer-upper," and the purchase price was $500,000. The contract of sale included a clause that the property was being sold "as is," meaning that the Defendant made no warranties regarding the condition of the property.After the purchase, Plaintiff discovered that the house had significant structural damage, including foundation issues and a leaking roof. Plaintiff hired an engineer to inspect the property and determine the extent of the damage. The engineer's report estimated the repairs would cost $150,000.Plaintiff brought a lawsuit against Defendant, alleging breach of contract and fraud. Plaintiff claimed that the Defendant had knowledge of the property's condition and intentionally misrepresented it as being in good repair. The Defendant denied these allegations and claimed that the "as is" clause released him from any liability.Issue:The issue before the court was whether the "as is" clause in thecontract of sale effectively released the Defendant from liability for the property's condition, or whether the Plaintiff could prove fraud or breach of contract.Decision:The court, after considering the evidence presented by both parties, found that the "as is" clause did not absolve the Defendant of liability. The court held that while the "as is" clause limited the Defendant's warranties, it did not completely release him from liability for fraudor material misrepresentations.The court found that the Defendant had indeed made material misrepresentations regarding the condition of the property. The evidence showed that the Defendant knew about the structural damage and roofleaks but failed to disclose them to the Plaintiff. The court further found that the misrepresentations were intentional and resulted in the Plaintiff purchasing the property under false pretenses.The court also found that the Defendant's actions constituted fraud. The court explained that fraud occurs when a party makes a false statementof material fact with the intent to induce another party to rely on that statement and to enter into a contract. In this case, the Defendant's failure to disclose the property's condition was a false statement that induced the Plaintiff to enter into the contract.Rationale:The court's decision was based on the principle that while "as is" clauses can limit the seller's warranties, they do not eliminate the seller's duty to disclose known defects. The court emphasized that the purpose of the "as is" clause is to inform the buyer that the seller is not making any representations about the condition of the property, but it does not grant the seller the right to misrepresent the property's condition.The court further held that the Defendant's misrepresentations were material, meaning that they were likely to influence the Plaintiff'sdecision to purchase the property. The court found that the structural damage and roof leaks were significant defects that would have been a major consideration for the Plaintiff in deciding whether to purchase the property.Remedy:The court awarded the Plaintiff damages in the amount of $150,000, which represented the cost of repairs to the property. The court also ordered the Defendant to pay the Plaintiff's attorney's fees and costs.Conclusion:This case serves as an important reminder that "as is" clauses in real estate contracts do not completely release sellers from liability for their actions. Sellers must still disclose known defects and cannot make material misrepresentations about the condition of the property. Buyers should be cautious when entering into "as is" contracts and should conduct thorough inspections to protect themselves against potential defects.Legal Principles:1. "As Is" Clauses: These clauses typically limit the seller's warranties and disclaim any representations about the condition of the property. However, they do not absolve the seller from liability for fraud or material misrepresentations.2. Fraud: Fraud occurs when a party makes a false statement of material fact with the intent to induce another party to rely on that statement and to enter into a contract.3. Material Misrepresentations: These are false statements that are likely to influence the decision of the buyer.4. Duty to Disclose: Sellers have a duty to disclose known defects to potential buyers.References:- California Civil Code § 1790.4- Restatement (Second) of Torts § 538- Prosser and Keeton on Torts § 107Additional Notes:This case is an example of the application of common law principles in a real estate context. The court's decision reflects the balance between protecting buyers from fraudulent sellers and upholding the intent of "as is" clauses. Buyers should be aware of their rights and obligations when entering into real estate transactions and should consult withlegal professionals to ensure their interests are protected.。
TERMINATION LETTER(解职信)name,address,date,referenceDear Mr/Ms/etc .........∙Further to our meeting of (date) I (regretfully) confirm that your employment with us is terminated with effect from (date)/with immediate effect. As stated at our meeting the reason(s) for terminating your employment with us is/are as follows: (Employer must clearly statereasons - transgressions美[trænz'ɡrɛʃən] and relevant policies if applicable美[ə'plɪkəbl])可适用的;可应用的;合适的∙(Employer must clearly state previous warnings, informal, formal, written etc., and circumstances and person's response and subsequent behaviour/performance for each warning.) (Clearly state requirements regarding return of documentation, equipment, car, submission投降;提交(物);服从;(向法官提出的)意见;谦恭美[səb'mɪʃən] of final expenses claims, andany other leaving administration issues.)∙(Clearly state actual leaving date, requirement or otherwise to serve period of notice, holiday pay, and other pay and pension details.)∙(Clearly state the position regarding the employee's right of appeal, and state the appeal美[ə'pil]呼吁,请求;吸引力,感染力;上诉;诉诸裁判∙process and timescales.)∙∙(Optional sign-off, for example: Thank you for your past efforts and all the best for your future endeavours.)∙Yours, etc.∙name and position∙(Optionally and recommended: attach, at the foot of the letter refer to, a copy of your written disciplinary process, and also attach and refer to copies of written/printed evidence gatheredduring the employee's case. This enables employees to understand clearly the case against them, and also the process and their rights during the disciplinary process, which are central to the principles of the employment dispute regulations.)(Optional section at foot of letter, requiring person to sign, confirming receipt of the letter and any attachment(s), by way of returning a signed copy of this letter.)∙TERMINATION LETTER TEMPLATETERMINATION LETTERPERSONAL & CONFIDENTIALDateDr. xxxxxAddress –lab location or homeDear Dr. xxxxx: As a result of laboratory organizational changes underDr. xxxxx, (reword as appropriate) this letter constitutes two months’ notice of termination as required under the University of Toronto’s “Policy for Post doctoral Fellows”,Termination clause.The termination of your engagement is effective,end of day, (enter date.)1)∙At this time, you will continue to be paid your monthly stipend美['staɪpɛnd]奖学金;固定薪金;定期津贴;养老金of $amount until date. 2)∙As a Postdoctoral博士后的;博士学位取得后的Fellow ,∙you are not eligible for a vacation entitlement权利;津贴and therefore no vacation buyout will be offered or given.∙3)You will continue to be responsible for your taxes and statutory美[stætjʊt(ə)rɪ]法定的;法令的;可依法惩处的deductions, as applicable.∙∙4)In accordance with U of T's Inventions Policy, you will disclose all intellectual property that you have created under existing funding (research contracts or grants美[grænt], 补助金;津贴(grant的复数);政府奖助∙v. [专利] 授予;同意;承认(grant的第三人称单数)∙salary support, etc.) up to and including the effective date of your termination.∙This obligation will survive the termination of your engagement at U of T. 5)call forwarding电话转接;呼叫转送\转发address for your mail and tax slips with the Business Officer.If you have any questions regarding this termination of engagement, please do not hesitate to contact name of Business Officer or Chair at the Dept or Unit department office.We thank you for your participation in the research activities of the Dept / Lab and wish you well in your future endeavours.Sincerely, Remove any items (above) if considered not appropriate, and especially check that item 4 is relevant.。
英文终止双方交易合同模版Terminating a Bilateral Business Contract: A Comprehensive GuideIn the dynamic world of commerce, business partnerships and contractual agreements are essential for success. However, there may come a time when one or both parties involved in a bilateral contract decide to terminate the agreement. This decision can be influenced by a variety of factors, ranging from changing market conditions to irreconcilable differences between the contracting parties. Regardless of the reasons, it is crucial to follow a well-defined process to ensure a smooth and legally compliant termination of the contract.This essay aims to provide a comprehensive guide on the key elements to consider when terminating a bilateral business contract. By understanding the necessary steps and legal implications, businesses can navigate the termination process effectively and protect their interests.Grounds for TerminationThe first step in terminating a bilateral contract is to carefully review the agreement and identify the valid grounds for termination. These grounds can typically be categorized into two main types: breach of contract and mutual agreement.Breach of ContractA breach of contract occurs when one party fails to fulfill their contractual obligations as stipulated in the agreement. This can include non-payment of fees, failure to deliver goods or services, or any other material violation of the contract terms. When a breach of contract is identified, the non-breaching party may have the right to terminate the agreement and seek appropriate legal remedies.It is important to note that the severity of the breach can play a significant role in determining the appropriate course of action. Minor or technical breaches may not necessarily warrant termination, and the non-breaching party may be required to provide the breaching party with a reasonable opportunity to rectify the issue before proceeding with termination.Mutual AgreementIn some cases, both parties may mutually agree to terminate the contract, regardless of whether a breach has occurred. This scenario can arise when the parties have collectively determined that the continuation of the contract is no longer beneficial or feasible fortheir respective business interests. Mutual agreement to terminate a contract is often the preferred approach, as it allows for a more amicable and collaborative resolution.Termination ProceduresOnce the grounds for termination have been established, the next step is to follow the proper termination procedures as outlined in the contract or applicable laws and regulations. These procedures typically include the following key elements:Notice of TerminationThe party initiating the termination must provide the other party with a formal written notice of termination. This notice should clearly state the reasons for the termination, the effective date of the termination, and any other relevant details. The notice period required for termination may be specified in the original contract and should be strictly adhered to.Termination ClausesMany bilateral contracts include specific termination clauses that outline the conditions and procedures for ending the agreement. These clauses may stipulate the required notice period, the grounds for termination, and any applicable penalties or consequences. It is crucial to review and comply with the termination clauses outlined in the contract.Dispute ResolutionIn the event of a dispute or disagreement regarding the termination, the contract may outline a specific dispute resolution process, such as mediation or arbitration. The parties involved should follow the designated dispute resolution procedures to address any outstanding issues and reach a mutually acceptable resolution.Obligations and LiabilitiesWhen terminating a bilateral contract, it is essential to consider the outstanding obligations and liabilities of both parties. This may include the settlement of any outstanding payments, the return of any property or assets, the transfer of intellectual property rights, and the handling of confidential information. The termination agreement should clearly outline the rights, responsibilities, and liabilities of each party upon the conclusion of the contract.Legal ConsiderationsTerminating a bilateral business contract can have significant legal implications, and it is crucial to ensure compliance with relevant laws and regulations. Some key legal considerations include:Applicable Laws and RegulationsDepending on the nature of the business and the jurisdiction in which the contract was formed, there may be specific laws andregulations that govern the termination process. It is essential to research and adhere to the applicable legal requirements to avoid any legal disputes or penalties.Contractual ProvisionsIn addition to the termination clauses, the contract may contain other provisions that must be considered during the termination process. These may include clauses related to confidentiality, non-compete agreements, and the assignment of rights and obligations.Documentation and Record-keepingThorough documentation and record-keeping are crucial throughout the termination process. This includes maintaining copies of all written communications, termination notices, and any other relevant documentation. These records can serve as crucial evidence in the event of any legal disputes or future audits.Termination AgreementTo formalize the termination of the bilateral contract, the parties may choose to create a separate termination agreement. This agreement should outline the specific terms and conditions of the termination, including the effective date, the settlement of outstanding obligations, and any other relevant details. The termination agreement should be carefully reviewed and signed by both parties to ensure a clear and legally binding conclusion to the contract.ConclusionTerminating a bilateral business contract is a complex process that requires careful planning, attention to detail, and adherence to legal requirements. By understanding the grounds for termination, following the proper procedures, and considering the legal implications, businesses can navigate the termination process effectively and protect their interests.Ultimately, the successful termination of a bilateral contract requires open communication, mutual understanding, and a commitment to a fair and amicable resolution. By approaching the termination process with a focus on professionalism and legal compliance, businesses can ensure a smooth transition and pave the way for future successful partnerships.。
t distribution 学生t分布t number t数t statistic t统计量t test t检验t1topological space t1拓扑空间t2topological space t2拓扑空间t3topological space 分离空间t4topological space 正则拓扑空间t5 topological space 正规空间t6topological space 遗传正规空间table 表table of random numbers 随机数表table of sines 正弦表table of square roots 平方根表table of values 值表tabular 表的tabular value 表值tabulate 制表tabulation 造表tabulator 制表机tacnode 互切点tag 标签tame 驯顺嵌入tame distribution 缓增广义函数tamely imbedded 驯顺嵌入tangency 接触tangent 正切tangent bundle 切丛tangent cone 切线锥面tangent curve 正切曲线tangent function 正切tangent line 切线tangent of an angle 角的正切tangent plane 切平面tangent plane method 切面法tangent surface 切曲面tangent vector 切向量tangent vector field 切向量场tangent vector space 切向量空间tangential approximation 切线逼近tangential component 切线分量tangential curve 正切曲线tangential equation 切线方程tangential stress 切向应力tangents method 切线法tape 纸带tape inscription 纸带记录tariff 税tautology 重言taylor circle 泰勒圆taylor expansion 泰勒展开taylor formula 泰勒公式taylor series 泰勒级数technics 技术technique 技术telegraph equation 电报方程teleparallelism 绝对平行性temperature 温度tempered distribution 缓增广义函数tend 倾向tendency 瞧tension 张力tensor 张量tensor algebra 张量代数tensor analysis 张量分析tensor bundle 张量丛tensor calculus 张量演算法tensor density 张量密度tensor differential equation 张量微分方程tensor field 张量场tensor form 张量形式tensor form of the first kind 第一张量形式tensor function 张量函数tensor of torsion 挠率张量tensor product 张量乘积tensor product functor 张量乘积函子tensor representation 张量表示tensor space 张量空间tensor subspace 张量子空间tensor surface 张量曲面tensorial multiplication 张量乘法term 项term of higher degree 高次项term of higher order 高次项term of series 级数的项terminability 有限性terminable 有限的terminal decision 最后判决terminal edge 终结边terminal point 终点terminal unit 级端设备terminal vertex 悬挂点terminate 终止terminating chain 可终止的链terminating continued fraction 有尽连分数terminating decimal 有尽小数termination 终止terminology 专门名词termwise 逐项的termwise addition 逐项加法termwise differentiation 逐项微分termwise integration 逐项积分ternary 三元的ternary connective 三元联结ternary form 三元形式ternary notation 三进制记数法ternary number system 三进制数系ternary operation 三项运算ternary relation 三项关系ternary representation og numbers 三进制记数法tertiary obstruction 第三障碍tesseral harmonic 田形函数tesseral legendre function 田形函数test 检验test for additivity 加性检验test for uniform convergence 一致收敛检验test function 测试函数test of dispersion 色散检验test of goodness of fit 拟合优度检验test of hypothesis 假设检验test of independence 独立性检验test of linearity 线性检验test of normality 正规性检验test point 测试点test routine 检验程序test statistic 检验统计量tetracyclic coordinates 四圆坐标tetrad 四元组tetragon 四角形tetragonal 正方的tetrahedral 四面角tetrahedral angle 四面角tetrahedral co ordinates 四面坐标tetrahedral group 四面体群tetrahedral surface 四面曲面tetrahedroid 四面体tetrahedron 四面形tetrahedron equation 四面体方程theorem 定理theorem for damping 阻尼定理theorem of alternative 择一定理theorem of identity for power series 幂级数的一致定理theorem of implicit functions 隐函数定理theorem of mean value 平均值定理theorem of principal axes 轴定理theorem of residues 残数定理theorem of riemann roch type 黎曼洛赫型定理theorem on embedding 嵌入定理theorems for limits 极限定理theoretical curve 理论曲线theoretical model 理论模型theory of automata 自动机理论theory of cardinals 基数论theory of complex multiplication 复数乘法论theory of complexity of computations 计算的复杂性理论theory of correlation 相关论theory of differential equations 微分方程论theory of dimensions 维数论theory of elementary divisors 初等因子理论theory of elementary particles 基本粒子论theory of equations 方程论theory of errors 误差论theory of estimation 估计论theory of functions 函数论theory of games 对策论theory of hyperbolic functions 双曲函数论theory of judgment 判断论theory of numbers 数论theory of ordinals 序数论theory of perturbations 摄动理论theory of probability 概率论theory of proportions 比例论theory of relativity 相对论theory of reliability 可靠性理论theory of representations 表示论theory of sets 集论theory of sheaves 层理论theory of singularities 奇点理论theory of testing 检验论theory of time series 时间序列论theory of transversals 横断线论theory of types 类型论thermal 热的thermodynamic 热力学的thermodynamics 热力学theta function 函数theta series 级数thick 厚的thickness 厚度thin 薄的thin set 薄集third boundary condition 第三边界条件third boundary value problem 第三边界值问题third fundamental form 第三基本形式third isomorphism theorem 第三同构定理third proportional 比例第三项third root 立方根thom class 汤姆类thom complex 汤姆复形three body problem 三体问题three dimensional 三维的three dimensional space 三维空间three dimensional torus 三维环面three eighths rule 八分之三法three faced 三面的three figur 三位的three place 三位的three point problem 三点问题three series theorem 三级数定理three sheeted 三叶的three sided 三面的three sigma rule 三规则three termed 三项的three valued 三值的three valued logic 三值逻辑three valued logic calculus 三值逻辑学threshold logic 阈逻辑time interval 时程time lag 时滞time series analysis 时序分析timesharing 分时toeplitz matrix 托普利兹矩阵tolerance 容许tolerance distribution 容许分布tolerance estimation 容许估计tolerance factor 容许因子tolerance level 耐受水平tolerance limit 容许界限tolerance region 容许区域top digit 最高位数字topological 拓扑的topological abelian group 拓扑阿贝耳群topological algebra 拓扑代数topological cell 拓扑胞腔topological circle 拓扑圆topological completeness 拓扑完备性topological complex 拓扑复形topological convergence 拓扑收敛topological dimension 拓扑维topological direct sum 拓扑直和topological dynamics 拓扑动力学topological embedding 拓扑嵌入topological field 拓扑域topological group 拓扑群topological homeomorphism 拓扑同胚topological index 拓扑指数topological invariant 拓扑不变量topological limit 拓扑极限topological linear space 拓扑线性空间topological manifold 拓扑廖topological mapping 拓扑同胚topological pair 拓扑偶topological polyhedron 曲多面体topological product 拓扑积topological residue class ring 拓扑剩余类环topological ring 拓扑环topological simplex 拓扑单形topological skew field 拓扑非交换域topological space 拓扑空间topological sphere 拓扑球面topological structure 拓扑结构topological sum 拓扑和topological type 拓扑型topologically complete set 拓扑完备集topologically complete space 拓扑完备空间topologically equivalent space 拓扑等价空间topologically nilpotent element 拓扑幂零元topologically ringed space 拓扑环式空间topologically solvable group 拓扑可解群topologico differential invariant 拓扑微分不变式topologize 拓扑化topology 拓扑topology of bounded convergence 有界收敛拓扑topology of compact convergence 紧收敛拓扑topology of uniform convergence 一致收敛拓扑toroid 超环面toroidal coordinates 圆环坐标toroidal function 圆环函数torque 转矩torsion 挠率torsion coefficient 挠系数torsion form 挠率形式torsion free group 非挠群torsion group 挠群torsion module 挠模torsion of a curve 曲线的挠率torsion product 挠积torsion subgroup 挠子群torsion tensor 挠率张量torsion vector 挠向量torsionfree connection 非挠联络torsionfree module 无挠模torsionfree ring 无挠环torus 环面torus function 圆环函数torus group 环面群torusknot 环面纽结total 总和total correlation 全相关total curvature 全曲率total degree 全次数total differential 全微分total differential equation 全微分方程total error 全误差total graph 全图total image 全象total inspection 全检查total instability 全不稳定性total inverse image 全逆象total matrix algebra 全阵环total matrix ring 全阵环total order 全序total predicate 全谓词total probability 总概率total probability formula 总概率公式total regression 总回归total relation 通用关系total space 全空间total stability 全稳定性total step iteration 整步迭代法total step method 整步迭代法total stiefel whitney class 全斯蒂费尔惠特尼类total subset 全子集total sum 总和total variation 全变差totally bounded set 准紧集totally bounded space 准紧空间totally differentiable 完全可微分的totally differentiable function 完全可微函数totally disconnected 完全不连通的totally disconnected graph 完全不连通图totally disconnected groupoid 完全不连通广群totally disconnected set 完全不连通集totally disconnected space 完全不连通空间totally geodesic 全测地的totally nonnegative matrix 全非负矩阵totally ordered group 全有序群totally ordered set 线性有序集totally positive 全正的totally positive matrix 全正矩阵totally quasi ordered set 完全拟有序集totally real field 全实域totally reflexive relation 完全自反关系totally regular matrix method 完全正则矩阵法totally singular subspace 全奇异子空间totally symmetric loop 完全对称圈totally symmetric quasigroup 完全对称拟群touch 相切tournament 竞赛图trace 迹trace form 迹型trace function 迹函数trace of dyadic 并向量的迹trace of matrix 矩阵的迹trace of tensor 张量的迹tracing point 追迹点track 轨迹tractrix 曳物线trajectory 轨道transcendence 超越性transcendence basis 超越基transcendence degree 超越次数transcendency 超越性transcendental element 超越元素transcendental equation 超越方程transcendental function 超越函数transcendental integral function 超越整函数transcendental number 超越数transcendental singularity 超越奇点transcendental surface 超越曲面transfer 转移transfer function 转移函数transfinite 超限的transfinite diameter 超限直径transfinite induction 超限归纳法transfinite number 超限序数transfinite ordinal 超限序数transform 变换transformation 变换transformation equation 变换方程transformation factor 变换因子transformation formulas of the coordinates 坐标的变换公式transformation function 变换函数transformation group 变换群transformation of air mass 气团变性transformation of coordinates 坐标的变换transformation of parameter 参数变换transformation of state 状态变换transformation of the variable 变量的更换transformation rules 变换规则transformation theory 变换论transformation to principal axes 轴变换transgression 超渡transient response 瞬态响应transient stability 瞬态稳定性transient state 瞬态transient time 过渡时间transition function 转移函数transition graph 转换图transition matrix 转移矩阵transition probability 转移函数transitive closure 传递闭包transitive graph 传递图transitive group of motions 可迁运动群transitive law 可迁律transitive permutation group 可迁置换群transitive relation 传递关系transitive set 可递集transitivity 可递性transitivity laws 可迁律translatable design 可旋转试验设计translate 转移translation 平移translation curve 平移曲线translation group 平移群translation invariant 平移不变的translation invariant metric 平移不变度量translation number 殆周期translation of axes 坐标轴的平移translation operator 平移算子translation surface 平移曲面translation symmetry 平移对称translation theorem 平移定理transmission channel 传输通道transmission ratio 传输比transport problem 运输问题transportation algorithm 运输算法transportation matrix 运输矩阵transportation network 运输网络transportation problem 运输问题transpose 转置transposed inverse matrix 转置逆矩阵transposed kernel 转置核transposed map 转置映射transposed matrix 转置阵transposition 对换transversal 横截矩阵胚transversal curve 横截曲线transversal field 模截场transversal lines 截线transversality 横截性transversality condition 横截条件transverse axis 横截轴transverse surface 横截曲面trapezium 不规则四边形trapezoid 不规则四边形trapezoid formula 梯形公式trapezoid method 梯形公式traveling salesman problem 转播塞尔斯曼问题tree 树trefoil 三叶形trefoil knot 三叶形纽结trend 瞧trend line 瞧直线triad 三元组trial 试验triangle 三角形triangle axiom 三角形公理triangle condition 三角形公理triangle inequality 三角形公理triangulable 可三角剖分的triangular decomposition 三角分解triangular form 三角型triangular matrix 三角形矩阵triangular number 三角数triangular prism 三棱柱triangular pyramid 四面形triangular surface 三角曲面triangulate 分成三角形triangulation 三角剖分triaxial 三轴的triaxial ellipsoid 三维椭面trichotomy 三分法trident of newton 牛顿三叉线tridiagonal matrix 三对角线矩阵tridimensional 三维的trigammafunction 三函数trigonometric 三角的trigonometric approximation polynomial 三角近似多项式trigonometric equation 三角方程trigonometric function 三角函数trigonometric moment problem 三角矩问题trigonometric polynomial 三角多项式trigonometric series 三角级数trigonometrical interpolation 三角内插法trigonometry 三角学trihedral 三面形的trihedral angle 三面角trihedron 三面体trilateral 三边的trilinear 三线的trilinear coordinates 三线坐标trilinear form 三线性形式trinomial 三项式;三项式的trinomial equation 三项方程triplanar point 三切面重点 ?triple 三元组triple curve 三重曲线triple integral 三重积分triple point 三重点triple product 纯量三重积triple product of vectors 向量三重积triple root 三重根triple series 三重级数triple tangent 三重切线triply orthogonal system 三重正交系triply tangent 三重切线的trirectangular spherical triangle 三直角球面三角形trisecant 三度割线trisect 把...三等分trisection 三等分trisection of an angle 角的三等分trisectrix 三等分角线trivalent map 三价地图trivector 三向量trivial 平凡的trivial character 单位特贞trivial cohomology functor 平凡上同弹子trivial extension 平凡扩张trivial fibre bundle 平凡纤维丛trivial graph 平凡图trivial homogeneous ideal 平凡齐次理想trivial knot 平凡纽结trivial solution 平凡解trivial subset 平凡子集trivial topology 密着拓扑trivial valuation 平凡赋值triviality 平凡性trivialization 平凡化trochoid 摆线trochoidal 余摆线的trochoidal curve 摆线true error 真误差true formula 真公式true proposition 真命题true sign 直符号true value 真值truncated cone 截锥truncated cylinder 截柱truncated distribution 截尾分布truncated pyramid 截棱锥truncated sample 截样本truncated sequence 截序列truncation 舍位truncation error 舍位误差truncation point 舍位点truth 真值truth function 真值函项truth matrix 真值表truth set 真值集合truth symbol 真符号truth table 真值表truthvalue 真值tube 管tubular knot 管状纽结tubular neighborhood 管状邻域tubular surface 管状曲面turbulence 湍流turbulent 湍聊turing computability 图灵机可计算性turing computable 图灵机可计算的turing machine 图录机turn 转向turning point 转向点twice 再次twice differentiable function 二次可微函数twin primes 素数对twisted curve 空间曲线twisted torus 挠环面two address 二地址的two address code 二地址代码two address instruction 二地址指令two body problem 二体问题two decision problem 二判定问题two digit 二位的two dimensional 二维的two dimensional laplace transformation 二重拉普拉斯变换two dimensional normal distribution 二元正态分布two dimensional quadric 二维二次曲面two dimensional vector space 二维向量空间two fold transitive group 双重可迁群two person game 两人对策two person zero sum game 二人零和对策two phase sampling 二相抽样法two place 二位的two point distribution 二点分布two point form 两点式two sample method 二样本法two sample problem 二样本问题two sample test 双样本检验two sheet 双叶的two sided condition 双边条件two sided decomposition 双边分解two sided divisor 双边因子two sided ideal 双边理想two sided inverse 双边逆元two sided module 双边模two sided neighborhood 双侧邻域two sided surface 双侧曲面two sided test 双侧检定two stage sampling 两阶段抽样法two termed expression 二项式two valued logic 二值逻辑two valued measure 二值测度two variable matrix 双变量矩阵two way array 二向分类two way classification 二向分类twopoint boundary value problem 两点边值问题type 型type problem 类型问题typenumber 型数typical mean 典型平均。
Terminating modal tableaux with sim-ple completeness proofOlivier Gasquet,Andreas Herzig,Mohamad Sahadeabstract.In this paper we study formal tools for the definition of tableauxsystems for modal logics.We begin by some preliminaries about graphs.Then we present graph rewriting rules together with control constructs fordefining strategies of application of these rules.We establish fair strategiesthat are sound,complete and terminating.These results cover almost allbasic modal logics such as K,T,S4,KB4,K+Confluence etc.,and canbe generalized for logics such LTL and PDL.Our framework provides thetheoretical basis for our generic tableau theorem prover LoTREC.Keywords:modal logics,tableau methods,automatic proof.1IntroductionBasic modal logics are normal modal logics defined by any combination of axioms D,T,B,4,5[4].For all these logics the complexity of satisfiability is in PSPACE,and research(in particular in thefield of description logics) has focussed on tableau decision procedures for them that stay within that boundary.For all these logics one canfind sound,complete and terminat-ing tableau algorithms in the standard textbooks[5,7],and implemented tableau theorem provers such as LogicWorkBench(LWB)[8]are readily available.This contrasts with the situation for more complex modal logics that are used to reason about mental states of artificial agents,about knowledge and communication,or norms and regulations.In these domains we have to deal with BDI logics,logics of knowledge,common knowledge and time, logics of action and obligation,etc.Such logics are not as‘stable’as the traditional modal logics wefind in textbooks:for example,it often happens that one would like to slightly modify the interplay between the modalities (preservation of beliefs through actions,influence of actions on obligations, interplay between belief and knowledge,etc.),or to add a new modality to an existing logic(add actions or time to a logic of belief,add goals to a logic of belief and action,add obligations to BDI logics,etc.)The variety and complexity of these logics make that in practice we do not always have at our disposal tableau algorithms that enjoy soundness,completeness and termination.The contributions of our paper are the following:Advances in Modal Logic,Volume6.c 2006,Olivier Gasquet,Andreas Herzig,Mo-hamad Sahade.4Olivier Gasquet,Andreas Herzig,Mohamad Sahade1.We provide a general formal framework for tableau systems,withinwhich virtually all tableau algorithms for modal logics can be imple-mented.This framework is based on rooted directed acyclic graphs(RDAGs)in order to stay close to possible worlds models,and itsbasic mechanism is graph rewriting.It provides a simple language todefine tableau rules and a simple language to define strategies.Weformally define the semantics of the usually rather implicit or purelyoperational notions of rule and strategy.2.Then we give simple and uniform completeness and soundness prooffor a large family of modal logics.3.We state general termination theorems for two classes of strategies.These classes contain strategies for the modal logics K,T,S4,S5,KB4,and even LTL and PDL.This paper presents the theoretical elements which are the basis of thegeneric tableau prover LoTREC that we develop in the LILaC group ofIRIT.LoTREC is available at:http://www.irit.fr/LILAC/LotrecIn particular there is a version of LoTREC that is executable via the web.A system description can be found in[6].2Modal logics and relational propertiesThe modal logics we consider are all obtained by extending the basic modallogic K by one or several of the well-known axioms T,B,4,5,D,and C(axiom of confluence:32p→23p).Thus KDC4denotes the modal logicobtained by adding the axioms D,C and4to the basic system K.With each of these axioms there is a associated relational property ofthe accessibility relation of the Kripke models.The axioms:T:2p→p,4:2p→22p,B:32p→p,5:32p→2p,D:2p→3p and C:32p→23p correspond respectively to:reflexivity(Ref),transitivity(Tr),symmetry(Sym),euclideanity(Eucl),seriality(Ser),confluence(Eucl).As a consequence of Sahlqvist’s theorem[11],a modal logic based on Kplus any combination of these axioms is characterized by the class of Kripkemodels whose accessibility relation satisfies the corresponding properties.Thus,KD4is characterized by Kripke models where the accessibility relationis both serial and transitive.Let group1be{Ref,Tr,Sym,Eucl}and group2be{Ser,Eucl}. DEFINITION1.Given a setρof relational properties among group1and 2,aρ-model is a Kripke model whose accessibility relation satisfiesρ.A formula isρ-satisfiable iffit is satisfiable in aρ-model.It isρ-valid iffit is valid in the class of allρ-models,which will be noted|=ρA.Thus A is a theorem of a system denoted by a setρof properties iffit isρ-valid.3Rooted Directed GraphIn the paper[3]we have advocated that tableaux systems should work onrooted directed acyclic graphs(RDAGs).Such a choice not only has theTerminating modal tableaux with simple completeness proof5 advantage of generality:as RDAGs are closer to Kripke models than sequent and trees,it also makes the way from an open tableau to a countermodel shorter.Here we pursue that approach.DEFINITION2.A labelledρ-directed graph is a triple N,Σ,for where:• N,Σ is a directed graph:N is some set andΣ⊆N×N• N,Σ satisfies all properties ofρ;•for is a function that associates a set of formulae with each of the nodes.A labelled rooted directed graphρ-(RGRAPH)is a couple G,root where G= N,Σ,for is a labelled directed graph,and root∈N is a root of G, that can access every other node in the transitive closure ofΣ.The elements x,x ofΣare also written xΣx .xΣx means that x is accessible from x via the relationΣ.For x,A ∈for we sometimes also write A∈for(x).A∈for(x)means that the formula A is associated with node x in the graph.Graphs are denoted by G,g,g ,....Small letters will be used for subgraphs.DEFINITION3.(rdag)A labelled rooted directed acyclic graph(ρ-RDAG) is a labelled RGRAPH whereΣcontains no cycles.4Closure of rgraphWe define the following closure operation on rgraph:DEFINITION4.Let G,root be an rgraph over a set andρa set of relational properties from group1;theρ-closure ofΣ(denoted byΣρ)is the least rgraph that containsΣand which satisfies every property ofρ.Thisρ-closure always exists for properties among{Ref,Tr,Sym,Eucl}.A very important point is that for properties of group1,the closure can be expressed in terms of the initial RGRAPH.For example,the transitive closure of an RGRAPH G,root is defined by:(x,y)∈ΣT r iff∃n≥1such that(x,y)∈Σn.Note that we do not consider here properties of group 2:is not so easy to define closure under a property of group2.This is the reason why they will be handled in a different way:as we will see later no propagation rule can simulate them.LEMMA5(Relational Closure Lemma).LetΣbe an RGRAPH over a set N of nodes:•(x,y)∈ΣRef iff(x,y)∈Σor x=y.•(x,y)∈ΣSym iff(x,y)∈Σor(y,x)∈Σ.•(x,y)∈ΣT r iff∃n≥1such that(x,y)∈Σn.•(x,y)∈ΣEucl iff(x,y)∈Σor∃u∈N∃n≥1∃m≥1such that (u,x)∈Σn and(u,y)∈Σm.6Olivier Gasquet,Andreas Herzig,Mohamad Sahade•(x,y)∈ΣRef,Sym iff(x,y)∈Σor x=y or(y,x)∈Σ.•(x,y)∈ΣRef,T r iff∃n≥0such that(x,y)∈Σn.•(x,y)∈ΣRef,Eucl iff∃n≥0∃x0=x,...x i,x i+1,...,x n=y:(x i,x i+1)∈Σor(x i+1,x i)∈Σ.•(x,y)∈ΣSym,T r iff∃n≥1∃x0=x,...x i,x i+1,...,x n:(x i,x i+1)∈Σor(x i+1,x i)∈Σ.•(x,y)∈ΣT r,Eucl iff∃u∈N∃n≥0∃m≥1such that(u,x)∈Σn and(u,y)∈Σm.LEMMA6.The remaining cases are reducible to those of the previous lemma:•ΣSym,Eucl=ΣSym,T r,Eucl=ΣSym,T r;•ΣRef,Sym,T r=ΣRef,T r,Eucl=ΣRef,Sym,Eucl=ΣRef,Sym,T r,Eucl=ΣRef,Eucl.The above lemmas will be a powerful tool for proving completeness:it will allow to define a model for a formula from an open tableau.But this is not the whole story.As we previously said,some properties are han-dled structurally;roughly speaking seriality and confluence are treated by generating the RDAG underlying of the tableau.LEMMA7(Structural Lemma).Letρ2be a subset of group2,ρ1a subset of group1and letΣbe aρ2-RGRAPH over a set N of nodes.ThenΣρ1is also aρ2-RGRAPH,and hence is a(ρ1∪ρ2)-RGRAPH.For the proof of the above three lemmas see([3]).5A simple graph-rewriting language RLAs we would like to handle complex modal logics,the tableau rules that we need must be able to perform at least the following operations on graphs: add a formula A to a node x,add a link with some label R between two nodes x,x ,create a new node x,duplicate the graph(to take into account disjunctions),mark and unmark a node x by an expression E,mark and unmark a formula A in a node x by an expression E.Note that links have to be labelled in order to be able to handle multiple modalities.Note also that marking is not needed for the simplest modal logics(the basic modal logics defined by any combination of axioms D,T,B, as well as K45,KD45and S5).Such rewriting steps are executed under some conditions.We need at least the following ones:node x contains a formula A(or not),node x is linked to node x by a link having label R(or not),node x is an ancestor of node x ,node x is marked by an expression E(or not).Note that only thefirst two conditions are necessary for the simplest modal logics.Terminating modal tableaux with simple completeness proof7 We here choose a minimal rule language that is restricted to exactly the above conditions and actions.It is also the rule language of LoTREC,with self-explaining conditions such as:if a node x contains a formula A,or does not contain a formula A,...and actions such as:add an expression E to a node x,link two nodes x,x ...DEFINITION8.(pattern)A pattern is a connected graph.DEFINITION9.A tableau rule r is a couple(p,{g1,...,g n})where p is a pattern(the pattern on which the rule can be applied)and g i,1≤i≤n are the graphs resulting from the application of the rule on patterns which match p.The rule r is analytic if every formula appearing in g i,1≤i≤n is a subformula of some formula appearing in p.r is strictly analytic if every formula appearing in g i,1≤i≤n is a strict subformula of some formula appearing in p.Note that in the sequel I use the terms:conditions part of a rule to rep-resent the pattern p and action part of a rule to represent the modifications done which lead to the new graphs{g1,...,g n}.Application of a rule r to a pattern p0is defined by:(p,{g1,...,g n})(p0)= {ν(g1),...,ν(g n)}if∃νsuch thatν(p)=p0⊥otherwiseNote thatνis an isomorphism on the structure of the graph,but it is a homomorphism on the set of formulas.We extend the partial function r to a total function(application of r on a graph)by stipulating one of the following definitions:1−r(G)={G∪( p⊆Gδ):δ∈r(p)}where∀p,r(p)is applicable 2−r(G)={(G∪( p⊆G r(p)=⊥δ)):δ∈r(p)}Note that a graph is a set of patterns(and may be only one pattern).Application of a rule r to a set of graphsΓis defined by:r(Γ)={Γ∪( G⊆Γr(G))As every rule(p,{g1,...,g n})is monotonic(in the sense that that p⊆g i,1≤i≤n),we only give the new expressions of g i,1≤i≤n.REMARK10.p and g i,1≤i≤n may contain node variables and formula variables(that are instantiated viaνwhen the rule is applied).We can think of such rules as the(infinite)set of their instances.REMARK11.Of course,r(p)may contain new nodes and/or edges that do not belong to p and that have to be named.We suppose that such new nodes will be denoted by groundfirst-order terms:a Skolem function denoted by the name of the rule(and possibly with an integer subscript8Olivier Gasquet,Andreas Herzig,Mohamad Sahadeif the rule introduces more that one new node)and with elements of g as arguments.As an example,if some node x contains a formula3A then applying the rule for3will introduce the new node3(x,A).6Tableau rulesIn order to define a tableau calculus for a modal logic denoted by KA1,...,A n corresponding to a set of propertiesρ1form group1and a set of properties ρ2of group2,we must associate a set of rules with it.All the tableaux calculi we are going to define contain:the classical rules,createSuccessor and propagateFormulas rules which will be written in RL as( {n0},∅,{(n0,{¬2A})} ,{ ∅,{(n0,n)},{(n,{¬A})} }),( {n0,n1},{(n0,n1)},{(n0,{2A})} ,{ ∅,∅,{(n1,{A})} })respectively(see also appendix B).As these rules are common to all tableaux calculi,we will henceforth omit them.More over we have some structural and propagation rules.A tableau calculus for a logic defined by(ρ1∪ρ2)is obtained by taking (in addition to classical rules,createSuccessor and propagateFormulas rules)the rules corresponding to properties of(ρ1∪ρ2);this correspondence is given in thefigure below.Properties RulesGroup1Ref propagateNec2ActualWorldSym addFormula2ParentTr propagateNec2ChildEucl copyNecIfLinked,copyNec2Parent,copyNec2BrothersGroup2Ser addPosTrueEucl confluenceIn the case of logics which contain the relational properties:Eucl or Tr, to stop the possibly infinite computation we perform an inclusion test called loop test.It tests if a node x is included in some ancestor.In this case we don’t create any successor for x and we stop the computation on the tableau which contain x.REMARKS12.It is worth noticing that our three rules handling euclideanity are rem-iniscent of the three sequent rules used in[2].This connection between a tableau approach and a sequent approach probably deserves to be investi-gated further.DEFINITION13.A(ρ1∪ρ2)-tableau for a formula A is the limit of a sequence of RDAGsΥ0,...,Υi,Υi+1,...such that1:•Υ0is an rdag consisting of only one node whose associated set of formulas is{A}:Υ0= root,∅,{(root,{A})};1this limit being defined as S i≥0ΥiTerminating modal tableaux with simple completeness proof9•Υi+1is obtained fromΥi by applying either a classical rule,or the createSuccessor rule,or the rule propagateFormulas,or a rule for (ρ1∪ρ2);•every applicable rule has been applied.Note that what we call a tableau here is usually called a tableau branch in the literature.While this is appropriate for classical logic,it might lead to confusions to have branches that are in fact graphs.DEFINITION14.A tableau is closed if some node in it contains False;it is open otherwise.A formula is(ρ1∪ρ2)-closed iffall its(ρ1∪ρ2)-tableaux are closed2.7A simple strategy languageSuppose we have defined a set of rewriting rules RULES for our logic L, and we now want to design a strategy and prove it terminates.Strategies define how the rules from RULES are applied,and in which order.Given a set of graphs,the simplest way of applying rule r∈RULES is to apply it to every formula in every node in every graph.It is also the most radical definition.It follows that in order to avoid loops(in the case of modal logics beyond the simplest ones),rule application has to be blocked by means of appropriate marking conditions.For example in S4, createSuccessor-rules application in node x is blocked if x is included in some ancestor node.7.1Combining rule applicationsOne of the simplest ways of combining rule applications is by means of regular expressions.We here provide a language that is made up of such expressions,and that is inspired by the concepts that are used in the Isabelle and HOL provers for higher-order logics[10].We choose the following constructs:allRules,firstRule and repeat to combine rules for defining strategies.They work as follows,where each r i is itself a strategy,possibly consisting of only one rule:•(allRules r1...r n)sequentially applies each of the applicable rules to the current graph,starting with r1.•(firstRule r1...r n)applies only thefirst applicable rule among the listed ones to the current tableau.•(repeat r1)repeatedly applies r1until afixpoint is reached.The semantic of our constructs allRules,firstRule and repeat for strategies is:DEFINITION15.(semantics of a strategy)2Due to the rule∨,which is equivalent to the rule NotAnd in the language of LoTREC, a formula may have several distinct tableaux.10Olivier Gasquet,Andreas Herzig,Mohamad SahadeallRules(r1...r k)(G)=(r k(...(r2(r1(G)))...)firstRule(nil)(G)=G,where nil is the empty sequence of rules firstRule(r1...r k)(G)=r1(G)if k≥1and r1(G)=G firstRule(r1...r k)(G)=firstRule(r2...r k)(G)otherwiserepeat(r)(G)=lfp(r,G)Thus repeat-loops build a leastfix-point.To apply a strategy to a formula A means to apply the strategy to the initial graph G=(root,∅,{(root,{A})}.Of course,a repeat-loop may make the resulting strategy loop forever if r is not designed carefully.8CompletenessIn this section we prove the completeness of our tableau calculi3.We show how,from a given open L-tableau for A we can construct aρ-model for A, whereρis the set of accessibility relations which characterize the logic L. DEFINITION16(End Node).x is an end node iffneither createSuccessor nor addPosTrue nor confluence rule are applicable.DEFINITION17(Loop Node).A node x is a loop node in a tableauΥiff:1.the rule createsuccessor or addPosTrue or confluence is applicableto x and2.∃y:yΣ+x and x⊆y(such y will be denoted by loop(x)). DEFINITION18(Looping Tableau).A tableauΥ=(N,Σ,for)is a loop-ing tableau iffeach of its leafs is either an end node or a looping node. The set of loop nodes ofΥis loop(Υ).Note that looping tableaux arefinite, since the computation is stopped either because no create new node rules are applicable or due to the inclusion test.DEFINITION19(Folded Tableau).LetΥbe a looping open L-tableau for A such that T r or Eucl are in L, where we stop the execution by making the inclusion test.The Folded Tableau FΥ=(FN,FΣ,F for)corresponding toΥis de-fined as follows:•FN=N−loop(Υ);•FΣ=Σ−{(x,y)/y∈loop(Υ)}∪{(x,loop(y))/y∈loop(Υ)}where x∈N;•F for is the restriction of for to FN.3We make the usual assumption of fairness.Terminating modal tableaux with simple completeness proof11 LetΥbe an open Folded L-tableau for A.Now let M=(W,R,τ)be the Kripke model defined as follows:DEFINITION20(Folded Model).•W=FN•R is theρ1-closure of FΣ,i.e.R=FΣρ1•for all w∈W,w∈τ(p)iffp∈F for(w).To prove the completeness of a L-tableaux we will consider for a formula A a L-folded open tableau FΥ=(FN,FΣ,F for)and then give a L-model for this formula fromΥ(as described in Definition20).Of course,folded models arefinite.Wefirst establish the following important lemma: LEMMA21(Box Lemma).Let FΥ=(FN,FΣ,F for)be a folded(ρ1∪ρ2)-tableau.Let x,y be such that(x,y)∈FΣρ1and2A∈x;then A∈y. Proof.There are several cases,depending on(ρ1∪ρ2);we only prove the lemma for some complex cases involving euclideanity:•ρ1={Eucl}:if(x,y)∈FΣρ1then by the Relational Closure Lemma (Lemma4)in the case of a folded tableau,we have:–either(x,y)∈Σand then A∈y by rule propagateFormulas–or∃u∃n≥1∃m≥1such that(u,x)∈FΣn and(u,y)∈FΣm.Hence∗∃x0=x,...,x i,x i+1,...,x n=u:for0≤i≤n we have either:·(x i+1,x i)∈Σ;then A,2A∈x i(by rulespropagateFormulas and CopyNec2Parent)·(x i+1,x i)∈FΣand(x i+1,x i)/∈Σ;then∃z∈N:(x i+1,z)∈Σand z⊆x i then A,2A∈z(by rulespropagateFormulas and CopyNecIfLinked)and2A∈x i since z⊆x i.In particular:A,2A∈x n=u.∗∃y0=u,...,y i,y i+1,...,y m=y:·if m=1then y=y1and A∈y by rulepropagateFormulas.We have2A∈y1(by rule copyNec2Brothers since2A∈x n−1).For1≤i≤m we have:·either(y i,y i+1)∈Σ;then A,2A∈y i(by rulespropagateFormulas and CopyNecIfLinked)·or(y i,y i+1)∈FΣ;and(y i,y i+1)/∈Σ:then∃z∈N:(y i,z)∈Σand z⊆y i+1then A,2A∈z(by rulespropagateFormulas and CopyNecIfLinked)and A,2A∈y i+1since z⊆y i+1.12Olivier Gasquet,Andreas Herzig,Mohamad SahadeSince2A∈x n=u=y0we have:2A∈y i for0≤i≤m.Hence A∈y i for1≤i≤m(by rule propagateFormulas),in particular A∈y.•ρ1={T r,Eucl}:if(x,y)∈FΣρ1then by the Relational Closure Lemma,we have∃u∈N∃n≥0∃m≥1such that(u,x)∈FΣn and (u,y)∈FΣm.This implies that:–∃n≥0∃x0=x,...,x i,x i+1,...,x n=u:∗(x i+1,x i)∈Σ;then2A∈x i implies2A∈x i+1(by ruleCopyNec2Parent)∗(x i+1,x i)∈FΣand(x i+1,x i)/∈Σ;then∃z∈N:(x i+1,z)∈Σand z⊆x i then2A∈x i implies2A∈z(by ruleCopyNecIfLinked)and2A∈x i+1since z⊆x i+1.In particular2A∈x0=u.–∃m≥1∃y0=u,...,y i,y i+1,...,y m+1=y::∗if m=1then y=y1and A∈y by rule propagateFormulas.∗(y i,y i+1)∈Σ;hence2A∈y i implies2A∈y i+1(by rulepropagateNec2Child).∗(y i,y i+1)∈FΣand(y i,y i+1)/∈Σ:then2A∈y i+1(sameproof as above)In particular,if2A∈y m−1,then A∈y(by rulepropagateFormulas).•In the case of logics with the axiom D,we build a serial model by adding to every leaf of the folded tableau a reflexive edge.We call this operation reflexive leaf closure(RLC).The folded tableau will be the tableau obtained from the original tableau by RLC.Now if(x,y)∈FΣ,we have:–either(x,y)∈Σ,so A∈y by rule propagateFormulas;–or(x,y)is a RLC edge;in this case x=y is a leaf and it cannotcontain a formula of the form2A.Because if it is the case,therule addPosTrue adds a3 to x and createSuccessor createsa successor to it.To achieve the completeness proof,it remains to prove that: for all B∈sf(A)if B∈x then M,x|=B for all L-models obtained from the tableaux of A.Where sf(A)is the set of subformulas of A.The following fundamental lemma brings us to the desired conclusion:LEMMA22(Fundamental Lemma).Let FΥbe an openρ1-tableau for A, let M be theρ1-model defined as in(Definition20)from FΥ,let B∈sf(A) and let x∈N−loop(FΥ).Then(i)B∈x iffM,x|=B.Terminating modal tableaux with simple completeness proof13 Proof.By induction on the structure of B.Without loss of generality suppose that B is written only with¬,⊥,∧,2.•Induction initialization:let B be an atom,(i)holds by the definition ofτ;•Induction step:1.B cannot be False,otherwiseΥwould be closed;2.If B is a propositional classical formula the proof is straightfor-ward.3.Let B=¬2C.¬2C∈x⇒∃y such that(x,y)∈FΣand¬C∈y(rule createSuccessoror because of the inclusion if x is a loop node)⇒∃y such that(x,y)∈R and C/∈y(FΣ⊆R)⇒∃y such that(x,y)∈R and M,y C⇒M,x|=¬2C4.Let B be2C and suppose(x,y)∈R;then by the Box Lemma21,C∈y.Then by induction hypothesis,M,y|=C for every ysuch that(x,y)∈R.Hence,M,x|=2C.As a direct consequence of the previous lemma,we have:COROLLARY23.If A has an open looping(ρ1∪ρ2)-tableau then it has a (ρ1∪ρ2)-model.We will show later that for all our logics,and for any formula,either all its tableaux are closed or it has at least one open looping tableau.In other words,a formula cannot have an infinite tableau.In thefirst case we will show that the formula is unsatisfiable(section9).If is not the case,we will show that it has an open fair loopingρ-tableau(section10).9SoundnessIn this section,we prove the soundness of our tableaux calculi:if a formula A is(ρ1∪ρ2)-satisfiable then there exists a open(ρ1∪ρ2)-tableau for A.We use pattern(r)to note the pattern described by the left hand side of a rule r.To prove the soundness we start by the following proposition: PROPOSITION24.LetΥbe a tableau.Suppose that RULES is subset of the rules given in appendix B.∀ρ∈RULES∀g∈pattern(r)we have:g⊆FΥ(folded tableau)iffthere is g ⊆Υsuch that g is a pattern isomorphic to g.Proof.The only new patterns are caused by the redirected edges,but these edges existed before the folding operation.14Olivier Gasquet,Andreas Herzig,Mohamad Sahade PROPOSITION25.LetΦbe a modal formula,we have the following im-plications:1.IfΦis(ρ1∪ρ2)-satisfiable then there is an infinite open tableau forΦ(without loop test);2.If there is an infinite open tableau forΦthen there is afinite opentableau forΦ(by using the looping test);3.If there is anfinite open tableau forΦthen there is an open foldedtableau forΦ.Proof.For thefirst implication see[3].The second implication is trivial since we stop the computation by the loop test,the resulting tableau is included in the infinite one and it can not be closed.The last implication results from Proposition24. 10TerminationThe setting we are concerned with is a well-delimited one(labelled graphs and monotonic rules,regular expression-like strategies),and it would be desirable to have general termination results.Unfortunately there seems to be no results in the literature that apply to such a case.For example techniques from rewriting theory such as Knuth-Bendix completion cannot be applied here(in particular because rewriting focusses on rule application in any order,Church-Rosser property etc,while we insist on the contrary).Here we give a formal termination proof of our strategies:In the case where our strategies are formed only by the constructors:allRules and firstRule the termination is trivial since in thefirst one there are afinite number of rules and every applicable one will be applied once,and in the second case only thefirst applicable rule will be applied.It remains to prove the termination in the case of repeat.We here give two termination criteria,which are enough to handle many logics.10.1Adding strict sub-formulas onlyOurfirst kind of strategies only adds strict subformulas in the action part. It might also be called strictly analytic.THEOREM26.Let d(x)be the length of the shortest path from the root to the node x,and let RULES be a set of rules such that for every r∈RULES,•r is strictly analytic(only strict subformulas of the original formula are added to nodes in the action part of r);•if the action part of r contains the createNewNode-action creating a node x then–the condition part of r checks for existence of at least one formulain any node among x0,...,x k,andTerminating modal tableaux with simple completeness proof15–d(x)>max(d(x0),...,d(x k)),i.e.new nodes are strictly fartherfrom the root(ensuring strict decrease of their contents), where x0,...,x k are the nodes referred to in the condition part of r. Let S be any strategy on RULES.Let G be afinite graph.Then the appli-cation of S to input G terminates,and S(G)is afinite graph.Proof.We prove that•application of S can never lead to RDAGs with infinite branching factor;•application of S can never lead to RDAGs of infinite depth.The proof is part of the proof of Theorem27.Theorem26covers the case of logics like K,KD,KT,K+confluence.More generally,such strategies can account for many modal logics without transi-tive accessibility relations,in particular those with symmetric accessibility relations such as KB,KDB,and KTB.10.2Testing loopsOur second criterion applies in particular to logics with transitive acces-sibility relations.In these logics we use a dynamic loop test(DLT):the createSuccessor-rule does not apply in a node if this node is included in some ancestor4.We say that createSuccessor-rule is enriched by DLT. We discuss how DLT is implemented in LoTREC in the conclusion. THEOREM27.Let RULES be a set of rules such that for every r∈RULES,•r is analytic(only subformulas of the original formula are added to nodes in the action part of r);•The createSuccessor-rule does not apply in a node which is included in some ancestor.Let S be a strategy that is built from RULES.Then the tableaux construc-tion terminates,in other words apply S to any formula leads to a looping tableau.Proof.First,observe that due to the subformula condition:•every node only contains afinite number of formulas,and•there can only be afinite number of nodes with differing associated set of formulas.4...i.e.we have x0(SΣ)+x,and every formula A appearing in x also appears in x0.。