A Tableau Compiled Labelled Deductive System for Hybrid Logic

tableau 字符串函数-回复如何使用Tableau的字符串函数。

Tableau是一款非常强大的可视化工具，允许用户通过对数据进行分析和可视化，发现数据背后的故事。

Tableau提供了许多函数和工具，可以在数据操纵和分析过程中帮助用户更好地理解和展示数据。

其中，字符串函数是Tableau中非常有用的一部分，可以在数据操作过程中对字符串进行转换、拆分、连接等操作。

本文将深入介绍Tableau的一些常用字符串函数，并演示它们的使用方法。

首先，我们来讨论Tableau中常用的字符串函数之一，即LEN函数。

LEN 函数用于计算字符串的长度。

在Tableau中，使用LEN函数非常简单。

只需在计算字段或计算表达式中输入以下公式：LEN([字段名称])其中，[字段名称]表示要计算长度的字段的名称。

例如，如果我们想要计算“产品名称”字段中每个产品名称的长度，可以使用以下公式：LEN([产品名称])接下来，让我们来介绍Tableau中的另一个字符串函数，即UPPER函数。

UPPER函数将字符串转换为大写字母。

使用UPPER函数也非常简单，只需在计算字段或计算表达式中输入以下公式：UPPER([字段名称])其中，[字段名称]表示要转换为大写的字段的名称。

例如，如果我们想要将“姓名”字段中每个人的姓名转换为大写字母，可以使用以下公式：UPPER([姓名])除了UPPER函数外，Tableau还提供了LOWER函数，可以将字符串转换为小写字母。

LOWER函数的使用方法与UPPER函数相似，只需将公式中的UPPER替换为LOWER即可。

此外，Tableau还提供了一些可以进行字符串连接和拆分操作的函数。

比如，CONCAT函数用于将多个字符串连接成一个字符串。

在计算字段或计算表达式中，我们可以使用以下公式：CONCAT([字符串1], [字符串2], ...)其中，[字符串1]、[字符串2]表示要连接的字符串。

例如，如果我们想要将“姓”和“名”两个字段连接为一个完整的姓名字段，可以使用以下公式：CONCAT([姓], [名])另一个常用的字符串函数是SPLIT函数，在Tableau中用于将字符串根据特定的分隔符拆分为多个部分。

tableau 注释公式Tableau注释公式是Tableau软件中的一个强大功能，它可以帮助用户在数据可视化过程中添加注释，提供更丰富的信息和解释。

本文将介绍Tableau注释公式的使用方法和应用场景。

在Tableau中，注释公式是一种用于向可视化图表中添加文本注释的功能。

通过注释公式，用户可以在图表中添加文字、标签、解释或其他可视化元素，以增加数据的解读能力和可视化效果。

注释公式可以在图表中的任意位置添加，可以与其他数据元素进行交互，也可以根据特定条件进行显示或隐藏。

使用注释公式的第一步是选择要添加注释的图表。

在Tableau中，用户可以选择各种图表类型，如柱状图、折线图、饼图等。

选择合适的图表后，用户可以通过右键单击图表中的任意位置，选择“添加注释”选项，进入注释公式编辑界面。

在注释公式编辑界面中，用户可以自定义注释的内容和样式。

用户可以输入文字、数字、日期等各种类型的数据，也可以选择特定的数据字段作为注释内容。

注释公式还支持HTML标记语言，用户可以使用HTML标记语言添加超链接、字体样式、颜色等特效。

除了基本的注释内容，注释公式还可以根据特定条件进行显示或隐藏。

例如，用户可以设置注释在某个特定数值范围内显示，或者根据数据字段的取值情况进行动态显示。

这样，用户可以根据数据的变化情况，自动更新注释内容，提供更准确、实时的解读。

除了在图表中添加注释，注释公式还可以与其他数据元素进行交互。

例如，用户可以将注释与特定的数据点绑定，使得注释随着数据点的选择而改变。

这样，用户可以通过交互式操作，自由探索数据的不同维度和角度，获得更深入的洞察和理解。

Tableau注释公式的应用场景非常广泛。

它可以用于数据可视化报告中，为数据图表提供详细的解释和说明。

在商业分析中，注释公式可以用于解读销售趋势、市场份额、产品性能等关键指标，帮助管理者做出更准确的决策。

在学术研究中，注释公式可以用于解读实验结果、数据模型、统计分析等，提供更全面、准确的研究结论。

A Tableau Calculus with Automaton-LabelledFormulae for Regular Grammar LogicsRajeev Gor´e1and Linh Anh Nguyen21The Australian National University and NICTA,Canberra ACT0200,AustraliaRajeev.Gore@.au2Institute of Informatics,University of Warsaw,ul.Banacha2,02-097Warsaw,Polandnguyen@.plAbstract.We present a sound and complete tableau calculus for theclass of regular grammar logics.Our tableau rules use a special featurecalled automaton-labelled formulae,which are similar to formulae of au-tomaton propositional dynamic logic.Our calculus is cut-free and hasthe analytic superformula property so it gives a decision procedure.Weshow that the known EXPTIME upper bound for regular grammar log-ics can be obtained using our tableau calculus.We also give an effectiveCraig interpolation lemma for regular grammar logics using our calculus.1IntroductionMultimodal logics(and their description logic cousins)are useful in many ar-eas of computer science:for example,multimodal logics are used in knowledge representation by interpreting[i]ϕas“agent i knows/believes thatϕis true”[7,15,1].Grammar logics are normal multimodal logics characterised by“inclu-sion”axioms like[t1]...[t h]ϕ⊃[s1]...[s k]ϕ,where[t i]and[s j]are modalities indexed by members t i and s j from somefixed set of indices.Thus[1][2]ϕ→[1]ϕcaptures“if agent one knows that agent two knowsϕ,then agent one knowsϕ”.Inclusion axioms correspond in a strict sense to grammar rules of the form t1t2...t h→s1s2...s k when the index set is treated as a set of atomic words and juxtaposition is treated as word composition.Various refinements ask whether the corresponding grammar is left or right linear,or whether the language gen-erated by the corresponding grammar is regular,context-free etc.Grammar logics were introduced by Fari˜n as del Cerro and Penttonen in[8] and have been studied widely[3,4,20,11,5].Baldoni et al.[3]gave a prefixed tableau calculus for grammar logics and used it to show that the general sat-isfiability problem of right linear grammar logics is decidable and the general satisfiability problem of context-free grammar logics is undecidable.But the techniques of Baldoni et al.cannot be easily extended to regular grammar logics.While trying to understand why the decidability proof by Baldoni et al. [3,2]cannot be naturally extended to left linear grammars,Demri[4]observedB.Beckert(Ed):TABLEAUX2005,LNAI3702,pp.138–152,2005.c Springer-Verlag Berlin Heidelberg2005A Tableau Calculus with Automaton-Labelled Formulae139 that although right linear grammars generate the same class of languages as left linear grammars,this correspondence is not useful at the level of regular grammar logics.By using a transformation into the satisfiability problem for propositional dynamic logic(PDL),Demri was able to prove that the general satisfiability problem of regular grammar logics is EXPTIME-complete and that the general satisfiability problem of linear grammar logics is undecidable.In[5], Demri and de Nivelle gave a translation of the satisfiability problem for grammar logics with converse into the two-variable guarded fragment GF2offirst-order logic,and showed that the general satisfiability problem for regular grammar logics with converse is in EXPTIME.The relationship between grammar logics and description logics was considered,among others,in[11,20].Thus,various methods have been required to obtain complexity results and decision procedures for regular grammar logics.We show that it is possible to give a(non-prefixed)tableau calculus which is a decision procedure for the whole class of regular grammar logics,and which also gives an estimate of the complexity of these logics.Efficient tableaux for propositional multimodal(description)logics are highly competitive with translation methods,so it is not at all obvious that the translation into GF2from[5]is the best method for deciding these logics.The naive way to encode inclusion axioms in a non-prefixed tableau calculus is to add a rule like([t])shown below at left.But such rules cannot lead to a general decision procedure because there are well-known examples like transitivity[t]ϕ⊃[t][t]ϕ,whose analogous rule is shown below at right,which immediately cause an infinite branch by adding[t][t]ϕ,and then[t][t][t]ϕ,and so on:([t])X;[t]ϕX;[t]ϕ;[s1][s2]...[s k]ϕ(4t)X;[t]ϕX;[t]ϕ;[t][t]ϕOur calculus uses a special feature called automaton-labelled formulae,which are similar to formulae of APDL[10].Informally,whenever a formula[t]ϕis true at a tableau node w,we add an automaton labelled formula that tracks the modal transitions from w.If a sequence of transitions leads to a tableau node u, and this sequence corresponds to a word s1s2...s k recognised by the automaton labelled formula,then we add the formulaϕto u.This captures the effect of the rule([t])above left in a tractable manner since the influence of[t]ϕbeing true at w can be computed directly from the content of the automaton labelled formulae in node u.Our tableau calculus is sound,complete,cut-free and has the analytic superformula property,so it is a decision procedure.As usual for tableau calculi, it allows efficient implementation and good complexity estimation.In Section2,we define regular grammar logics and automaton-labelled for-mulae.In Section3,we present our tableau calculus for regular grammar logics, and prove it sound.In Section4,we prove it complete.In Section5,we prove that the general satisfiability problem of regular grammar logics is in EXPTIME by using our tableau rules in a systematic way.In Section6,we use our calculus to prove effective Craig interpolation for regular grammar logics.Further work and concluding remarks are in Section7.The Appendix contains an example.140R.Gor´e and L.A.Nguyen2Preliminaries2.1Definitions for Multimodal LogicsOur modal language is built from two disjoint sets:MOD is afinite set of modal indices and PROP is a set of primitive propositions.We use p and q for elements of PROP and use t and s for elements of MOD.Formulae of our primitive language are recursively defined using the BNF grammar below:ϕ::=p|¬ϕ|ϕ∧ϕ|ϕ∨ϕ|ϕ⊃ϕ|[t]ϕ| t ϕA Kripke frame is a tuple W,τ,{R t|t∈MOD} ,where W is a nonempty set of possible worlds,τ∈W is the current world,and each R t is a binary relation on W,called the accessibility relation for[t]and t .If R t(w,u)holds then we say that the world u is accessible from the world w via R t.A Kripke model is a tuple W,τ,{R t|t∈MOD},h ,where W,τ,{R t|t∈MOD} is a Kripke frame and h is a function mapping worlds to sets of primitive propositions.For w∈W,the set of primitive propositions“true”at w is h(w).A model graph is a tuple W,τ,{R t|t∈MOD},H ,where W,τ,{R t|t∈MOD} is a Kripke frame and H is a function mapping worlds to formula sets. We sometimes treat model graphs as models with H restricted to PROP.Given a Kripke model M= W,τ,{R t|t∈MOD},h and a world w∈W, the satisfaction relation|=is defined as usual for the classical connectives with two extra clauses for the modalities as below:M,w|=[t]ϕiff∀v∈W.R t(w,v)implies M,v|=ϕM,w|= t ϕiff∃v∈W.R t(w,v)and M,v|=ϕ.We say thatϕis satisfied at w in M if M,w|=ϕ.We say thatϕis satisfied in M and call M a model ofϕif M,τ|=ϕ.If we consider only Kripke models,with no restrictions on R t,we obtain a normal multimodal logic with a standard Hilbert-style axiomatisation K n.Note:We now assume that formulae are in negation normal form,where⊃is translated away and¬occurs only directly before primitive propositions.Every formulaϕhas a logically equivalent formulaϕ which is in negation normal form.2.2Regular Grammar LogicsRecall that afinite automaton A is a tuple Σ,Q,I,δ,F ,whereΣis the alphabet (for our case,Σ=MOD),Q is afinite set of states,I⊆Q is the set of initial states,δ⊆Q×Σ×Q is the transition relation,and F⊆Q is the set of accepting states.A run of A on a word s1...s k is afinite sequence of states q0,q1,...,q k such that q0∈I andδ(q i−1,s i,q i)holds for every1≤i≤k.It is an accepting run if q k∈F.We say that A accepts word w if there exists an accepting run of A on w.The set of all words accepted/recognised by A is denoted by L(A).Given two binary relations R1and R2over W,their relational composition R1◦R2={(x,y)|∃y∈W.R1(x,y)&R2(y,z)}is also a binary relation over W.A Tableau Calculus with Automaton-Labelled Formulae 141A grammar logic is a multimodal logic extending K n with “inclusion axioms”of the form [t 1]...[t h ]ϕ⊃[s 1]...[s k ]ϕ,where {t 1,...t h ,s 1,...s k }⊆MOD .Each inclusion axiom corresponds to the restriction R s 1◦...◦R s k ⊆R t 1◦...◦R t h on accessibility relations where the corresponding side stands for the identity relation if k =0or h =0.For a grammar logic L ,the L-frame restrictions are the set of all such corresponding restrictions.A Kripke model is an L-model if its frame satisfies all L -frame restrictions.A formula ϕis L-satisfiable if there exists an L -model satisfying it.A formula ϕis L-valid if it is satisfied in all L -models.An inclusion axiom [t 1]...[t h ]ϕ⊃[s 1]...[s k ]ϕcan also be seen as the gram-mar rule t 1...t h →s 1...s k where the corresponding side stands for the empty word if k =0or h =0.Thus the inclusion axioms of a grammar logic L capture a grammar G (L ).Here we do not distinguish terminal symbols and nonterminal symbols.G (L )is context-free if its rules are of the form t →s 1...s k ,and is reg-ular if it is context-free and for every t ∈MOD there exists a finite automaton A t that recognises the words derivable from t using G (L ).A regular grammar logic L is a grammar logic whose inclusion axioms cor-respond to grammar rules that collectively capture a regular grammar G (L ).A regular language is traditionally specified either by a regular expression or by a left/right linear grammar or by a finite automaton.The first two forms can be transformed in PTIME to an equivalent finite automaton that is at most polynomially larger.But there is no syntactic way to specify the class of regular (context-free)grammars,and checking whether a context-free grammar gener-ates a regular language is undecidable (see,e.g.,[14]).Hence,we cannot compute these automata if we are given an arbitrary regular grammar logic.We therefore assume that for each t ∈MOD we are given an automaton A t recognising the words derivable from t using G (L ).These are the automata specifying L .Lemma 1.Let L be a regular grammar logic and let {A t |t ∈MOD}be the automata specifying L.Then the following conditions are equivalent:(i)the word s 1...s k is accepted by A t (ii)the formula [t ]ϕ⊃[s 1]...[s k ]ϕis L-valid(iii)the inclusion R s 1◦...◦R s k ⊆R t is a consequence of the L-frame restrictions.Proof.The equivalence (ii )⇔(iii )is well-known from correspondence theory[19].The implication (i )⇒(ii )follows by induction on the length of the deriva-tion of s 1...s k from t by the grammar G (L ),using substitution,the K-axiom[t ](ϕ⊃ψ)⊃([t ]ϕ⊃[t ]ψ)and the modal necessitation rule ϕ/[t ]ϕ.The im-plication (iii )⇒(i )follows by induction on the length of the derivation of R s 1◦...◦R s k ⊆R t from the L -frame restrictions.See also [3,4]for details.Example 1.Let MOD ={1,...,m }for a fixed m .Consider the grammar logic with the inclusion axioms [i ]ϕ⊃[j ][i ]ϕfor any i,j ∈MOD and [i ]ϕ⊃[j ]ϕif i >j .This is a regular grammar logic because the set of words derivable from i using the corresponding grammar is {1,...,m }∗.{1,...,i }.This set is recognised by the automaton A i = MOD ,{p,q },{p },δi ,{q } with δi ={(p,j,p )|1≤j ≤m }∪{(p,j,q )|1≤j ≤i }.Note that the corresponding grammar is not “linear”in that at most one symbol in the right hand side of a rule can be nonterminal.142R.Gor´e and L.A.Nguyen2.3Automaton-Labelled FormulaeIf A is afinite automaton,Q is a subset of the states of A,andϕis a formula in the primitive language then(A,Q):ϕis an automaton-labelled formula.Fix a regular grammar logic L and let{A t= MOD,Q t,I t,δt,F t |t∈MOD}be the automata specifying L.Letδt(Q,s)={q |∃q∈Q.(q,s,q )∈δt} be the states which can be reached from Q via an s-transition using A t.The intuitions of automaton labelled formulae are as follows:Tagging:A formula of the form[t]ϕin a world u is represented by(A t,I t):ϕ. Tracking:If(A t,Q):ϕoccurs at u and R(u,v)holds then we add the formula(A t,δt(Q,s)):ϕto v.In particular,if(A t,I t):ϕappears in world u andR s(u,v)holds then we add(A t,δt(I t,s)):ϕto the world v. Acceptance:If(A t,Q):ϕoccurs at u and Q contains an accepting state of A t, then we addϕto u.The formal semantics of automaton-labelled formulae are defined as fol-lows.Letεbe the empty word and define δt(Q,ε)=Q and δt(Q,s1...s k)=δt (δt(Q,s1),s2...s k).If M is a Kripke model,w is a world of M,and A t=MOD,Q t,I t,δt,F t is an automaton,then M,w|=(A t,Q):ϕiffthere ex-ist worlds w0,...,w k=w(of M)and indices s1,...,s k∈MOD such that M,w0|=[t]ϕ,R s i(w i−1,w i)holds for1≤i≤k,and δt(I t,s1...s k)=Q. Pictorially:M,w|=(A t,Q):ϕiffw0s1w1s2···w k−1s k w k=w[t]ϕ δ(I t,s1)··· δ(I t,s1...s k−1) δ(I t,s1...s k)=QWe can see the soundness of these definitions by the following inter-derivable sequence of validities of multimodal tense logic which use the residuation prop-erties of s −1and[s]shown at right where s −1is the converse of s :[t]ϕ⊃[s1][s2]...[s k]ϕs1 −1[t]ϕ⊃[s2]...[s k]ϕ...s k −1... s1 −1[t]ϕ⊃ϕϕ⊃[s]ψ s −1ϕ⊃ψThat is,if we want to ensure that[t]ϕ⊃[s1][s2]...[s k]ϕis valid,then it suffices to ensure that s k −1... s1 −1[t]ϕ⊃ϕis valid instead.But converse modalities are not part of our official language so we use the occurrence of[t]ϕat w0to start an automaton A t which tracks the following constraint:ϕmust be true at any world w reachable from w0by the path/word s1...s k.Our automaton-labelled formulae are similar to formulae of automaton propositional dynamic logic(APDL)[10].A formula involving automata inA Tableau Calculus with Automaton-Labelled Formulae143 APDL is of the form[A]ϕ,where A is afinite automata with one initial state and one accepting state.An automaton labelled formula like our(A t,Q):ϕwith Q={q1,q2,...,q k}can be simulated by the APDL formula[B1]ϕ∨...∨[B k]ϕwhere each B i is the automaton A t restricted to start at the initial state q i.Thus our formulation uses a more compact representation in which APDL formulae that differ only in their initial state are grouped together.Moreover,we do not treat different“states”of an automaton as different automata.Our compact representation not only saves memory but also increases efficiency of deduction.From now on,by a formula we mean either a formula in the primitive lan-guage(as defined in Section2.1)or an automaton-labelled formula.2.4Definitions for Tableau CalculiAs in our previous works on tableau calculi[9,17],our tableau formulation trace their roots to Hintikka via[18].A tableau ruleσconsists of a numerator N above the line and a(finite)list of denominators D1,D2,...,D k(below the line)separated by vertical bars.The numerator is afinite formula set,and so is each denominator.As we shall see later,each rule is read downwards as“if the numerator is L-satisfiable,then so is one of the denominators”.The numerator of each tableau rule contains one or more distinguished formulae called the principal formulae.A tableau calculus C L for a logic L is afinite set of tableau rules.A C L-tableau for X is a tree with root X whose nodes carryfinite formula sets obtained from their parent nodes by instantiating a tableau rule with the proviso that if a child s carries a set Z and Z has already appeared on the branch from the root to s then s is an end node.Let∆be a set of tableau rules.We say that Y is obtainable from X by applications of rules from∆if there exists a tableau for X which uses only rules from∆and has a node that carries Y.A branch in a tableau is closed if its end node carries only⊥.A tableau is closed if every one of its branches is closed.A tableau is open if it is not closed.Afinite formula set X in the primitive language is said to be C L-consistent if every C L-tableau for X is open.If there is a closed C L-tableau for X then we say that X is C L-inconsistent.A tableau calculus C L is sound if for allfinite formula sets X in the primitive language,X is L-satisfiable implies X is C L-consistent.It is complete if for all finite formula sets X in the primitive language,X is C L-consistent implies X is L-satisfiable.Letσbe a rule of C L.We say thatσis sound w.r.t.L if for every instanceσ ofσ,if the numerator ofσ is L-satisfiable then so is one of the denominators ofσ .Any C L containing only rules sound w.r.t.L is sound.3A Tableau Calculus for Regular Grammar LogicsFix a regular grammar logic L and let{A t= MOD,Q t,I t,δt,F t |t∈MOD} be the automata specifying L.Recall that formulae are in negation normal form. We use X for a formula set,and semicolon to separate elements of a formula set.We have deliberately used“s.t.”for“such that”in the set notation of the144R.Gor´e and L.A.Nguyen(⊥)X ;p ;¬p ⊥(∧)X ;ϕ∧ψX ;ϕ;ψ(∨)X ;ϕ∨ψX ;ϕ|X ;ψ(label )X ;[t ]ϕX ;(A t ,I t ):ϕ(add )X ;(A t ,Q ):ϕX ;(A t ,Q ):ϕ;ϕif Q ∩F t =∅(trans )X ; t ϕ{(A s ,δs (Q,t )):ψs.t.(A s ,Q ):ψ∈X };ϕFig.1.Tableau Rulesdenominator of the (trans )-rule because we use colon in automaton-labelled for-mulae and the alternative |indicates a branching rule!The tableau calculus C L is given in Figure 1.The first five rules are static rules,and the last rule is a transitional rule.An example is in the appendix.A tableau calculus C L has the analytic superformula property iffto every finite set X we can assign a finite set X ∗C L which contains all formulae that may appear in any tableau for X .We write Sf (ϕ)for the set of all subformulae of ϕ,and Sf (X )for the set ϕ∈X Sf (ϕ)∪{⊥}.Our calculus has the analytic superfor-mula property,with X ∗C L =Sf (X )∪{(A t ,Q ):ϕs.t.[t ]ϕ∈Sf (X )and Q ⊆Q t }.Lemma 2.The tableau calculus C L is sound.Proof.We show that C L contains only rules sound w.r.t.L as follows.Suppose that the numerator of the considered rule is satisfied at a world w in a model M = W,τ,{R t |t ∈MOD},h .We have to show that at least one of the denominators of the rule is also satisfiable.For the static rules,we show that some denominator is satisfied at w itself.For the transitional rule (trans ),we show that its denominator is satisfied at some world reachable from w via R t .(⊥),(∧),(∨):These cases are obvious.(label ):If M,w |=X ;[t ]ϕthen M,w |=(A t ,I t ):ϕby definition.(add ):Suppose that M,w |=X ;(A t ,Q ):ϕand Q ∩F t =∅.By definition,there exist worlds w 0,...,w k −1,w k =w and indices s 1,...,s k ∈MOD such that M,w 0|=[t ]ϕ,and R s i (w i −1,w i )holds for 1≤i ≤k ,and δt (I t ,s 1...s k )=Q .Since Q ∩F t =∅,the word s 1...s k is accepted by A t .By Lemma 1,it follows that M,w 0|=[s 1]...[s k ]ϕ.Since w =w k and R s i (w i −1,w i )holds for 1≤i ≤k ,we must have M,w |=ϕ.(trans ):Suppose that M,w |=X ; t ϕ.Then there exists some u such thatR t (w,u )holds and M,u |=ϕ.For each (A s ,Q ):ψ∈X ,we have M,w |=(A s ,Q ):ψ,and by the semantics of automaton-labelled formulae,it follows that M,u |=(A s ,δs (Q,t )):ψ.Hence,the denominator is satisfied at u .4CompletenessWe prove completeness of our calculus via model graphs following [18,9,16,17]by giving an algorithm that accepts a finite C L -consistent formula set X in the primitive language and constructs an L -model graph (defined in Section 4.2)for X that satisfies every one of its formulae at the appropriate world.A Tableau Calculus with Automaton-Labelled Formulae1454.1SaturationIn the rules(∧),(∨),(label)the principal formula does not occur in the denomi-nators.For any of these rulesδ,letδ denote the rule obtained fromδby adding the principal formula to each of the denominators.Let SC L denote the set of static rules of C L with(∧),(∨),(label)replaced by(∧ ),(∨ ),(label ).For every rule of SC L,except(⊥),the numerator is included in each of the denominators.For afinite C L-consistent formula set X,a formula set Y is called a C L-saturation of X if Y is a maximal C L-consistent set obtainable from X by ap-plications of the rules of SC L.A set X is closed w.r.t.a tableau rule if applying that rule to X gives back X as one of the denominators.Lemma3.Let X be afinite C L-consistent formula set and Y a C L-saturation of X.Then X⊆Y⊆X∗C L and Y is closed w.r.t.the rules of SC L.Furthermore, there is an effective procedure that,given afinite C L-consistent formula set X, constructs some C L-saturation of X.Proof.It is clear that X⊆Y⊆X∗C L.Observe that if a rule of SC L is ap-plicable to Y,then one of the corresponding instances of the denominators is C L-consistent.Since Y is a C L-saturation,Y is closed w.r.t.the rules of SC L.We construct a C L-saturation of X as follows:let Y=X;while there is some ruleδof SC L applicable to Y such that one of its corresponding denominator instance Z is C L-consistent and strictly contains Y,set Y=Z.At each iteration, Y⊂Z⊆X∗C L.Hence the above process always terminates.It is clear that the resulting set Y is a C L-saturation of X.4.2Proving Completeness via Model GraphsA model graph is an L-model graph if its frame is an L-frame.An L-model graph W,τ,{R t|t∈MOD},H is saturated if every w∈W satisfies:And:ifϕ∧ψ∈H(w)then{ϕ,ψ}⊆H(w);Or:ifϕ∨ψ∈H(w)thenϕ∈H(w)orψ∈H(w);Box:if[t]ϕ∈H(w)and R t(w,u)holds thenϕ∈H(u);Dia:if t ϕ∈H(w)then there exists a u∈W such that R t(w,u)andϕ∈H(u).A saturated model graph is consistent if no world contains⊥,and no world contains{p,¬p}.Our model graphs merely denote a data structure,while Raut-enberg’s model graphs are required to be saturated and consistent.Lemma4.If M= W,τ,{R t|t∈MOD},H is a consistent saturated L-model graph,then M satisfies all formulae of H(τ)which are in the primitive language. Proof.By provingϕ∈H(w)implies M,w|=ϕby induction on the length ofϕ.Given afinite C L-consistent set X in the primitive language,we construct a consistent saturated L-model graph M= W,τ,{R t|t∈MOD},H such that X⊆H(τ),thereby giving an L-model for X.146R.Gor´e and L.A.Nguyen4.3Constructing Model GraphsIn the following algorithm,the worlds of the constructed model graph are marked either as unresolved or as resolved .Algorithm 1Input:a finite C L -consistent set X of formulae in the primitive language.Output:an L -model graph M = W,τ,{R t |t ∈MOD},H satisfying X .1.Let W ={τ},H (τ)be a C L -saturation of X ,and R t =∅for all t ∈MOD .Mark τas unresolved.2.While there are unresolved worlds,take one,say w ,and do the following:(a)For every formula t ϕin H (w ):i.Let Y ={(A s ,δs (Q,t )):ψs.t.(A s ,Q ):ψ∈H (w )}∪{ϕ}be the result of applying rule (trans )to H (w ),and let Z be a C L -saturation of Y .ii.If ∃u ∈W on the path from the root to w with H (u )=Z ,then add the pair (w,u )to R t .Otherwise,add a new world w ϕwith content Z to W ,mark it as unresolved,and add the pair (w,w ϕ)to R t .(b)Mark w as resolved.3.Let R t be the least extension of R t for t ∈MOD such that W,τ,{R t |t ∈MOD} is an L -frame.This algorithm always terminates:eventually,for every w ,either w contains no t -formulae,or there exists an ancestor with H (u )=Z at Step 2(a)ii because all C L -saturated sets are drawn from the finite and fixed set X ∗C L .Lemma 5.Suppose R t (w,u )holds via Step 3.Then there exist w 0,...,w k in M with w 0=w ,w k =u ,and indices s 1,...,s k ∈MOD such that R s i (w i −1,w i )holds for 1≤i ≤k ,and R s 1◦...◦R s k ⊆R t follows from the L-frame restrictions.Proof.By induction on number of inferences in deriving R t (w,u )when extending R s to R s for s ∈MOD ,with L -frame restrictions of the form R t 1◦...◦R t h ⊆R s .4.4Completeness ProofLemma 6.Let X be a finite C L-consistent set of formulae in the primitive lan-guage and M = W,τ,{R t |t ∈MOD},H be the model graph for X constructed by Algorithm 1.Then M is a consistent saturated L-model graph satisfying X .Proof.It is clear that M is an L -model graph and for any w ∈W ,the set H (w )is C L -consistent.We want to show that M is a saturated model graph.It suffices to show that,for every w,u ∈W ,if [t ]ϕ∈H (w )and R t (w,u )holds then ϕ∈H (u ).Suppose that [t ]ϕ∈H (w )and R t (w,u )holds.By Lemma 5,there exist worlds w 0,...,w k with w 0=w ,w k =u and indices s 1,...,s k ∈MOD such that R s i (w i −1,w i )holds for 1≤i ≤k and R s 1◦...◦R s k ⊆R t is a consequence of the L -frame restrictions.Since H (w )is a C L -saturation,we have that (A t ,I t ):ϕ∈H (w ).By Step 2a of Algorithm 1,(A t , δt (I t ,s 1...s i )):ϕ∈H (w i )for 1≤i ≤k .Thus (A t , δt (I t ,s 1...s k )):ϕ∈H (u ).Since R s 1◦...◦R s k ⊆R t is a consequence of the L -frame restrictions,by Lemma 1,the word s 1...s k is accepted by A t .Hence δt (I t ,s 1...s k )∩F t =∅.It follows that ϕ∈H (u ),since H (u )is a C L -saturation.A Tableau Calculus with Automaton-Labelled Formulae147The following theorem follows from Lemmas2and6.Theorem1.The tableau calculus C L is sound and complete.5ComplexityThe satisfiability problem of a logic L is to check the L-satisfiability of an input formulaϕ.The general satisfiability problem of a class C of logics is to check L-satisfiability of an input formulaϕin an input logic L∈C.Demri[4]proved that the general satisfiability problem of regular grammar logics is EXPTIME-complete by a transformation into satisfiability for PDL.We now obtain the upper bound EXPTIME using our tableaux calculus.We need a rule(∪)to coalesce(A t,Q):ϕand(A t,Q ):ϕinto(A t,Q∪Q ):ϕ(∪)X;(A t,Q):ϕ;(A t,Q ):ϕX;(A t,Q∪Q ):ϕObserve that X;(A t,Q):ϕ;(A t,Q ):ϕis C L-consistent iffX;(A t,Q∪Q ):ϕis C L-consistent.This follows from the facts thatδt(Q,s)∪δt(Q ,s)=δt(Q∪Q ,s) and((Q∩F t=∅)∨(Q ∩F t=∅))≡((Q∪Q )∩F t=∅).Thus,rule(∪)can be added to C L as a static rule,and used whenever possible without affecting soundness and completeness.Let C L u be C L plus(∪).Allowing(∪)requires a change in the semantics of automaton-labelled formu-lae to:if M is a Kripke model,w is a world of M,and A t= MOD,Q t,I t,δt,F t is an automaton,then M,w|=(A t,Q):ϕifffor every q∈Q there exist worlds w0,...,w k−1,w k=w(of M)and indices s1,...,s k∈MOD such that M,w0|=[t]ϕ,and R s i(w i−1,w i)holds for1≤i≤k,and q∈ δt(I t,s1...s k)⊆Q.Let L be a regular logic and X afinite formula set in the primitive language. Let n be the sum of the sizes of the formulae in X and the sizes of the automata specifying L.To check whether X is L-satisfiable we can search for a closed C L u-tableau for X,or equivalently,examine an and-or tree for X constructed by using Algorithm1to apply our C L u-tableau rules in a systematic way.In such a tree,and-branching is caused by all possible applications of rule(trans),while or-branching is caused by an application of rule(∨).The other C L u-tableau rules,including(∪),are applied locally for each node whenever possible.There are at most O(n)unlabelled subformulae of X,and at most2O(n) different labels.By using the rule(∪)whenever possible,each subformula of X occurs in a node with at most two labels,so a node contains at most2n i.e.O(n) labelled formulae.Hence there are at most(2O(n))O(n)=2O(n2)different nodes. Without the rule(∪),there are at most22O(n)different nodes,which breaks EXPTIME worst-case complexity,so the(∪)rule is absolutely essential.But it is not necessary for the satisfiability problem of afixed logic.Algorithm1terminates in general because it checks for repeated ancestors: this check is built into the definition of an end-node,and also in Step2(a)ii.Thus the same node can appear on multiple branches.In the worst case,Algorithm1 therefore requires22O(n2)time.We therefore refine it into Algorithm2below:。

数据可视化工具Tableau的高级使用方法Tableau是一款功能强大的数据可视化工具，它提供了丰富的功能和灵活的操作方式，可以帮助用户更好地理解和分析数据。

在本文中，我将为您介绍Tableau的高级使用方法，帮助您更好地利用这个工具进行数据可视化和分析。

一、数据连接与准备在使用Tableau之前，首先需要将数据导入到Tableau中，并进行必要的数据清洗和转换。

Tableau支持多种数据源，包括Excel、CSV、数据库等，您可以根据数据的来源选择相应的数据连接方式。

一旦连接成功，您可以使用Tableau的数据预览功能来查看导入的数据并进行初步的数据清洗和筛选。

在数据准备过程中，Tableau的高级使用方法包括使用计算字段、参数和集合。

计算字段允许您根据已有字段创建自定义的计算结果，参数允许您动态地调整Dashboard中的数值和指标，集合则使您能够将多个字段或数据聚合为一个单独的对象。

通过这些高级功能，您可以更精确地对数据进行分析，并创建更丰富和有针对性的可视化效果。

二、创建高级可视化效果Tableau拥有丰富的可视化选项和工具，使用户能够创建多种类型的图表和图形。

除了常见的柱状图、折线图和散点图之外，Tableau还支持创建地图、热力图、树状图等多种图表类型，以满足不同数据分析和展示的需求。

在创建可视化效果时，Tableau的高级使用方法包括使用过滤器、参数和动作来增强和改进可视化效果。

过滤器允许您根据特定条件筛选数据，参数可以用于动态调整图表中的数值和指标，动作则允许您创建交互式的图表效果，在用户与可视化结果进行交互时更直观地展示数据和分析结果。

三、设计交互式仪表盘Tableau的仪表盘（Dashboard）功能使用户能够将多个可视化效果组合在一起，并创建交互式的报表和分析工具。

在设计仪表盘时，您可以使用Tableau的布局和样式功能来调整和美化仪表盘的外观和格式，使其更符合您的需求和风格。