Completeness of Higher-Order Duration Calculus
数学分析
数学分析有理数无理数集合函数绝对值不等式三角形区域邻域确界原理确界上确界下确界开区间闭区间有界集自变量因变量符号函数定义域值域复合函数反函数初等函数常量函数幂函数指数函数对数函数三角函数反三角函数有界函数单调函数奇函数偶函数周期函数数列极限收敛数列发散数列唯一性有界性保号性保不等式性Mathematical analysisRational numberirrationalA collection offunctionThe absolute valueinequalitytriangleareaneighborhoodWorld indeed principleWorld indeedsupremuminfimumOpen intervalClosed intervalBounded setThe independent variablesThe dependent variableSymbolic functiondomaindomainComposite functionInverse functionElementary functionConstant functionPower functionExponential functionLogarithmic functionTrigonometric functionsInverse trigonometric functionBounded functionMonotonic functionOdd functionEven functionsPeriodic functionSequence limitConvergent sequenceDivergent seriesuniquenessboundednessProtecting,Protecting the inequality保不等式性迫敛性四则运算法则子列单调数列单调有界定理自然对数致密性定理柯西收敛准则函数极限单侧极限局部有界性海涅定理无穷小量有界量高阶无穷小量等价无穷小量无穷大量渐近线连续性可去间断点跳跃间断点分段函数介值性定理一致连续性导数和微分单侧导数导函数极大值极小值费马定理稳定点链式法则光滑曲线高阶导数莱布尼茨公式微分可微函数高阶微分微分中值定理罗尔定理拉格朗日中值定理Protecting the inequalityForced convergence propertyFour algorithmsThe child columnsMonotone sequenceMonotony is definedNatural logarithmCompactness theoremCauchy convergence criteriaFunction limitUnilateral limitLocal boundednessHeine theoremdimensionlessA bounded amountHigh order dimensionlessEquivalent infinite smallinfinityasymptotecontinuityTo go discontinuitiesJump discontinuity pointPiecewise functionIntermediate value theoremUniform continuityDerivative and differentialUnilateral derivativeDerived functionThe maximumminimumFermat's theoremThe stable pointThe chain ruleSmooth curveHigher derivativeLeibniz formuladifferentialDifferential functionHigh order differentialDifferential mean value theoremRoller's theoremLagrange mean value theorem导数极限定理达布定理柯西中值定理不定式极限泰勒公式佩亚诺型余项泰勒多项式麦克劳林公式极值凸性拐点凸函数凹函数詹森不等式实数的完备性区间套定理魏尔斯特拉斯聚点定理有限覆盖定理开覆盖无限开覆盖有限开覆盖上极限下极限最大聚点最小聚点不定积分原函数换元积分法分部积分法有理函数待定系数法三角函数有理式欧拉变换定积分分割模积分和黎曼和黎曼可积牛顿—莱布尼茨公式The derivative limit theoremDarboux theoremCauchy mid-value theorem are obtained Infinitive limitTaylor formulaMore than jarno typeTaylor polynomialMcLaughlin formulaThe extremeconvexityInflection pointConvex functionConcave functionJensen's inequalityCompleteness of real NumbersNested interval theoremChris whales, lars accumulation point theorem Limited covering theoremOpen coverThe infinite open coverLimited open coverOn the limitUnder the limitThe biggest accumulation pointThe week-long pointIndefinite integralThe functionIntegral method in yuanDivision of integral methodRational functionMethod of undetermined coefficients Trigonometric function rational expression Euler transformationDefinite integralsegmentationdieIntegral andRiemann andRiemann integralNewton-Leibniz formula可积条件充要条件可积函数类振幅狄利克雷函数黎曼函数积分第一中值定理积分第二中值定理积分型余项上积分下积分施瓦茨不等式闵可夫斯基不等式弧长曲率微元法反常积分无界函数瑕积分绝对收敛柯西判别法狄利克雷判别法阿贝尔判别法阿基米德性数项级数敛散性级数收敛比式判别法根式判别法积分判别法交错级数绝对收敛级数条件收敛级数函数列一致收敛性内闭一致收敛函数项级数收敛域魏尔斯特拉斯判别法Integrable conditionNecessary and sufficient condition Integrable function classThe amplitudeDirichlet functionRiemann functionThe first mean value theorem for integrals Second mean value theorem for integrals More than integral typeThe integralThe integralSchwartz inequalityMinkowski inequalityArc lengthThe curvatureThe micro element methodImproper integralUnbounded functionDefect integralAbsolute convergenceCauchy criterionDirichlet criterionAbel criterionArchimedeanitySeveral seriesDivergence sexThe series convergenceThan type criterionRadical criterionIntegral criterionAlternating seriesAbsolutely convergent series Conditions of convergent series Function columnUniform convergenceWithin the closed uniform convergence Series expressed by function terms Convergence domainChris whales, criterion收敛区间收敛半径傅里叶级数周期三角级数收敛定理按段光滑贝塞尔不等式多元函数余集边界开集闭集连通性闭域点集直径区域有界点集累次极限重极限有界闭域微分学偏导数连续可微法线内函数外函数全微分方向导数梯度凸区域中值公式中值定理黑赛矩阵隐函数显函数隐函数定理隐含数组一一映射逆映射切平面法线Convergence rangeRadius of convergenceFourier seriescycleTrigonometric seriesConvergence theoremAccording to the period of smoothBessel inequalityMultivariate functionI setThe borderOpen setClosed setconnectivityClose your domainPoint setThe diameter ofareaA bounded set of pointsThe iterated limitWeight limitBounded closed regionDifferential calculusPartial derivativeContinuously differentialnormalWithin the functionOutside the functionTotal differentialDirectional derivativeThe gradientConvex regionThe median formulaMean value theoremThe black race matrixImplicit functionExplicit functionThe implicit function theoremImplicit arrayOne-to-one mappingThe inverse mappingThe tangent planenormal含参量积分欧拉积分伽玛函数贝塔函数曲线积分重积分可求面积二重积分三重积分格林公式单连通区域复连通区域极坐标变换被积函数积分变量柱坐标变换球坐标变换反常二重积分无界区域曲面积分高斯公式斯托克斯公式向量场梯度场散度场旋度场向量函数微分学Integral containing parametersEuler integralThe gamma functionBeta functionCurvilinear integralDouble integralAsk for the areaDouble integralTriple integralGreen's theoremSimply connected regionAfter connected areaPolar coordinate transformationintegrandThe integral variableColumn coordinate transformationThe ball coordinate transformationAbnormal double integralUnbounded regionSurface integralGauss formulaStokes formula of vector fieldGradient fieldThe divergence fieldThe curl fieldVector function differential calculus专业名词集锦矩阵{数} matrix; array矩阵变换法matrix transform method;矩阵表示matrix notation; matrix representation;矩阵表示法matrix representation;矩阵代数matrix algebra; algebra of matrices;矩阵定理matrix theorem;矩阵法matrix method; matrix technique;矩阵范数matrix norm;矩阵方程matrix equation;矩阵方法matrix method;矩阵分块partitioning of matrix;矩阵分数matrix fraction;矩阵分析matrix analysis;矩阵符号matrix notation;矩阵函数matrix function; function of a matrix;矩阵环matrix ring;矩阵迹traces of matrix;矩阵计算matrix computation; matrix calculation; matrix calculus; 矩阵记号matrix notation;矩阵阶order of a matrix;矩阵解法matrix solution;矩阵列rectangular array;矩阵论matrix theory;矩阵群matrix group;矩阵特征根characteristic root of a matrix;矩阵特征值matrix eigenvalue;矩阵微分方程matrix differential equation;矩阵元素{数} matrix element;矩阵运算matrix operation;矩阵组set of matrices正则{数} {物} canonical; regular正则边界regular boundary;正则变换regular transformation;正则表示regular representation;正则参数regular parameter;正则点regular point;正则方程式regular equation;正则分布canonical distribution;正则函数regular function;正则化regularization;正则矩阵regular matrix;正则空间regular space;正则扩张regular extension;正则理想regular ideal;正则列regular column;正则奇点regular singular point; regular singularity;正则群regular group;正则-P群regular P-group;正则算子regular operator;正则微分方程regular differential equation;正则系统canonical system;正则形式canonical form;正则性regularity;正则映射regular mapping;正则值regular value;正则坐标canonical coordinates微分{数} differential:全微分total differential;偏微分partial differential;二项式微分binomial differential; differentiation微分包含differential inclusion;微分表示differential representation;微分参数differential parameter;微分-差分方程differential-difference equation;微分代数differential algebra;微分动力系统differentiable dynamical system;微分动态规划differential dynamic programming;微分多项式differential polynomial;微分方程（式）{数} differential equation;微分方程解法solution of differential equation;微分方程组simultaneous differential equations;微分符号differential sign;微分积分方程式differential-integral equation;微分几何（学）differential geometry;微分矩阵differential matrix;微分曲线differential curve;微分算符differentiating operator; differential operator; 微分算子differentiating operator; differential operator; 微分系数differential coefficient; differential quotient; 微分线性differential linearity微分形式differential form;微分学differential calculus;微分映射differentiable mapping;积分{数} integral; integrate; integration:定积分definite; integral;不定积分indefinite integral积分变换integral transformation; integral transform;积分变换法integral-transform method;积分变量integration variable;积分变数integral variable; variable of integration;积分表table of integrals;积分表示integral representation;积分不等式integral inequality;积分差分方程integro-difference equation;积分常数integral constant; integration constant;积分对数integral logarithm;积分法integration;积分方程（式）integral equation;积分公式integral formula; formula of integration;积分号sign of integration;积分几何学integral geometry;积分区间integrating range;积分曲线integral curves;积分算子integral operator;积分微分方程integral differential equation;积分-微分算子integro-differential operator;积分微分运算integral differential operation;积分学integral calculus;积分因数(子) integrating factor;积分域domain of integration; field of integration;正交{数} {物} {电} orthogonal; perpendicular; normal; orthogonality; quadrature正交变换{数} orthogonal transformation;正交变换器quadrature transformer;正交表orthogonal array;正交表示orthogonal representation;正交补{数}orthogonal complement;正交单位向量组{数} orthogonal vectors;正交多项式{数} orthogonal polynomial;正交关系orthogonality relation;正交轨线{数} orthogonal trajectory;正交函数orthogonal function;正交函数系orthogonal function system;正交函数展开式orthogonal function expansion;正交函数族orthogonal family of function;正交矩阵{数} orthogonal matrix;正交曲面{光} normal surface; orthogonal surface;正交曲线系orthogonal system of curves;正交射影orthogonal projection; orthographic;正交投影{测} rectangular projection;正交系{数} orthogonal system;正交（晶）系rhombic system; orthorhombic system;正交线{数} cross line;正交向量{数} orthogonal vectors;正交性{数} orthogonality;正交元素{数} orthogonal elements; orthogonal quantities;正交阵列orthogonal array;正交直线组{数} orthogonal lines;正交坐标orthogonal coordinates;正交坐标系orthogonal coordinate system特征characteristic; feature; properties; aspect; trait:特征变量characteristic variable;特征参数characteristic parameter;特征常数characteristic constant;特征点characteristic point;特征多项式characteristic polynomial;特征二次型{数} characteristic quadratic form;特征泛函characteristic functional;特征法characteristic method;特征方程characteristic equation; proper equation;特征根characteristic root; latent root;特征函数characteristic function; eigenfunction; proper function; 特征集feature set;特征矩阵eigenmatrix; characteristic matrix;特征空间feature space;特征数characteristic;特征线characteristic curve; characteristic line; characteristics;特征线法method of characteristic curves; characteristics method; 特征向量characteristic vector; feature vector;特征因数characteristic factor;特征元characteristic element;特征子群characteristic subgroup有界{数} bounded有界变差 bounded variation; limited variation;有界变差函数 functions of bounded variation;有界变数 bounded variable;有界表示 bounded representation;有界泛函 bounded functional;有界函数 bounded function;有界集 bounded aggregate; bounded set; limited set;有界量 bounded quantity;有界区域 bounded domain;有界数集 bounded set of numbers;有界算子 bounded operator;有界投影 bounded projection;有界性 boundedness;有界序列 bounded sequence线性{数}linear;linearity线性表示：linear expression线性常微分方程：linear ordinary differential equation；线性代数：linear algebra；线性代数方程：linear algebraic equation；线性代数方程组：simultaneous linear algebraic equations；线性度量空间：linear metric spaces；线性泛函：linear functional；线性赋范空间：linear normed spaces；线性空间：linear space；线性偏微分方程：linear partial differential equation；线性算子：linear operator；线性微分方程：linear differential equation；线性无关组：linearly independent quantities；线性相关：linear correlation；线性子空间：linear subspaces；解solution通解：general solution特解：particular solution；particular integral；初值：初值定理：initial-value theorem；初值方法：initial-value method；初值问题：initial-value problem；初值参数：initial parameter；纯量纯量函数：scalar function；纯量积：scalar product；纯量矩阵：scalar matrix算子：operator空间：space（子空间：subspace）范数：norm矩阵：matrix度量：metric。
Resolution
http://www.lsv.ens−cachan.fr/Publis/Research Report LSV−02−8, Lab. Spécification et Vérification, CNRS & ENS de Cachan, France, Jul. 2002A Note on the Completeness of Certain Refinements ofResolutionJean Goubault-LarrecqLSV/CNRS UMR8643,ENS Cachan61,av.du pr´e sident-Wilson94235Cachan Cedex,FranceAbstract.Resolution and many refinements thereof have now been known fora long pleteness is usually proved by semantic means(e.g.,semantictrees,Bachmair-Ganzinger forcing),or by syntactic tricks(Bledsoe’s excess lit-eral technique).The purpose of this paper is to note that there is a completelyproof-theoretic way of proving completeness for several refinements of resolu-tion,resembling Gentzen’s method of cut-elimination.While this technique hasa number of shortcomings compared to the semantic arguments cited above,it isvaluable in that the completeness proofs for different refinements are the same.We have found this proof technique to be effective in teaching the ins and outs ofrefinements of resolution to masters level students.This can also be used to ex-tract propositional proofs in one resolution format from resolution proofs in someother format automatically;in thefirst-order case,the same technique allows oneto extract ordered resolution or hyperresolution proofs from proofs in any otherresolution format mechanically.1IntroductionContext and related work.Resolution[10]has several refinements,among which hy-perresolution,semantic resolution,ordered strategies,and combinations thereof,mostof which being complete[2].The classical proofs of their completeness is different ineach case,and sometimes feel ad hoc.Amongst these techniques,wefind Kowalski andHayes’semantic tree technique[9],which works nicely for ordered resolution,hyper-resolution and semantic resolution;Bachmair and Ganzinger’s forcing technique[1],which is often used for ordered resolution with selection and provides an explicit modelconstruction in case no refutation can be found.These semantic methods have severaladvantages,including the fact that it is easy to show that several deletion strategies(tautology elimination,subsumption)as well as additional rules(e.g.,splitting[5]orcondensation[8])can be used without destroying the completeness of the base calcu-lus.One syntactic method of proving completeness is Boyer’s excess literal technique,which applies notably to Boyer’s locking and to semantic resolution,including hyper-resolution:see[2].It is then usually somewhat awkward to teach students a course in resolution tech-niques,as refinements look like a hodge-podge of ad hoc tricks,with specific semanticor syntactic completeness arguments.The purpose of this paper is threefold:–First,to show one unifying intuition behind several refinements of resolution:res-olution,at least in the propositional case,is Gentzen’s cut rule,and there are many ways that a proof using only cuts can be rearranged by permuting and distribut-ing cuts across each other.Normal proofs will then be of specific forms,allowing one to reduce non-determinism in proof search.This is very much in the spirit of Gentzen’s Hauptsatz for sequent calculi[13].While permuting and distributing cuts is easy,showing that this process terminates is slightly more challenging.(Note that weak termination would be enough,but it is not much harder to show strong termination:all rewrites terminate.)–Second,to provide a simple argument by which termination is ensured,hence from which completeness follows.This simple argument(condition(5)below)is enough to retrieve some of the most well-known refinements of resolution.–Third,to provide effective translations between resolution formats.For example, our argument allows one to rewrite any ordered resolution refutation into a positive hyperresolution refutation from the same set of clauses.This can be used to provide human-readable proofs from machine-generated proofs;positive hyperresolution derivations,notably,tend to be more readable than ordered resolution proofs.On the other hand,it is not the purpose of this paper to show that resolution refine-ments are still complete in the presence of such or such deletion strategy,for which semantic trees or Bachmair and Ganzinger’s technique are preferable.This is mostly an orthogonal concern.For example,it is still possible to show that subsumed clauses and tautologies can be eliminated,when they can,by syntactic methods[7].Neither is it the purpose of this paper to introduce new refinements of resolution,or to introduce a uni-versal completeness proof.In particular,it seems that certain refinements of resolution, e.g.,ordered resolution with free selection of negative literals,are not easily amenable to the technique described here.Outline.Because we can always rest on lifting arguments,we mostly deal with propo-sitional resolution in this paper—that is,until Section5.Wefix notations and recall the resolution principle in Section2.We then introduce our proof transformation rules in Section3,and give sufficient conditions for them to terminate,thus implying complete-ness.Section4illustrates a number of known refinements that can be shown complete by this technique.Although the stress is put on the propositional case in this paper,we deal with thefirst-order case in Section5;this is more difficult to tackle without going through liftingfirst.This effort pays up:our technique provides an effective translation from anyfirst-order resolution refutation to ordered,or to hyperresolution refutations. We conclude in Section6.2ResolutionLet usfix a vocabulary of atoms,,...;literals are either positive atoms or negative atoms.Clauses,,...,arefinite sets of literals,seen as disjunctions.2On propositional formulas,which is the case we deal with except in Section5,the resolution rule is nothing else than Gentzen’s Cut rule:where comma denotes union,and disjoint union in premises(in particular,, ).We write the empty clause,the negation of literal,i.e.,,.In the Cut rule above,is called the cut formula.A resolution proof of a clause from the set of clauses is anyfinite tree of resolution inferences(instance of Cut) whose leaves(at the top)are clauses in and whose root(at the bottom)is.A refutation from is a resolution proof of from.The completeness of resolution,i.e.,that there is a refutation from whenever is inconsistent,can be established by semantic means,or by appealing to the syntactic device of cut elimination:let denote Gentzen’s sequent calculus for classical logic augmented with non-logical axioms taken from(clauses being read as sequents), then eliminating cuts from any-proof of the empty sequent yields one where the only rule is(Cut)[6,7].That resolution is complete will be assumed in the sequel. To show that some refinement of resolution is complete,we only need to rewrite any given refutation of into a refutation of that obeys the constraints of the refinement. We shall do this by using rewrite rules(–below)that express all possible ways of permuting one cut past another.(1)(2)The,,,marks have been added for future reference.For example,we shall say that the-cuts are those marked with(the topmost cuts in each rule,on the right).Recall that we assume that in premises such as,the literals and are distinct and not in.These rules do not terminate in general.However we shallfind a series of condi-tions that ensure termination in the next section,and demonstrate that several known refinements of resolution obey these conditions.33Completeness via Selection FunctionsLet us specify a refinement of resolution by means of a selection function mapping each clause to a subset of literals that we are allowed to take as cut formulas in the (Cut)rule.For example,if returns the set of all-maximal literals for some ordering,and compares literals by comparing the underlying atoms,then we get ordered resolution. We shall give more examples of functions in Section4.We ask the selection function to obey the following axioms.First,selects from literals in the clause:(3) Then,should select the unique literal from each unit(one-literal)clause:(4) for every literal.That is,it is not allowed to select nothing from a unit clause.Before we introduce the last condition,define the-cut rule as the restriction of the(Cut)rule where and.A resolution proof is a -resolution proof if and only if it only uses-cuts.We then require the following condition:In(1),(2),if the-cut is a-cut,and the-cut is not,then the-cuts are-cuts.(5)If is a non--resolution refutation from,there must be a lowest instance of (Cut)that is not a-cut.This lowest non--cut cannot be the last one.Indeed,because for every unit clause,anyfinal instance of(Cut)in a refutation,which derives and therefore must be a cut between two unit clauses,is a-cut.So there must be a-cut below the lowest non--cut.That is,contains the following configuration:(6)non--cut-cutwhich we call a redex.Condition(5)says that(1)and(2)rewrite redexes to configu-rations where the-cut was permuted upwards:all topmost cuts in right-hand sides are required to be-cuts.Since the non--cut in the redex(6)is an instance of(Cut),the clause must equal.Then either contains but not(or by symmetry does not but does),or both and contain:these yield the left-hand sides of rules(1)and(2) respectively.We claim that under the assumptions(3),(4),(5),these rewrite rules(1)and(2) restricted to apply to redexes(6)terminate.It will follow that we can always transform any resolution refutation into a-resolution refutation;in particular,-resolution will be complete.4To establish termination,define an interpretation of proofs asfirst-order terms built on one constant and two binary function symbols and,representing-cuts and non--cuts respectively.Recall that is the set of clauses that we start from:-cutnon--cutWe assume and to be commutative,i.e.,to avoid ambiguity in this translation.(We might also untie the knot by imposing that be,say,the translation of the premise where the cut formula is positive,but this would unnecessarily duplicate the cases to handle.)Translating(1)and(2)through yields the following rewrite rules:(7)(8) where ranges over.That is,by rules(1)or(2)impliesby rules(7)or(8);this is an easy check,using condition(5).Lemma1(Termination).The rewrite system(7),(8)terminates.Proof.Let be the set of terms that are terminating,i.e.such that every rewrite starting from isfinite.Let the contexts be terms with one hole,denoted( where ranges over terms);denotes with the hole replaced by the term .Similarly,let be the context obtained by replacing the hole of by the context .Define-contexts inductively by:is an-context,and if is an-context and,then is an-context.Finally,say that a term is reducible if and only if for every-context;the set of reducible terms is written .Observe that:(a).This is because is an-context.Note also:(b)if and,then.Indeed,for ev-ery-context,is an-context by construction and,so .We also have:(c)if and then.Indeed,for any -context,,so the one-step reduct is in,too.We claim that:(d)if,then.To this end,let be any-context,and let us show that.Tofix ideas,write as,and let us show the claim by induction on ordered lexicographically,where terms,,,...,are com-pared via the relation—which is well-founded on and on5.Then look at one-step reducts from.Some of them are obtained by contracting a redex in,in,or in some,:then by(c)and the induction hypothesis the obtained one-step reduct is in;another is obtained by contracting the redex,provided.If this reduces by(8),the reduct is where. Note that and are in by(b).So,if,we may con-clude by the induction hypothesis(with decreased by,and replaced by);if ,then is in becauseis an-context(in particular is in by(c)and(a))and. Similarly,every one-step-reduct obtained by contracting by(7)is in :if,is in by induction hypothesis,using the fact that and by(b);if,is in since is an-context(using)and. Since every one-step reduct of is in,is in,too.Since is arbitrary,is reducible.It is easy to see that.Indeed,for every,is indeed in(an easy induction on).It follows that every term is reducible,by structural induction on.We have just dealt with the base case,and the inductive cases are dealt with by(d)when ,by(b)and(a)when.Since every term is reducible,by(a) every term is in.Most standard methods in rewriting fail to prove Lemma1.In particular,the recur-sive path ordering[3]cannot deal with rules(7)or(8)when.In fact,this rewrite system is not simply terminating,and being included in a recursive path ordering im-plies simple termination.Recall that a rewrite system is simply terminating if and only if plus the simplification rules,is terminating.A counter-example is:by(8)by simplification Condition(5)is also maximal in that any liberalization leads to non-termination; in fact even weak termination(existence of normal forms)fails with any liberalized form of condition(5).Allowing some-cuts to be non--cuts while the-cut is a-cut would mean creating redexes on the right-hand side,leading immediately to non-terminating behavior.More subtly,allowing some-cuts as well as the-cut to be non--cuts,which might seem a benign extension,also leads to non-termination. Consider the case of rule(1)for example,then allowing the latter would enable the fol-lowing non-terminating behavior.We have elided the actual clauses,which are unim-portant.Firstnon--cutnon--cut -cut rewrites to-cutnon--cut-cut-cut6by(2),then to-cut non--cutnon--cut-cutif we allow for the indicated liberalization of rule(1).The latter derivation is then again a-cut under a non--cut under a non--cut,which allows us to start this cycle of reductions all over again.Having proved Lemma1,we are now done:Theorem1.Every resolution refutation from can be effectively transformed into a -resolution refutation from,provided(3),(4),(5)hold.Corollary1.Every refinement of resolution based on a selection function satisfying (3),(4),(5)is complete.4ApplicationsWefirst show that Corollary1allows us to justify some standard refinements of resolu-tion.As announced in the introduction,we won’t deal with every known refinement of resolution.In particular,free selection functions[1],even in the propositional case that we are now considering,do not seem tofit well in this framework.4.1Ordered resolution.Let be a strict ordering on atoms.Ordered resolution is the case where is the set of all literals such that is maximal in:i.e.,there is no in,with a possibly different sign,such that.Clearly,(3)and(4)hold.For(5),assume that,,and is a -cut in(1):In other words,(a)is maximal in,and(b)no atom in is greater than in.If is not a-cut,then there must be an atom greater than in.It cannot be in by(b),so we must have.So is maximal in, otherwise there would be a greater atom in:then,contradicting (b).must also be maximal in,otherwise there would be a greater atom in ,so,contradicting(b)again.7The argument for rule(2)is similar:Since is a-cut,(a)is maximal in,and(b)is maximal in. If some-cut is not a-cut,say the left one by symmetry,then is not maximal in ,so by(b).Then is maximal in both premises of.Therefore (5)holds.In particular by Corollary1ordered resolution is complete.There was in fact an easier syntactic proof of termination here.Modify the trans-lation so that:where for each atom there is a new binary commutative function symbol.Then the termination of the rewrite system on proofs of Section3in this case can be shown by using a multiset path ordering[3]with the precedence iff.This applies since there are onlyfinitely many atoms in any given refutation,therefore is well-founded.4.2Positive hyperresolution.Let be the set of negative atoms in if any,otherwise.A-cut between two parent clauses and is then such that only contains positive atoms,otherwise cannot be selected.Conversely,every positive hyperres-olution step defined as(Cut)where one premise is a positive clause(a clause containing only positive atoms)is a-cut with this definition of.Again,conditions(3)and(4)are clear.If the left-hand side of(1)(or(2))is a redex, we claim that must be a positive clause.Indeed,since the-cut is a positive hyperresolution step,the only other possibility is that,resp.,is a positive clause;then one of the premises of the-cut must be positive,so the-cut would be a-cut,contradicting the fact that we have got a redex.So is a positive clause,hence the-cut is a positive hyperresolution step,i.e.,a-cut.Similarly for (2).Since is a positive clause in any case,condition(5)holds.Therefore positive hyperresolution is complete.In fact,we can spell out the rewrite rules(1),(2)in this case as follows:(9)8(10) where it should be clearer that the-cuts must be positive hyperresolution steps.Here is positive,however we shall use the same rules in Section4.4without this restriction.The alternate form of hyperresolution where macro-steps with premises are used,consisting of a non-positive clause(the nucleus)and positive clauses(the elec-trons)[2]which are resolved in steps to yield a new positive clause is complete,too. It is enough to notice that in a positive hyperresolution refutation,as defined above,if some-cut has a non-positive conclusion,then going down the refutation we eventu-ally reach a positive clause:at the latest,is a positive clause.Hence we can extract a refutation consisting entirely of macro-steps from any-resolution refutation.We let the interested reader check that the only role of in the argument of Sec-tion3is to show that the last instance of(Cut)in a refutation is a-cut.In positive hyperresolution we may generalize:any resolution derivation of any positive clause must end in a positive hyperresolution step(a-cut).We can then replay Section3: every resolution derivation of can be effectively transformed into a positive hy-perresolution derivation of.In particular,we get the well-known fact that positive hyperresolution derives exactly the same positive clauses as unconstrained resolution. 4.3Negative hyperresolution,semantic resolution.Negative hyperresolution is obtained similarly by letting be the set of all positive atoms in if any,otherwise.In general,-resolution,a.k.a.semantic resolution,where is a set of atoms,that is,a Herbrand interpretation,is obtained by letting be the set of all literals that are true in if any,otherwise.Just as in the case of positive hyperresolution,in rules(1)and(2)must be restricted to be false in.We do not need to use Corollary1here,though:completeness of-resolution fol-lows from that of positive hyperresolution by renaming every atom to in clauses whenever,and noticing that such a renaming preserves the existence of refuta-tions.4.4Semi-ordered hyperresolutionWhile imposing an ordering constraint on both premises of positive hyperresolution steps destroys completeness,it is well-known that imposing that be maximal only in the positive clause leads to a complete refinement of resolution.This is called semi-ordered hyperresolution in[7]to distinguish it from the aforementioned incom-plete ordered refinement of hyperresolution.This is obtained by letting be the set of all negative atoms in if any,otherwise is the set of maximal(positive) atoms in.9Curiously,Corollary1does not apply directly.The reason is that condition(5)is not satisfied.Indeed,note that-cuts are cuts where one premise is positive and the cut formula is maximal.Then in rule(1)it might be the case that is a positive clause with and maximal in(whence is a-cut),but is also positive with but is not maximal in(whence is not a-cut);in this case there is no reason why should be a-cut:neither nor is positive,in particular.It might be that there is still a way of showing that(1)and(2)terminate in this case,too.However,an easier way of converting any resolution refutation into a semi-ordered positive hyperresolution refutation is to proceed in two steps.First convert into an ordered refutation as in Section4.1,then convert into a positive hyperreso-lution refutation as in Section4.2,using rules(9)and(10).Then apply the following lemma.Call an instance of(Cut)-ordered if and only the cut formula is maximal in the one premise where is positive.A resolution derivation is-ordered if every step in it is.It is clear that ordered resolution derivations are-ordered,while positive hyperres-olution derivations that are-ordered are exactly the semi-ordered positive hyperres-olution derivations.For convenience,assume that is total here:if is maximal in then is greater than or equal to all atoms in.Lemma2.If by rules(9)or(10)and is-ordered,then is-ordered. Proof.The-cuts are clearly-ordered.It remains to show that the-cuts are,too. If is a positive literal in(9),then by assumption;since, in particular,so the-cut is-ordered.If is a negative literal in(9)then by assumption,so again the-cut is-ordered.In the case of rule(10),by symmetry we may assume that;then by assumption, so since;so the-cut is-ordered.It follows that the end refutation obtained in the two-step process above is both a positive hyperresolution refutation and-ordered,so it is a semi-ordered positive hyperresolution refutation.Again,similar arguments show that semi-ordered negative hyperresolution or semi-ordered-resolution are complete,and give an effective procedure to transform any resolution refutation in one of the required format.5The First-Order CaseWhile this is not completely immediate,the rewrite rules(1)and(2)generalize to the first-order case,where the resolution rule reads:provided is the most general unifier of.For any substitution ,we write the result of applying the substitution to the atom,and we use10similar notations for substituting in clauses.As is traditional,we leave implicit the fact that the parent clauses and arefirst renamed so that they have no free variable in common.For short,write for the set.Such sets will always be assumed to be non-empty,i.e.,.Let denote the most general uni-fier of.Extend this to literals and to several equality signs:is the most general common substitution(if any) that instantiate to the same literal,and...,andto the same literal.Let,the domain of,be the set.We shall always assume that most general unifiers are idempotent,i.e.,for every,is not free in. If is idempotent,letting be the set of all equations,,then ;furthermore,if and are idempotent and,then(we write the substitution mapping every variable to ),and if two sets of equations and have the same unquantified equational consequences,then.In particular if then.The rewrite rules(1)and(2)change as follows in thefirst-order case.The left-hand side of rule(1)now reads:(11)Now,because clauses are renamed apart,.Also,.It follows in particular thatexists,that exists and that.We can then rewrite the redex above to:(12)where.Since,the conclusion of(12)is the same as that of(11).The case of rule(2)is slightly more complicated.The left-hand side is:(13)11where denotes the union of the sets and.Recall that both and are assumed not empty.Again,and.Let now be a renaming(one-to-one,mapping variables to variables)substitution so that as no free variable in common with.Then is a unifier of ,so exists;is a unifier of,soexists.Then Note that and have disjoint domains.So the union makes sense,and we claim that it unifies.Indeed, unifies and(because is their common most general unifier),and also and.In particular,unifies,and is therefore an instance of.(For any atom,clause or substitution,an instance of is any atom,clause or substitution, for any substitution.We then say that is more general than if and only if is an instance of.)We may then generate the derivation:(14) Now because all clauses are renamed apart,the bottom clause in(14)is alsoThen,and we have already noticed that is an instance of this.In general,is not the same as,however,contrarily to the case for rule(1).For example,take be the atom,take to be, to be,to be.Let us say that is,and assume that every two variables with different names are distinct.Then,and therefore also ,map,,,to the same term.On the other hand,only equates with and with.With the rules(11)(12)and(13)(14)we can now define a transformation on derivations as follows:if and only if either is of the form(11)and is(12), or is of the form(13)and is(14),or is(15)12and,where derives a clause that is more general than—say, and—,and is:(16) Note indeed that unifies,hence the substitution mapping every variable free in to and every variable free in to unifies.So exists and is more general than.Since,is also more general than.It follows that the relation is well-defined and rewrites derivations of clauses into derivations of more general clauses.In particular,it rewrites refutations into refutations.As in Section3,let be a selection function,and call afirst-order resolution step a-step if and only if all literals resolved upon(and in(15)for example)are selected in their respective clauses.The last difficulty that awaits us in adapting the arguments of Section3is that the resolution step of(15)might be a-step while that of(16)fails to be,or conversely.It turns out that converting a non--step into a-step is benign,while the converse leads to non-termination.The former requires us to add the following rewrite rule to(7),(8):(17)Then the termination Lemma1extends smoothly:Lemma3.The rewrite system(7),(8),(17)is terminating.Proof.As for Lemma1.The only additional case is in claim(d),where inthe subterm rewrites to,with, and,.Then the contractum is,where is in and is clearly an-context, so that.At this point,the technique of Section3applies almost without modification.We only have to replace condition(5)by the condition that,whenever the bottom resolution step in(11)is a-step and the top step is not,then the top resolution step in(12)is a -step;and similarly,that whenever the bottom resolution step in(13)is a-step and the top step is not,then the top two resolution steps in(14)are-steps.By extension, call this condition(5)again.Then:Theorem2.Let the selection function be stable:for every literal,clause,and substitution,if then.Assume that conditions(3),(4)and(5) hold.Then every resolution derivation of from can be effectively transformed into a-resolution derivation of some clause more general than from.13Proof.If(15)is a-step,then.Letting as above be and be,it obtains,so by stability.Therefore(16) is a-step ing the obvious adaptation of the interpretation,terminates if the rewrite system(7),(8),(17)does.Then apply Lemma3.Finally,observe that-normal forms consist only of-steps,using condition(4).This requires showing that the last resolution step in a refutation is a-step,and taking bottommost non--steps,as in Section3.And indeed the last resolution step must be a resolution between clauses and,with;by condition(4),since this is a unit clause,and similarly,so by stability and, therefore the last resolution step is a-step.This can be used to extract ordered refutations from any resolution refutation: Corollary2.Let be a stable ordering,i.e.,implies for every sub-stitution.Then everyfirst-order resolution refutation can be effectively transformed into an ordered resolution refutation(wrt.)from the same set of clauses.Proof.Apply a similar argument as in Section4.1.Consider rule(11)(12),and as-sume that(a)every literal in is maximal in,and(b)every literal in is max-imal in.By stability,(b’)every literal in is maximal in. If the topmost resolution step in(12)is not ordered,then,because of(a),some literal in is less than some literal in.By(b’)this literal in must be less than some literal in:(c).We claim that every literal in is maximal in.Otherwise there would be a literal in that is less than some literal in.First,cannot be in:since instantiates all of to the same atom,if then by stability,contradicting.Second,cannot be in:otherwise implies by stability;since by(c)and stability, ,contradicting(b).Third,cannot be in:otherwise(by stability)(by(c)and stability), a contradiction.So every literal in is indeed maximal in,therefore the topmost resolution step in(12)is a-step.The case of rule(13)(14)is similar.Now since defined as the set of maxi-mal literals in is stable in the sense of Lemma3,the result follows.It is probably more interesting to rearrange resolution refutations into hyperreso-lution proofs instead.Recall that positive hyperresolution derivations can be seen as proofs deriving new facts from old facts[2]:arguably these proofs look like standard mathematical proofs,proceeding from assumptions to theorems.The following corol-lary can then be used to improve the readability of proofs obtained by any resolution theorem prover,provided it keeps a trace of the proof obtained,by extracting a positive hyperresolution proof from it.Corollary3.Everyfirst-order resolution refutation can be effectively transformed into a positive hyperresolution refutation from the same set of clauses.Proof.As in Section4.2,this is the case where is a positive clause in(11),(12), (13),(14).Condition(5)is then verified.Moreover,the corresponding selection func-tion,which selects literals based on signs,is clearly stable.14。
时间计算方法
Architecture and Performance Methods ofA Knowledge Support System ofUbiquitous Time ComputationYinsheng ZhangInstitute of Scientific & Technical Information of China, Beijing, ChinaCity University of Hong Kong,Hong Kong, ChinaEmail: zhangyinshengnet@Abstract— An architecture and main performance methods of a knowledge support system of ubiquitous time computation based on relativity are proposed. As main results, modern time theories are described as certain relations of term-nodes in a tree, and some space-time computation models in a large scale and time computation models in different time measurement systems (institutions) are programmed as interfaces for time computation in complex conditions such as time-anisotropic movement systems or gravity-anisotropic environments.Index Terms—Space-Time, Relativity, Real Time Communication, Time Ontology, Time MeasurementI.I NTRODUCTIONTime computation is so ubiquitous nowadays, not only in analyzing texts with time terms, but also in real time computation even in circumstance across time zones or in quantum application such as satellite positioning systems, time-anisotropic movement systems, gravity-anisotropic environments, or space scale in the cosmos. As the relativity theory and quantum mechanics, which we call modern time theories, have made great advances, time computation is desirable to be made on the new time knowledge. It is well known that an ontology made up of specific terms in relations can succinctly represent knowledge homogeneously structured in syntactic pattern and stratified in entailments or in contents with stem-branch relations, and easily be applied to navigate knowledge by relational calculus, so a time knowledge support system based on time ontology with some computational models is proposed here to suffice requirement of time computation based on modern time theories.II.E XTENSION OF T IME E XPRESSIONTime mostly is expressed in a form of natural number and suitable for a unified time measure system in the Earth. For example, Dan Ionescu & Cristian Lambiri[1], E.-R.Orderog & H.Dierks[2], and Merlin [3] respectively gave time definitions or expressions for the real-time system, which, however, relativity of time, time computation models which define how to calculate time units, are omitted. In contrast to some software application fields’ research, some time science organizations give serial time expressions based on modern time theories, among which the International Astronomical Union (IAU,1991) made time definition widely accepted in a reality frame [4] . Thus we need to integrate these definitions and expressions in a complete and standard form for ubiquitous time. To do this, we give a time expression as follows.The physical quantity of time can be expressed as a 4-tuple:T=< D,U,M,I > (1) where,D: Data about time in quantity, it may be numbers or circle physical signals indicating time, or symbols expressing a time in quantity; that is, D∈{ time reading, tick, time number expression}.U: Unit, the measure unit such as “second”,” day”.M: Model, the mathematical formulae, using which you get a time quantity by mathematical computations.I: Institution, it may be indicated by a code which stipulates what unit U is meaningful, from which start time point S an interval can be fixed, according to what model M about time can be computed. So we use I( ) to indicate determining a time physical quantity by some parameters.For example, you say “2 seconds”, you might refer to two units of the Universal Time i.e., of coordinated universal time (CUT, or UTC) set by IAU and the finally arbitrated by the International Telecommunication Union (ITU). Of course, you probably might not refer to that, but to an atomic time (AT), as it may. Both the quantities can be computed by the corresponding models issued by the related organizations. Here, the institution determines the meanings of the time as a physical quantity and gives the computation methods, so we can give an expression similar with a programming expression as T=I(D,U,M), here, T serves as a return value ,and I, a function for the other parameters.Clearly, to set up a knowledge support system, we need to consider this time expression, its elements in the tuple will constitute the main profiles.© 2013 ACADEMY PUBLISHER doi:10.4304/jsw.8.11.2947-2955Figure 1. The architecture of the knowledge support system ofubiquitous time computation.III. A RCHITECTURE OF THE K NOWLEDGE S UPPORTS YSTEM We designed such an architecture for the knowledge support system developed by the author for the time computation in the complex systems.The system mainly made up of the 4 components that ①Time Knowledge Navigation, ② Time Measurement and Computation Models, ③ Time Expression Semantics Computation Models,④ Time Institution Knowledge Texts.Component ① accepts users’ requests for knowledge relating to the time measuring data, for example, a user requests for a model for computing the derivation between its time readings and a time unit in another space or in a time measurement system. The kernel of Component ① is a tree describing time knowledge profiles, say its branches are classifications of the time knowledge in certain relations. It is a catalogue of classification and relations of time knowledge, and also mappings between the classification and the knowledge in Component ② and Component ③. It contains institutions I in (1), which determines Component ② and Component ③ in logic, however, Component ② and Component ③ are listed for directing call not through the nodes of institutions.Component ② is the mathematical models for time measurement and computation, written in software programs and can be called for other time computation programs.Component ③ and ④ are discussed in number V and VI.IV. T IME ONTOLOGY.4. 0 General Description sThe tree in Component ① is a time ontology based on modern time theories for logically showing and savingall the knowledge term nodes in certain relations.These relations are potential information for deeper application such as inference based on relational calculus. On time ontology, most studies focus on time expressions and computations of relations between these expressions. For example, Moen’s time ontology is about time concepts in linguistics [5][6].; Frank etc. came up with a plan and principles building space-time in 4 dimensions and 5 tiers [7]. The typical extant time ontology see WordNet in the part of time, DAML time sub-ontology [8],Time Ontology in OWL built by W3C [9] ,and NASASWEET (Semantic Web for Earth an Environmental Terminology)[10]. In addition, ISO 19111 [11] and ISO 19112[12] set out the conceptual schema for spatial references based on geographic identifiers. This work shows various profiles of data structure of time description, yet has the limitations that(1) Time it describes is in the periphery of the Earth, but not in cosmos large scales;(2) The time properties are unraveled only on non-symmetry (non-back as an arrow), a little on relativity, singularity and quantum property.This might lead to difficulties in computations based on modern time theories.In contrast with this work, the time knowledge tree in Component ① is a time ontology based on modern time theories (hereafter “TOboMTT”, the main branches see attachment) .The nodes between any two levels in top-bottom constitute relations which are propositions (note that when we say “A and B in a certain relation”, it just says a proposition) stating the main frame of modern time theories. So, in essence, we have :TOboMTT={N,R }={Propositions} （2）here, N,R refer to nodes and relations respectively.The root (0- level) and the nodes in the next (1-level) are as followingz TimeSpace-Time Type Time Type Time Property Time Measure Time ExpressionThe root “Time” constitutes “has ” relations with the nodes in the 1-level. That is, “Time has the Space-Time Types”, “Time has the Time Types”, “Time has the Time Properties”, “Time has the Time Measures”, “Time has the Time Expressions”. These relations are basic profiles of the up-to-date study on time.The relations of the nodes between the 1 and 2 levels continue such propositions of those relations between 0 and 1 levels, for example, we can say “Time has the Space-Time Types like Euclid Space-Time”, here, “Euclid Space-Time” just is a node in the 2nd level. Thus,© 2013 ACADEMY PUBLISHERthe relations between the 1 and 2 levels are “includes ”, like “Space-Time Type includes Euclid Space-Time”. In the following contexts, we intuitively explain the main nodes which express some important assertions of modern time theories.4. 1 SPACE-Time TYPEAccording to Einstein’s field equation, space andtime are integrated. So we must take space as a parameterof time considering the space-time type. Einstein’s fieldequation see (3) [13]1()+=82R Rg g T αβαβαβαβ−Λπ （3）Here, α and β are space-time dimensions, i.e., α, β=0,1,2,3 and 0 denotes time for the left expression; R αβ is Ricci tensor, it is a 4×4 matrix of the 16 components ofsecond order space-time curvature, R is scalar curvature, g αβ is a 4×4 matrix of metric tensor, Λ is cosmological constant, T αβ is energy-momentum tensor, a 4×4 matrixtoo.From (3), we get (4), i.e., the differentiation of square of space-time intervals:2=ds g dx dy αβαβ (4) here, x,y are curvilineal coordinates, s is space-time interval. (4) adopts Einstein summation convention, normally like in physics, that a repeated index (α or β ) implies summation over all values of that indexed. (3) and (4) are well confirmed by some experiments in the scale 10-13 cm (the radius of a fundamental particle) to 1028 cm (the radius of the universe). A space-time type normally defined by a solution of the equations (3) or (4).See some basic nodes: Space-Time Type Euclidean space-time (absolute time) Riemannian space-time Inertial reference frame space-time Non-inertial reference frame space-time Friedmann- Walke space-time…… If (3) or (4) are determined as the nonlinear partial differential equations about g αβ , we call s is Riemannian space-time, which means space-time is of curvature and might not be flat (flatness is just a special instance, i.e., Minkowski space-time, in which gravity is neglected, it is regarded as inertial). In (3) or (4), if the time in different space places is described as absolutely not different , and independently from its different places and velocities, the space-time is Euclidean space-time or Newton space-time. Friedmann-Lemaître-Robertson-Walker space-time, simply Robertson-Walker space-time [14][15] , put forwarded by Robertson and Walker, and meet the inference of Friedman [16] and Lamaitre [17] , describes homogeneous and isotropic space-time in a non-inertial system, for which, cosmological curvature k and cosmological time t are introduced into (3) or (4). k takes 3 constants 0,1,-1 representing 3 possible space-time types: flatness, positive curvature and negative curvature. If R in (3) is a constant, Robertson-Walker space-time will become some special instance: when R =0, itwill be Minkowski space-time; R >0, de_Sitter space-time;R <0, anti-de_Sitter space-time. Bianchy I space-time is more general than Robertson-Walker that the space-time is homogeneousbut might be anisotropic [18]. Taub-NUT space-time adds magnetic and electric parameters into (3) or (4) [19]. Godel space-time adds rotationally symmetric axis into (3) or (4) [20]. Rindler space-time expresses such space-time determined by inertial system and non-inertial system [21][22]. In some special cases, R is not easy to be determined. To solve (3) or (4), some parameters are given for specialtypes of space-time. These special types include spherical and axial space-time, and time’s elapse may be neglected for a space spot. For (4), Schwarzschild space-time [23] isspherically symmetric beyond a mass sphere. A spherewith great mass and a radius less than Schwarzschild radius is a black hole, which is thought to bear only 3 kinds of information of mass, charge and angular momentum. Schwarzschild black hole is considered as one with only mass, while Ressner-Nordstrom black hole, named as Ressner-Nordstrom space-time, with mass and charge [24][25]; Kerr black hole, named as Kerr space-time with mass and angular momentum [26]; Kerr-Newmanblack hole, named as Kerr-Newman space-time [27], simultaneously have information of mass, charge and angular momentum. Some spherically symmetric space-time like Vaidya space-time [28] and Tolman space-time [29] consider time as the variable of the function of mass and curvature. As an axial metric space-time, Weyl-Levi-Civita space-time [30] is typical. . 4. 2 Time TYPEWhen we solely study time, we can primarily dividetime into the 3 types: Proper time Coordinate time Cosmological time Proper time is the elapsed between two events as measured by a clock that passes through both events. In other words, proper time value is from the real readings of the clock set by an observer in a definite space spot (ifthe measured body moves, then the clock spot and the moved body’s end spot are considered as one area for the two spots are so near for a large scale space). © 2013 ACADEMY PUBLISHERCoordinate time is integrated time under a coordinate system. It is not a real readings for a special spot (the difference between the different spots in the system is neglected), but a stipulated (calculated that it should be) time in the system. Proper time multiplied by (1- v2/c2)-2 is coordinate time (v is the velocity of the body, in which an implied observer is, c is light velocity). If we set a clock in a universe coordinate system indicating the integrated time, it would indicate the universal time (t in Robertson-Walker equation).The proper time in the Earth can be expressed in various forms as the follows.Ephemeris Time (ET) [31] was defined in principle by the orbital motion of the Earth around the Sun. Here, ephemeris is based on Julian calendar which had been reformed to be Gregorian calendar lasted to the nowadays.True solar time (apparent solar time) is given by the daily apparent motion of the true, or observed, Sun. It is based on the apparent solar day, which is the interval between two successive returns of the Sun to the local meridian [32].Mean solar time is the mean values of measured time of the intervals between two Sun passing an identical meridian [33].Sidereal Time is based on a sidereal day; a sidereal day is a time scale that is based on the Earth's rate of rotation measured relative to the fixed stars, normally to the Sun [34]. Sidereal time may be Greenwich Sidereal Time (GST) which calculated by Greenwich Royal Observatory in mean data or Local Sidereal Time (LST) which is computed by adding or subtracting the numbers of timezone [35] .Universal Time (UT) is computed by truly measured time data based on rotation of the Earth, it is a Greenwich Mean Time (GMT) and computed from the start of a midnight of Prime Meridian at Greenwich, and it has different versions such as UT0,UT1,UT2 and Coordinated Universal Time (UTC) for the computations from varying data on non-exact time scales of the Earth rotation. UT0 is Universal Time determined at an observatory by observing the diurnal motion of stars or extragalactic radio sources. It is uncorrected for the displacement of Earth's geographic pole from its rotational pole. This displacement, called polar motion, causes the geographic position of any place on Earth to vary by several metres, and different observatories will find a different value for UT0 at the same moment.UT1 is the principal form of Universal Time. While conceptually it is mean solar time at 0° longitude, precise measurements of the Sun are difficult. UT1R is a smoothly tuned version of UT1, filtering out periodic variations due to tides. UT2 is a smoothed version of UT1, filtering out periodic seasonal variations. UTC is an atomic timescale that approximates UT1. It is the international standard on which civil time is based [36].Atomic time applies the principle of stimulated atom radiation in a constant frequency. The Thirteenth General Conference of Weights and Measures define a second that "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom [37] ". That is a unit of International Atomic Time (ATI).The results of atomic time computed by different local laboratories are called local atomic time.Dynamical Time (DT) [38] is inferred from the observed position of an astronomical object via a theory of its motion, ET is a DT based on revolution of the Earth in replace of UT based on rotation of the Earth meet Newton’s time theory; to meet Einstein’s time theory IAU builds two versions of ET respectively in the system of Terrestrial Dynamic Time (TDT) Barycentric Dynamical Time (TDB).Local civil time is the corrected version of UTC by adding timezone numbers and adjusting daylight saving time [35] .Coordinate time includes centroid coordinate time and Earth-centered coordinate time, they are set by IAU.4. 3 Time propertyThe time properties are divided into 4 kinds as follows.Time PropertyAsymmetryRelativitySingularityQuantum propertyAsymmetry is the property human first discovered, it refers to what seems to be an arrow went out in one direction and not back.Relativity means anisotropy against gravity or in a light-like velocity.Singularity is the property of some places, where the present physical laws break down, or it can be thought of as the property of edge of space-time [39].The quantum property of time refers to that of time in the particle-scale, where time appears the stranger phenomena far from the macro-scale as we see. For example, the former -latter sequence in macro-scale might be isochronous in the quantum –scale [40].4. 4 Time measure4. 4.1 CoordinatorThe space-time expressed in (3) or (4) can’t always be indicated by Cartesian system, mostly due to some properties which are difficult to be indicated by Cartesian system, and also due to the singularity in the space-time which normally cannot be indicated by the real number system. So two kinds of coordinates are mainly introduced, they are general coordinates and special coordinates. The former are popular in common sense, and transforming them for a special purpose we get the latter----special coordinates, which mainly for describing some new metrics -solutions of (3), (4) with some© 2013 ACADEMY PUBLISHERsingularity variables, or for some particular space-time areas.The coordinates special for the metrics are introduced as follows.Schwarzschild coordinate indicates spherical symmetry, it sometimes becomes degeneratation of some more general conditions. Schwarzschild coordinate uses sphere coordinate with the radius r≠2GM/C2 and r≠0, here , G is universal gravitational constant, M is the mass. The coordinate is divided into two areas by r >2GM/C2 and r <2GM/C2 and leads to the two metrics in (3): g00= - (1-2GM/rC2) and g11= (1-2GM/rC2) -1.In Schwarzschild coordinate, there is not the expression that r=2GM/C2 (this is a singularity), but tortoise coordinate covers this singularity.Eddington coordinate does not diverge in r=2GM/C2 and r=0 by the linear transformation of the variables.Kruskal coordinate covers r=2GM/C2 and r=0 too, and more general in indicating space-time than tortoise and Eddington coordinate .Lemaitre coordinate covers r=2GM/C2 with a different method to Kruskal coordinate.Rindler coordinate indicates the space-time determined by both inertial and non-inertial system.Weyl coordinate indicates the function of metric and allows to indicate imaginary numbers.Fermi normal coordinate indicates space-like geodesic which is the trajectory that its covariant differential is 0 for (4). “space-like” denotes the velocity in the area is far less than light speed. And its time axis indicates proper time for a non-inertial or locally inertial conditions.Harmonic coordinate indicates harmonic conditions that coordinates in curved space satisfy a D' Alembert equation, it is a Cartesian-coordinate-like one in curved space.Local inertial coordinate indicates Minkowski space-time.The special coordinates for the particular space-time areas are introduced as follows.Centroid coordinate (center-of-mass coordinate system) is one taking the centre of a space area as the origin of coordinate. These coordinates include non-rotating geocentric reference system, rotating geocentric reference system, Barycentric Celestial Reference System (BCRS), International Celestial Reference System (ICRS).Non-rotating geocentric reference system takes the Earth centre as the origin of coordinate . IAU provides the metric and methods for computing proper time.Rotating geocentric reference system is supposed as rotated with the Earth together, its X3 axis is the rotation axis of the Earth, and it is taken as International Terrestrial Reference System (ITRS) by IAU. For the rotation direction is not considered, the time in non-rotating geocentric reference system and rotating geocentric reference system is the same.Barycentric Celestial Reference System (BCRS) is recommended by IAU, its origin is the mass centre of the solar system,its third axis is approximately the rotation axis of the Earth.International Celestial Reference System is a centroid coordinate, it is made up of circle of right ascension and circle of declination of approximate 600 quasars, the coordinates are provided by International Earth Rotation and Reference Systems Service (IERS) Most general coordinates are introduced by the mathematical textbooks, so they are omitted here.4. 4.2 Measure UNITThe frame of time measure unit is as follows:Measure of timeUnits of measureTime intervalDynamical time intervalDuration fixed time intervalTime interval with the duration fixedby an ephemerisIntegral time scaleDynamical time scale is referred to as measured values of time parameters by physical quantities in a physical system. Basically, a proper time interval is a dynamical time scale.The main units of dynamical time scales in the ontology are concerned with ephemeris time units. A second in ephemeris time is defined as the fraction 1/31,556,925.9747 of the tropical year in Julian calendar for 1900 January 0 at 12 hours ephemeris time by International Committee for Weights and Measures (CIPM), from this unit, Julian century, year, week and day can be worked out.An integral time scale is accumulated value copied from a contracted time start point, for example, atomic time scale. So it may be proper time or coordinate time.V. T IME MEASURE AND COMPUTATION MODELSComponent ②is the set of the measure and computation models, which are from two resources: one is from the institutions put forward by some organizations such as IAU stipulating how to measure and computation, another resource is from the exact solutions of the (3) or (4).The models are programmed in Mathematica as the Application Programming Interface (API) so that a users’ programs can call these API.EXAMPLE 1[41]: a model (group) to compute a coordinated universal timeUTC (t) – TAI(t) = ns (5)UTC (t) –UT 1(t)=<0.9s; (6) Here, UTC(t) is a time expressed in coordinated universal time’ institution unit, TAI(t) means a time of Atomic Time International, n is natural number; s is the second, UT 1(t) is a time expressed in UT 1.© 2013 ACADEMY PUBLISHEREXAMPLE 2 is calling from a user’s application for the interface of a model, which is drawn from reference [42] and re-wrote by the author, to get an exact solution of Einstein’s field equation given Roberson-Walker Metric:1 /*An application from users in pseudo-code callingthe model-interface. See the tree in the attachment*/2 e num Space-Time in non- inertial system3 {4 B ianchi I Space-Time,5……6R obertson-WalkerSpace-Time7 /*Here, all the 16 Space-Time in non- inertialsystem in the tree enumerated */8 } Metric[16];9 for(i=0;i<16;i++){10 switch(Metric [i])11 case Robertson-Walker Space-Time:12 input and assign vector:13 v = {t, r, e, phi};141516 M = {-1, R[t]^2/(1 - K (r^2), (r^2) (R [t]^2), (r^2) (Sin[e]^2) (R [t]^2)};Call Einstein [M, v]}“Einstein.m”1 E instein [g_, v_] := Block[2 {invsg, dg1, dg2, dg3, Christf2, dChristf2, Ruv1,Ruv2, Ruv3, Ruv4, RicciTensor, R, EMTensor} 3 EMTensor = {}; (*Save return value.*)(*Calculate the inverse metric of g.*)4 g=DiagonalMatrix[M];5 invsg = Inverse[g];(*Calculate the affine connection.*)6 dg1 = Outer[D, g, v];7 dg2 = Transpose[dg1, {1, 3, 2}];8 dg3 = Transpose[dg1, {2, 3, 1}];9 Christf2 = (1/2) invsg.(dg1 + dg2 - dg3);(*Calculate the Ricci tensor.*)10 dChristf2 = Outer[D, Christf2, v];11 Ruv1 = Table[Sum[dChristf2[[k, i, k, j]], {k, 4}],{i, 4}, {j, 4}];12 Ruv2 = Table[Sum[dChristf2[[k, i, j, k]], {k, 4}],{i, 4}, {j, 4}];13 R uv3 = Table[Sum[Christf2[[k, i, j]] Christf2[[h,k, h]], {k, 4}, {h, 4}], {i, 4}, {j, 4}];14 Ruv4 = Table[ Sum[Christf2[[k, i, h]] Christf2[[h,j, k]], {k, 4}, {h, 4}], {i, 4}, {j, 4}];15 RicciTensor = Ruv1 - Ruv2 - Ruv3 + Ruv4;(*Calculate the Curvature Scalar.*)16R = Sum[invsg[[i, i]] RicciTensor[[i, i]], {i, 4}];(*Calculate the field equation left part.*)17EMTensor = RicciTensor - (1/2) g R ;18return [EMTensor]19]20End[]21EndPackage[]This program is divided into two parts: the first part is user’s input for computation, which is space-time dimensions v in a spherical coordinator, in which, t is the cosmological time (see 4. 2 Time Type), M is Roberson-Walker Metric. Users can input similar metrics for calling the function Einstein[ ],which is saved in the second part, a document Einstein.m, starting from the sentence BeginPackage["Einstein`"]. mathlink.h in VC++ enables to run Mathematica programs in VC++ environment The section Block[] is a function of local variables for calling.Outer[] is to give the partial derivative ∂f/∂x.Transpose[dg1, {1, 3, 2}] is to transposes dg1 so that the k th level in dg1 is the n k th level in the result.D [] is to get partial differential.Table [] is to generate a list of the expression Sum[].Sum[] is to get sum.The line 19 is the computation result of left part of (3), yet the cosmological constant is omitted. The right part of (3) is considered as zero.VI. M ECHANISM AND R UNNING OF T HE A RCHITECTURETOboMTT is designed to be a tree not only for satisfying the structure and classification of knowledge of time, but also for developing the knowledge in Web Ontology Language (OWL), which is based on Resource Description Framework (RDF) in a tree. Thus we can divide TOboMTT into some sub-trees and further expressed them in OWL or RDF. Figure 2 is a sample of Class—SubClass relation in RDF. As a result, navigation of knowledge of time, based on TOboMTT, become navigation of resources and serves, based on eXtensible Markup Language (XML) compatible with both OWL and RDF.A query for a sub-class or property value will give the corresponding answer by rational calculus on a XML scheme. For the example in Figure 2, “Space-Time Type includes Euclid Space-Time ” will be the answer for the query “What kind does the Space-Time Type include?” Therefore, query and answer is the first and direct results of navigation of knowledge of time by TOboMTT.<?xml version="1.0"?>© 2013 ACADEMY PUBLISHER。
无穷级数-正整数平方倒数和的求法
as required. This is taken from an article in the Mathematical Intelligencer by Apostol in 1983. Proof 2: We start in a similar fashion to Proof 1, but we use (2). We get
.
= 1−
sin2 u sin2 v cos2 u cos2 v = 1 − x2 y 2 . 3 ζ (2) = 4
Hence
du dv
A
where A = {(u, v ) : u > 0, v > 0, u + v < π/2} has area π 2 /8, and again we get ζ (2) = π 2 /6. This is due to Calabi, Beukers and Kock. Proof 3: We use the power series for the inverse sine function:
S
du dv 1 − u2 + v 2
where S is the square with vertices (0, 0), (1/2, −1/2), (1, 0) and (1/2, 1/2). Exploiting the symmetry of the square we get
1/2 u 0 1/2
Evaluating ζ (2)
Robin Chapman Department of Mathematics University of Exeter, Exeter, EX4 4QE, UK rjc@ 30 April 1999 (corrected 7 July 2003)
bolzano-weierstrass定理英文
bolzano-weierstrass定理英文The Bolzano-Weierstrass theorem is an important result in real analysis that guarantees the existence of convergent subsequences in bounded sequences. It is named after the mathematicians Bernard Bolzano and Karl Weierstrass, who independently discovered and proved the theorem in the 19th century. The theorem has various applications in different areas of mathematics and is widely used in fields such as calculus, optimization, and functional analysis.The Bolzano-Weierstrass theorem states that every bounded sequence in Euclidean space has a convergent subsequence. To understand the implications of this theorem, let's first define some key terms. A sequence is said to be bounded if there exists a real number M such that all the elements in the sequence are within the interval [-M, M]. In other words, the sequence does not diverge to positive or negative infinity but remains within a finite range.The concept of a convergent subsequence is equally important. Given a sequence (an), a subsequence is formed by selecting a strictly increasing sequence of indices (nk) such that each element of the subsequence is obtained by indexing the original sequence, i.e., (an1, an2, an3, ...). A subsequence is said to converge if its limit exists, which essentially means that as the subsequence progresses, its terms get arbitrarily close to a fixed value.The Bolzano-Weierstrass theorem asserts that for any bounded sequence in Euclidean space, it is always possible to extract a convergent subsequence. This result provides a powerful tool to analyze and understand the behavior of sequences, especially whentheir properties are not immediately obvious. It guarantees that even within a bounded sequence, we can always find subsequences that tend towards certain values or exhibit specific patterns.To understand the proof of the Bolzano-Weierstrass theorem, one typically uses the concept of dichotomy. The idea is to bisect the interval in which the bounded sequence lies and show that at least one of the halves contains infinitely many terms of the sequence. By recursively applying this process, we obtain a nested sequence of closed intervals. Using the completeness property of the real numbers, we can then conclude that the intersection of the nested intervals contains a single point, which serves as the limit of the convergent subsequence.The Bolzano-Weierstrass theorem has numerous applications in mathematics. In calculus, it is often employed to prove important results such as the Extreme Value Theorem or to establish the existence of critical points. In optimization, the theorem helps identify optimal solutions when dealing with bounded parameter spaces. In functional analysis, it is used to demonstrate compactness of certain spaces and to establish convergence properties of functions.In summary, the Bolzano-Weierstrass theorem is a fundamental result in real analysis that guarantees the existence of convergent subsequences in bounded sequences. It is a powerful tool used in various areas of mathematics to analyze and understand the behavior of sequences. The theorem provides a bridge between the discrete world of sequences and the continuous world of limits,enabling mathematicians to explore and comprehend the properties of bounded sequences in a rigorous and systematic manner.。
Syllabus_Mathematical Economics
Mathematical EconomicsFall 2006InstructorDesheng Ouyang,E-mail:deshengouyang@Telephone: 65903805Office: 教学行政大楼 709 室Office Hours: TBACourse Description:This course is designed to get the students familiarized with the mathematical ideas and tools used in modern economics, as well as to train them in their skills of reading and constructing proofs. It will cover selected topics mainly from mathematical analysis, real analysis and optimization theory. A rudimentary knowledge of integral and differential calculus (both one variable and multivariable), elementary linear algebra is a prerequisite for this course.Students are strongly recommended to read Chapters 2 to 4 of Simon’s book (or chapters of other books containing similar topics) on their own in the first 2 weeks as a review. Required Textbooks:Carl P. Simon, Lawrence Blume, Mathematics for Economists, W.W. Norton, New York, 1994.Morton I. kamien,Nancy L.Schwartz, Dynamic optimization, North-Holland, 1981. Other References:Wing Suen，The Structure of Economics（经济学的结构），上海财经大学出版社，2005 H. L. Royden, Real analysis, 3rd Edition,Mas-Colell, Whinston, and Green, Microeconomic TheoryStokey and Lucas, Recursive Methods in Economic Dynamics, Harvard University Press. Problem Sets, Exams and GradingThere will be problem sets issued almost once per week. And there will be one midterm and one final.The final exam counts for 40%, midterm exam for 30%, and problem sets for 30% of the grade.Outline of TopicsPart I1.Linear Algebra: (1st-2nd week)Vector spaces, Matrices, Characteristic Value Problem and Matrix’s Negative(Semi)Definiteness2.Introduction to Real analysis: (3rd-9th week )Basic concepts in Real Analysis, Continuity , Weierstrass Theorem, Intermediate Value Theorem , the Impl icit Function Theorem, Homogeneity and Euler’s Theorem,Homotheticity, Convex Sets and Separating Hyperplane Theorem, Concavity, Quasi-ConcavityMetric Spaces, Normed Vector Spaces, Open Sets, Compactness, Convergence and Completeness, Cauchy Sequences, Correspondences, Fixed Point Theorems( Brouwer’s Fixed Point Theorem, Kakutaki’s Fixed Point Theorem, Tarsky’s Fixed Point Theorem , Contract Mapping Theorem)Part II1.Static Optimization(unconstrained and constrained): (9th-11th week )Kuhn-Tucker Formulation, Second Order Conditions, Comparative Statics andEnvelope Theorems, Linear Programming and the Duality Theorem2.Dynamic Optimization:(12th-17th week )A Review of Differential Equations, Calculus of Variations and Euler Equation,Optimal Control Theory and Pontryagin Maximum Principle, Dynamic programming and Bellman Optimality Principle。
无穷级数-正整数平方倒数和的求法
But f (0) = ζ (2) and f (π ) =
∞ n 2 n=1 (−1) /n
5
Alternatively we can put
∞
D(z ) =
n=1
zn , n2
the dilogarithm function. This is uniformly convergent on the closed unit disc, and satisfies D (z ) = −(log(1 − z ))/z on the open unit disc. Note that f (t) = Re D(e2πit ). We may now use arguments from complex variable theory to justify the above formula for f (t). This is just the previous proof with the Fourier theory eliminated. Proof 7: We use the infinite product
∞
r =0
1 = (2r + 1)2 2
1 0 0
1
dx dy . 1 − x2 y 2
We make the substitution (u, v ) = so that (x, y ) = The Jacobian matrix is ∂ (x, y ) = ∂ (u, v ) cos u/ cos v sin u sin v/ cos2 v 2 sin u sin v/ cos u cos v/ cos u tan−1 x 1 − y2 , tan−1 y 1 − x2 sin u sin v , cos v cos u 1 − x2 1 − y2
Phone
implemented CLP systems, like the freeze predicate of Prolog, the delay mechanism of CLP(R) 13], or the constraint propagation schemes of CLP(FD). It also opens, to some extent, constraint programming to a new eld of applications which are traditional in concurrent programming, like reactive systems and protocol speci cations. >From the logic programming tradition however, the operational aspects of CC programming should also be closely connected to a logical semantics for reasoning about programs at di erent levels of abstraction, getting rid of useless details of the execution. The monotonic evolution of the store during CC computations provides CC languages with a simple denotational semantics in which agents are identi ed to closure operators on the semi-lattice of constraints 32, 15]. Such denotational semantics are used in 5] to obtain a complete calculus for partial correctness assertions where the rules of the proof system mirror the equations of the denotational semantics. In this article, we explore another route based on Girard's intuitionistic linear logic (ILL) 10]. We review the semantics of CC languages in the logic programming paradigm based on linear logic and we investigate the use of the phase semantics for proving safety properties of CC programs. Section 2 presents a natural extension of CC languages in this context, namely Linear CC (LCC) where the constraint system is axiomatized in linear logic. LCC is an extension of CC, somewhat similar to 3] or 30], but where constraints are consumed by ask agents without dependency maintenance or recomputation. Linear constraint systems have also been proposed in 31] in a higher-order setting which will not be considered here. >From an operational point of view, LCC extends CC in a fundamental way by introducing some forms of imperative programming, particularly useful for reactive systems. Classical CC programs can however be recovered by the usual translation of intuitionistic logic into linear logic 10]. Section 3 settles the basic soundness and completeness results of CC and LCC operational semantics w.r.t. intuitionistic linear logic. Results similar to those of this section are part of the folklore on CC languages 19, 31] but have not been published. Here we prove that 1) the stores of CC computations can be characterized in intuitionistic logic, and 2) both the stores and the successes of CC and LCC computations can be characterized in intuitionistic linear logic. Completeness results show that ILL can be used to prove liveness properties of LCC programs, i.e. properties expressing that something good will eventually happen. This is developed for both \may" and \must" properties. Then we show in section 4.2 how safety properties of CC and LCC programs (i.e. that some derivations never happen) can be proved using the phase semantics of linear logic. The method relies on the soundness theorem of LCC computations w.r.t. linear logic, and on the soundness theorem of linear logic w.r.t. the phase semantics. Completeness results simply say in this context that for various classes of observable properties of the program, if the property holds then such a \phase semantical proof" exists. The method is illustrated with 3