An algebraic framework for urgency

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我正在做学习计划英语Making a Study PlanStudying can sometimes feel overwhelming, especially when you have a lot of material to cover or when you're trying to learn something new. However, creating a study plan can help you stay organized and focused, and can make the learning process more manageable and enjoyable. In this article, we'll discuss the importance of making a study plan and provide some tips for creating an effective one.Importance of Making a Study PlanA study plan is a roadmap that outlines what you need to learn, when you need to learn it, and how you will do so. By creating a study plan, you can:1. Stay organized: A study plan helps you keep track of what you need to learn and when you need to learn it. This can prevent you from feeling overwhelmed and ensure that you cover all the material you need to.2. Set goals: By breaking down your learning objectives into smaller, manageable tasks, a study plan can help you set and achieve specific goals. This can boost your motivation and confidence as you see your progress over time.3. Allocate time effectively: A study plan helps you allocate your time to different subjects or topics, based on their importance and your own strengths and weaknesses. This can prevent you from spending too much time on one subject at the expense of others.4. Reduce procrastination: When you have a clear plan for what you need to learn and when you need to learn it, it can be easier to get started and stay focused. This can reduce procrastination and increase productivity.Tips for Creating an Effective Study PlanNow that we've discussed the benefits of making a study plan, let's look at some tips for creating an effective one:1. Set Specific Goals: Start by setting specific, achievable goals for what you want to accomplish. Do you want to improve your grades in a particular subject? Do you want to learn a new skill or concept? Having clear goals will help you stay focused and motivated.2. Assess Your Current Situation: Before creating a study plan, it's important to assess your current knowledge and skills. This will help you identify areas where you need to focus more, as well as areas where you can afford to spend less time.3. Prioritize Your Tasks: Once you've assessed your current situation, prioritize your tasks based on their importance and urgency. This will help you allocate your time and resources effectively.4. Break It Down: Break down your learning objectives into smaller, manageable tasks. This will make your goals more achievable and will give you a clear roadmap for what you need to learn and when.5. Create a Schedule: Create a study schedule that includes specific times for studying, as well as breaks. Be realistic about the amount of time you can dedicate to studying each day, and try to stick to your schedule as much as possible.6. Use a Variety of Resources: Use a variety of resources, such as textbooks, online tutorials, and study groups, to help you learn and understand the material. This can help you gain different perspectives and reinforce your learning.7. Stay Flexible: While it's important to have a plan, it's also important to stay flexible. If you find that a particular study method isn't working for you, don't be afraid to adjust your plan and try something different.Example of a Study PlanNow let's look at an example of a study plan for a high school student who wants to improve their grades in math:Goal: Improve math grades from a C to an A.Current Situation: The student struggles with algebra and geometry, but is confident in their understanding of arithmetic and basic mathematical concepts.Prioritizing Tasks: The student will prioritize studying algebra and geometry, while still maintaining their understanding of arithmetic and basic mathematical concepts.Breaking It Down: The student will break down their learning objectives into smaller tasks, such as mastering specific algebraic equations and theorems in geometry, and practicing solving problems in each area.Creating a Schedule: The student will create a study schedule that includes 1-2 hours of math study each day, with breaks in between. The student will also schedule regular review sessions to reinforce their learning.Using a Variety of Resources: The student will use their textbook, online video tutorials, and study groups to help them learn and understand the material.Staying Flexible: The student will stay flexible and adjust their study plan as needed, based on their progress and any challenges they encounter.In conclusion, creating a study plan can help you stay organized, set goals, allocate time effectively, and reduce procrastination. By following the tips provided in this article, you can create an effective study plan that will help you achieve your learning objectives and improve your academic performance. Good luck!。

Global development via local observational construction steps

Global Development via Local ObservationalConstruction Steps⋆Michel Bidoit1,Donald Sannella2,and Andrzej Tarlecki3 1Laboratoire Sp´e ciﬁcation et V´e riﬁcation,CNRS&ENS de Cachan,France2Laboratory for Foundations of Computer Science,University of Edinburgh,UK 3Institute of Informatics,Warsaw University and Institute of Computer Science,Polish Academy of Sciences,Warsaw,PolandAbstract.The way that reﬁnement of individual“local”componentsof a speciﬁcation relates to development of a“global”system from aspeciﬁcation of requirements is explored.Observational interpretation ofspeciﬁcations and reﬁnements add expressive power andﬂexibility whilebringing in some subtle problems.The results are instantiated in thecontext of Casl architectural speciﬁcations.1IntroductionThere has been a great deal of work in the algebraic speciﬁcation tradition on formalizing the rather intuitive and appealing idea of program development by stepwise reﬁnement,including[EKMP82,Gan83,GM82,Sch87,ST88b];for a re-cent survey,see[EK99]).There are many issues that make this a diﬃcult prob-lem,and some of them are rather subtle,one example being the relationship between speciﬁcation structure and program structure.There are diﬃcult inter-actions and tradeoﬀs,an obvious one being between the expressive power of a speciﬁcation formalism and the ease of reasoning about speciﬁcations.Diﬀerent approaches give more or less prominence to diﬀerent issues.An overview that covers most of our own contributions is[ST97],with some more recent work addressing the problem of how to prove correctness of reﬁnement steps[BH98], the design of a convenient formalism for writing speciﬁcations[ABK+03,BST02], and applications to data reﬁnement in typedλ-calculus[HLST00].A new angle that we explore here is the“global”eﬀect of reﬁning individual “local”components of a speciﬁcation.This involves a well-known technique from algebraic speciﬁcation,namely the use of pushouts of signatures and amalgama-tion of models to build large systems by composition of separate interrelated components.The situation becomes considerably more subtle when observa-tional interpretation of speciﬁcations and reﬁnements is taken into account.Part of the answer has already been provided,the main references being Schoett’s thesis[Sch87,Sch90]and our work on formal development in the Ex-tended ML framework[ST89];the general ideas go back at least to[Hoa72].2We have another look at these issues here,in the context of the Casl spec-iﬁcation formalism[ABK+03]and in particular,its architectural speciﬁcations [BST02].Architectural speciﬁcations,for describing the modular structure of software systems,are probably the most novel feature of Casl.We view them here as a means of making complex reﬁnement steps,by deﬁning well-structured constructions to be used to build the overall system from implementations of individual units(these also include parametrized units,acting as constructions providing some local construction steps to be used in a more global context).We begin by introducing in Sect.2some details of the underlying logical system we will be working with,and our assumptions concerning speciﬁcations built using this system.Our basic view of program development by means of consecutive local reﬁnement steps is presented in Sect.3.Then,an observational view of speciﬁcations is motivated and recalled in Sect.4.The principal core of the work is in Sect.5,where we combine the ideas of the previous two sections and discuss program development by local reﬁnement steps with respect to an observational interpretation of the speciﬁcations involved.Section6introduces a simpliﬁed version of Casl architectural speciﬁcations,while Sect.7sketches their observational semantics and shows how the ideas of Sect.5are instantiated in this context.Further work and possible generalizations are discussed in Sect.8. Due to lack of space we have been unable to include concrete examples that illustrate the deﬁnitions and results,but we plan to provide such material in a future extended version.2Signatures,Models and SpeciﬁcationsA basic assumption underpinning algebraic speciﬁcation and derived approaches to software speciﬁcation and development is that software systems are modeled as algebras(of some kind)and their static properties are captured by alge-braic signatures(again,adapted as appropriate).This leads to quite aﬂexible framework,which can be tuned as desired to cope with various programming features of interest by selecting the appropriate variation of algebra and signa-ture.Thisﬂexibility has been formalized via the notion of institution[GB92] and related work on the theory of speciﬁcations and formal program develop-ment[ST88a,ST97,BH93].However,rather than exploiting the full generality of institutions,to keep things simple and illustrative we will in this paper base our considerations on a very basic logical framework,leaving to a more extensive pre-sentation elsewhere the required generalization and adaptation to a fully-ﬂedged formalism such as Casl.So,we will deal here with the usual notions of many-sorted algebraic sig-natures and signature morphisms;we will assume that all signatures contain a distinguished Boolean part:a sort bool with two constants true and false pre-served by all signature morphisms.This yields the category AlgSig—it is cocomplete,and we will assume that it comes with some standard construction of pushouts.3For each algebraic signature Σ,Alg (Σ)stands for the usual category of Σ-algebras and their homomorphisms —we restrict attention to algebras with a ﬁxed,standard interpretation of the Boolean part of the signature.As usual,each signature morphism σ:Σ→Σ′determines a reduct functor σ:Alg (Σ′)→Alg (Σ).This yields a functor Alg :AlgSig op →Cat .We refer to [ST99]for a more detailed presentation of the technicalities and for the standard notations we will use in the following.It can easily be checked that Alg is continuous,i.e.,maps colimits of algebraic signatures to limits of (algebra)categories (the initial signature,containing the Boolean part only,is mapped to the category having as its only object the algebra providing the ﬁxed interpretation for the Boolean part).In particular,the following amalgamation property holds:Lemma 2.1.Given a pushout in the category of algebraic signatures AlgSig :ΣΣ1Σ′Σ′1-ι6γ′for any algebras A 1∈|Alg (Σ1)|and A ′∈|Alg (Σ′)|such that A 1ιthere exists a unique algebra A ′1∈|Alg (Σ′1)|such that A ′1γ′=A ′;and similarly for algebra homomorphisms.Given a signature Σ,terms and ﬁrst-order formulae with equality are deﬁned as usual.Σ-sentences are closed ﬁrst-order formulae.Given a Σ-algebra A ,a set of variables X and a valuation of variables v :X →|A |,the value t A [v ]of a term t with variables X in A under v and the satisfaction A [v ]|=φof a formula φwith variables X in A under v are deﬁned as usual.We will also employ a generalized notion of terms,modeling a pretty general idea of how a value may be determined in an algebra.Given a signature Σ,a conditional term of sort s with variables X is of the form p =((φi ,t i )i ≥0,t ),where for i ≥0,φi are formulae with variables X ,and t i and t are terms of sort s with variables X .Given a Σ-algebra A and a valuation v :X →|A |,the value p A [v ]of such a conditional term p is (t k )A [v ]for the least k ≥0such that A [v ]|=φk ,or t A [v ]if no such k ≥0exists.This allows for a further generalization of derived signature morphisms [SB83],where we allow such a morphism δ:Σ→Σ′to map function symbols f :s 1×...×s n →s to conditional terms of sort s with variables {x 1:s 1,...,x n :s n }.Ev-idently,such a derived signature morphisms δ:Σ→Σ′still determines a reduct function δ:|Alg (Σ′)|→|Alg (Σ)|on algebra classes (which in general does not extend to a reduct functor between algebra categories).We will not need to know much about the formalism used for writing spec-iﬁcations.We just assume that some class of speciﬁcations is deﬁned,equipped with a semantics that for any speciﬁcation SP determines its signature Sig (SP )∈|AlgSig |and its class of models Mod (SP )⊆|Alg (Sig (SP ))|.We also assume4that the class speciﬁcations is closed under translation along signature mor-phisms,i.e.,for any speciﬁcation SP and signature morphismσ:Sig(SP)→Σ′,we have a speciﬁcationσ(SP)with Sig(σ(SP))=Σ′and Mod(σ(SP))= {A′∈|Alg(Σ′)||A′5'%SP 0∼∼∼>κ1'%SP 1∼∼∼>κ1 κ2κn •It seems only natural to separate the ﬁnished parts from the speciﬁcation of what remains to be done.This gives the following picture:'%SP 0κ1∼∼∼>'%SP 1κ2∼∼∼> SP 2κ3∼∼∼>···κn ∼∼∼>•SP n =EMPTY where for i =1,...,n ,the speciﬁcations SP i now describe the part of the sys-tem that remains to be implemented,while each κi is a parametrized program[Gog84]which semantically amounts to a (possibly partial)function on alge-bras κi :|Alg (Sig (SP i ))|⇀|Alg (Sig (SP i −1))|which we will call a construction .Now,given speciﬁcations SP and SP ′and a construction κ:|Alg (Sig (SP ′))|⇀|Alg (Sig (SP ))|,we deﬁne:SP κ∼∼∼>SP ′⇐⇒Mod (SP ′)⊆dom (κ)and κ(Mod (SP ′))⊆Mod (SP )This deﬁnition captures the correctness requirements we impose on the indi-vidual reﬁnement steps,which guarantee that given a successful development sequence:SP 0κ1∼∼∼>SP 1κ2∼∼∼>...κn ∼∼∼>SP n =EMPTY we obtain the algebra:κ1(κ2(...κn (empty )...))∈Mod (SP 0)where EMPTY is the empty speciﬁcation over the “empty”signature (i.e.the initial object in AlgSig ,containing the Boolean part only)and empty is its unique standard realization.Even though our presentation suggests a “top-down”development process,starting from the requirements speciﬁcation and proceeding towards a situation where nothing is left to be implemented,this need not be the case in general.We can instead proceed “bottom-up”,starting with EMPTY and successively providing constructions which add in bits and pieces in an incremental fashion until an implementation of the original speciﬁcation is obtained.Or we can com-bine the two techniques,and proceed in a “middle-out”fashion.What matters is that at the end a chain of correct reﬁnement steps emerges which links the requirements speciﬁcation with EMPTY .6Another point about the above presentation is that it relies on a global view of speciﬁcations and their reﬁnement:constructions are required to work on the whole system (represented as a model of the reﬁning speciﬁcation)and produce a whole system (represented as a model of the reﬁned speciﬁcation).Good practice suggests that there should be a way to develop such complex constructions in a well-structured way.In Sect.6we will present a speciﬁc view of how constructions may be built from smaller pieces,and how to decompose a development task into a number of subtasks via multi-argument constructions.For now,let us concentrate on one aspect of this,and discuss how to make reﬁnement steps “local”—that is,how to use only part of the system built so far to implement some remaining parts of the requirements speciﬁcation,and then incorporate the result in the system as a whole.Technically,this means that we need to look at constructions that map Σ-algebras to Σ′-algebras,but apply them to parts cut out of “larger”ΣG -algebras,where this “cutting out”is given as the reduct with respect to a signature mor-phism γ:Σ→ΣG that ﬁts the local argument signature into its global context.W.l.o.g.we can assume that constructions are persistent :the argument of a con-struction is always fully included in its result,without modiﬁcation 4—note that this assumption holds for all constructions that can be declared and speciﬁed in Casl ,see Sect.6.In fact,we generalize this somewhat by considering arbitrary signature morphisms rather than just inclusions.Throughout the rest of the paper,we will repeatedly refer to the signatures and morphisms in the following pushout diagram:ΣΣG Σ′Σ′G -ι6γ′where the local construction is along the bottom of the diagram,“cutting out”its argument from a larger algebra uses the signature morphism on the left,and the resulting global construction is along the top.Deﬁnition 3.1.Given a signature morphism ι:Σ→Σ′,a local construc-tion along ιis a persistent partial function F :|Alg (Σ)|⇀|Alg (Σ′)|(for each A ∈dom (F ),F (A )4Otherwise we would have to explicitly indicate “sharing”between the argument and result of each construction,and explain how such sharing is preserved by the vari-ous ways of putting together constructions,as was painfully spelled out in [ST89].If necessary,superﬂuous components of algebras constructed using persistent con-structions can be discarded at the end using the reduct along a signature inclusion.7above,using the amalgamation property:if Gι′=G and F G(G)γ);otherwise F G(G)is undeﬁned.This determines a global construction F G:|Alg(ΣG)|⇀|Alg(Σ′G)|,which is persistent alongι′:ΣG→Σ′G.This way of“lifting”a persistent function to a larger context via a“ﬁtting mor-phism”using signature pushout and amalgamation is well established in the algebraic speciﬁcation tradition,going back at least to“parametrized speciﬁca-tions”with free functor semantics,see[EM85].We will not dwell here on how particular(local)constructions are deﬁned. Free functor semantics for parametrized speciﬁcations is one way to proceed,with the persistency requirement giving rise to additional proof obligations[EM85]. Perhaps closer to ordinary programming is to give explicitly a“deﬁnitional”derived signature morphismδ:Σ′→Σthat deﬁnesΣ′-components in terms ofΣ-components.The induced reduct functionδ:|Alg(Σ)|→|Alg(Σ′)|is a local construction along a signature morphismι:Σ→Σ′wheneverι;δ=idΣ.5 Suppose now that a local construction F alongι:Σ→Σ′comes with a “semantic”speciﬁcation of its input/output properties,given as a speciﬁcation SP with Sig(SP)=Σof the requirements on its arguments together with a speciﬁcation SP′with Sig(SP′)=Σ′of the guaranteed properties of its result. Again w.l.o.g.we require that Mod(SP′)∈Mod(SP),and so Gγ)∈Mod(SP′).Consequently F G(G)∈Mod(γ′(SP′))∩Mod(ι′(SP G)).⊓⊔γInformally,this captures directly a“bottom-up”process of building implementa-tions,whereby we start with SP G,ﬁnd a local construction F∈Mod(SPι−→SP′)8with aﬁtting morphismγthat satisﬁes theﬁrst condition,and deﬁne SP′G such that the second condition is satisﬁed(e.g.take SP′G=γ′(SP′)andι′(SP G)). When proceeding“top-down”,we start with the global requirements speciﬁca-tion SP′G.To use a local construction F∈Mod(SPι−→SP′),we have to decide which part of the requirements it is going to implement by providing a signature morphismγ′:Sig(SP′)→Sig(SP′G),then construct the“pushout complement”γ:Sig(SP)→ΣG,ι′:ΣG→Sig(SP′G)forιandγ′,andﬁnally devise a speciﬁ-cation SP G with Sig(SP G)=ΣG such that both conditions are satisﬁed.Then we can proceed with SP G as the requirements speciﬁcation for the components that remain to be implemented.4Observational EquivalenceSo far,we have made few assumptions about the formalism used for writing speciﬁcations.Intuitively,it is clear that any such formalism should admit basic speciﬁcations given as sets of axioms over someﬁxed signature.The usual inter-pretation then is to take as models for such a basic speciﬁcation all the algebras that satisfy the axioms.However,in many practical examples this turns out to be overly restrictive.The point is that only a subset of the sorts in the signature of a speciﬁcation are typically intended to be directly observable—the oth-ers are treated as internal,with properties of their elements made visible only via observations leading to the observable sorts.This calls for a relaxation of the interpretation of speciﬁcations,as advocated in numerous“observational”or “behavioural”approaches,going back at least to[GGM76,Rei81].The starting point is that given an algebraic signature,one has toﬁx a set of observable sorts. Then,roughly,two approaches are possible:–introduce an internal observational indistinguishability relation between al-gebra elements,and re-interpret equality in the axioms as indistinguishabil-ity,–introduce an external observational equivalence on algebras,and re-interpret speciﬁcations by closing their class of models under such equivalence.It turns out that under some acceptable technical conditions,the two approaches are closely related and coincide for most basic speciﬁcations[BHW95,BT96]. However,the former approach seems more diﬃcult to extend to structured spec-iﬁcations and parametrization.Hence,we follow here the latter possibility.Deﬁnition4.1.Consider a signatureΣwith observable sorts OBS⊆sorts(Σ). We always assume that bool∈OBS.A correspondence between two algebras A,B∈|Alg(Σ)|,writtenρ:A⊲⊳B,is a relationρ⊆|A|×|B|that is closed under the operations6and is the identity on|A|bool=|B|bool.It is observational if it is bijective on observable sorts.Two algebras A,B∈|Alg(Σ)|are observationally equivalent,written A≡OBS B,if there exists an observational correspondence between them.9 This formulation is due to[Sch87](cf.“simulations”in[Mil71]and“weak ho-momorphisms”in[Gin68])and is equivalent to other standard ways of deﬁning observational equivalence between algebras,where a special role is played by observable equalities,i.e.,equalities between terms of observable sorts.It is easy to check that identities are correspondences and the class of corre-spondences is closed under composition and reducts w.r.t.signature morphisms.Correspondences may in fact be identiﬁed with certain spans of homomor-phisms:a correspondenceρ:A⊲⊳B is a span(h A:C→A,h B:C→B)where, for each sort s distinct from bool,|C|s is a subset of the Cartesian product |A|s×|B|s,|C|bool=|A|bool=|B|bool,the homomorphisms are the projections for all sorts s=bool,and the identity on the carrier of the sort bool.Such a span is observational if the homomorphisms are bijective on observable sorts.This directly implies that the reduct of a correspondence along a signature morphism is a correspondence.More interestingly,for observational correspondences this extends to derived signature morphisms with observable conditions.Consider a signatureΣwith observable sorts OBS⊆sorts(Σ).A conditional term((φi,t i)i≥0,t)is OBS-admissible if for all i≥0,φi are quantiﬁer-free formulae with observable equalities only.A derived signature morphismδ:Σ′→Σis OBS-admissible if it mapsΣ′-operations to OBS-admissible terms. Lemma4.2.Letδ:Σ′→Σbe an OBS-admissible derived signature morphism, A and B be twoΣ-algebras,andρ:A⊲⊳B be an observational correspondence. Thenρδ⊲⊳B10sort as observable by introducing an “equality predicate”on it.Moreover,this choice will not prevent us from manipulating an explicit set of observable sorts (always keeping bool among them though)when considering “local”signatures for veriﬁcation purposes.We will consider observational equivalence of “global”models with respect to the single observable sort bool —we write ≡{bool }simply as ≡.For any “global”speciﬁcation SP G with Sig (SP G )=ΣG ,we deﬁne its observational interpretation by abstracting from the standard interpretation as follows:Abs ≡(SP G )={G ∈|Alg (ΣG )||G ≡H for some H ∈Mod (SP G )}.5Observational Reﬁnement StepsThe most obvious way to re-interpret correctness of reﬁnement steps SP ′κ∼∼∼>SP to take advantage of the observational interpretation of speciﬁcations indicatedin the previous section is to relax the earlier deﬁnition by requiring Abs ≡(SP )⊆dom (κ)and κ(Abs ≡(SP ))⊆Abs ≡(SP ′).This works,but misses a crucial point:when using a realization of a speciﬁcation,we should be able to pretend that it satisﬁes the speciﬁcation literally,even if when actually implementing it we are permitted to supply an algebra that is correct only up to observational equiva-lence.This leads to a new notion of observational reﬁnement :given speciﬁcations SP and SP ′and a construction κ:|Alg (Sig (SP ′))|⇀|Alg (Sig (SP ))|,we deﬁne:SP ≡κ∼∼∼∼>SP ′⇐⇒Mod (SP ′)⊆dom (κ)and κ(Mod (SP ′))⊆Abs ≡(SP )This relaxation has a price:observational reﬁnements do not automatically com-pose!The crucial insight to resolve this problem comes from [Sch87],who noticed that well-behaved constructions satisfy the following stability property.Deﬁnition 5.1.A construction κ:|Alg (Σ)|⇀|Alg (Σ′)|is stable if it pre-serves observational equivalence of algebras,i.e.,for any algebras A,B ∈|Alg (Σ)|such that A ≡B ,if A ∈dom (κ)then B ∈dom (κ)and κ(A )≡κ(B ).Now,if all the constructions involved are stable then from a successful chain of observational reﬁnementsSP 0≡κ1∼∼∼∼>SP 1≡κ2∼∼∼∼>...≡κn ∼∼∼∼>SP n =EMPTY we obtain:κ1(κ2(...κn (empty )...))∈Abs ≡(SP 0).The rest of this section is devoted to an analysis of conditions that ensure stability of constructions and observational correctness of reﬁnement steps when the constructions arise via the use of local constructions,as in Sect.3.The problem is that we want to restrict attention to conditions that are essentially local to the local constructions involved,rather than conditions that refer to all the possible global contexts in which such a construction can be used.Let us start with the stability property.Deﬁnition5.2.A local construction F alongι:Σ→Σ′is locally stable if for anyΣ-algebras A,B∈|Alg(Σ)|and correspondenceρ:A⊲⊳B,A∈dom(F)if and only if B∈dom(F)and moreover,if this is the case then there exists a correspondenceρ′:F(A)⊲⊳F(B)that extendsρ(i.e.,ρ′:Gγ,so Gγ∈γdom(F),and consequently G∈dom(F G)iﬀH∈dom(F G).Suppose G)⊲⊳F(Hι=γρGι′=ρG.⊓⊔Corollary5.5.If F is a locally stable construction alongι:Σ→Σ′then for any signatureΣG andﬁtting morphismγ:Σ→ΣG,the induced global construction F G:|Alg(ΣG)|⇀|Alg(Σ′G)|alongι′:ΣG→Σ′G is stable.Proof.Let G,H∈|Alg(ΣG)|be such that G≡H.Then there is a correspon-denceρG:G⊲⊳H.By Lemma5.4,if G∈dom(F G)then H∈dom(F G)and there is a correspondenceρ′G:F G(G)⊲⊳F G(H),which proves F G(G)≡F G(H).⊓⊔This establishes a suﬃcient local condition which ensures that a local con-struction induces a stable global construction in every possible context of use.The following is a corollary of Lemma4.2.Corollary5.6.Letδ:Σ′→Σbe a{bool}-admissible derived signature mor-phism andι:Σ→Σ′be a signature morphism such thatι;δ=idΣ.Then the reduct F=δ:Mod(Σ)→Mod(Σ′))is a local construction that is locally sta-ble.⊓⊔The above corollary supports the point put forward in[Sch87]that stable constructions are those that respect modularity in the software construction process.That is,such constructions can use the components provided by their imported parameters,but they cannot take advantage of their particular internal properties.This is the point of the requirement thatδbe{bool}-admissible:any branching in the code must be governed by directly observable properties.This turns(local)stability into a directive for language design,rather than a condition to be checked on a case-by-case basis:in a language with good modularization facilities,all constructions that one can code should be locally stable.Let us turn now to the issue of correctness w.r.t.given speciﬁcations.Deﬁnition 5.7.A local construction F along ι:Sig (SP )→Sig (SP ′)is observa-tionally correct w.r.t.SP and SP ′if for every model A ∈Mod (SP ),A ∈dom (F )and there exists a model A ′∈Mod (SP ′)and correspondence ρ′:A ′⊲⊳F (A )such that ρ′ι=A and A ′≡ι(sorts (Σ))F (A ),whichin turn is in general stronger than F (A )∈Abs ≡ι(sorts (Σ))(SP ′).It follows thatif F ∈Mod lc (SP ι−→SP ′)then there is some F ′∈Mod (SP ι−→SP ′)such thatdom (F ′)=dom (F )and for each A ∈Mod (SP ),F ′(A )≡ι(sorts (Σ))F (A ).How-ever,in general Mod (SP ι−→SP ′)⊆Mod lc (SP ι−→SP ′),as strictly correct localconstructions need not be stable.Moreover,it may happen that there are no sta-ble observationally correct constructions,even if there are strictly correct ones:that is,we may have Mod lc (SP ι−→SP ′)=∅even if Mod (SP ι−→SP ′)=∅.This was perhaps ﬁrst pointed out in [Ber87],in a diﬀerent framework.Counterexample 5.8.Let SP 1include a non-observable sort s with two constants a,b :s ,and let SP 2enrich SP 1by an observable sort o ,two constants c,d :o and axiom c =d ⇐⇒a =b .Then Mod (SP 1→SP 2)is non-empty,with any construction in it mapping models satisfying a =b to those that satisfy c =d ,and models satisfying a =b to those that satisfy c =d .But none of these constructions is stable!Lemma 5.9.Consider a local construction F along ι:Sig (SP )→Sig (SP ′)that is observationally correct w.r.t.SP and SP ′.Then,for every global signature ΣG and ﬁtting morphism γ:Sig (SP )→ΣG ,for every G ∈Mod (γ(SP ))we have G ∈dom (F G )and there is some G ′∈Mod (γ′(SP ′))such that G ′γ∈Mod (SP ),and so Gγ)with identity reduct ρ′ι′=G and G′γ)amalgamate to a correspondenceρ′G :G ′⊲⊳F G (G ),which proves that F G (G )≡G ′∈Mod (γ′(SP ′)).⊓⊔If F ∈Mod lc (SP ι−→SP ′)and γ:Sig (SP )→ΣG ,then by Lemma 5.9we obtain γ′(SP ′)≡F G ∼∼∼∼>γ(SP ),and since F G is stable by Cor.5.5,we can use this in the observational development process.Given two “global”speciﬁcations SP G with Sig (SP G )=ΣG and SP ′G with Sig (SP ′G )=Σ′G ,we have SP ′G ≡F G ∼∼∼∼>SP G whenever Mod (SP G )⊆Abs ≡(γ(SP ))and Mod (γ′(SP ′))⊆Abs ≡(SP ′G ).But while the former requirement is quite acceptable,the latter is in fact impossible to achieve in practice since it implicitly requires that all the global requirements must follow (up to observational equivalence)from the result speciﬁcation for the local construction.More practical requirements are obtained by generalizing Thm.3.3to the observational setting:Theorem5.10.Given a local construction F∈Mod lc(SPι−→SP′),speciﬁca-tion SP G withﬁtting morphismγ:Sig(SP)→Sig(SP G),and speciﬁcation SP′G with Sig(SP′G)=Σ′G,if(i)Mod(SP G)⊆Abs≡(SP G andγ(SP)),and(ii)Mod(γ′(SP′)andι′(SP G))⊆Abs≡(SP′G)then for every G∈Mod(SP G),we have G∈dom(F G)and F G(G)∈Abs≡(SP′G). Consequently:∼∼∼∼>SP G.SP′G≡F GProof.Let G∈Mod(SP G).Then G≡H for some H∈Mod(SP G)∩Mod(γ(SP)) by(i).By Lemma5.9,F G(H)≡H′for some H′∈Mod(γ′(SP′))with H′⊆Mod(SP)),which isγoften easiest to verify.However,(i)is strictly stronger in general than the perhaps more expected Mod(SP G)⊆Abs≡(γ(SP)).This weaker condition turns out to be suﬃcient(and in fact,equivalent to(i))if we additionally assume that the two speciﬁcations involved are behaviourally consistent[BHW95],that is,closed under observational quotients.When this is not the case,then the use of this weaker condition must be paid for by a stronger version of(ii):Abs≡(γ′(SP′))∩Mod(ι′(SP G))⊆Abs≡(SP′G),which seems even less convenient to use than(i).Overall,we need a way to pass information on the global context from SP G to SP′G independently from the observational interpretation of the local construction and its correctness,and this must result in some inconvenience of veriﬁcation on either the parameter or the result side.6Architectural SpeciﬁcationsUsing local constructions for global implementations of speciﬁcations,we have moved only one step away from the monolithic global view of speciﬁcations and constructions used to implement them.The notion of architectural speciﬁ-cation[BST02]as introduced for Casl takes us much further.An architectural speciﬁcation prescribes a decomposition of the task of implementing a require-ments speciﬁcation into a number of subtasks to implement speciﬁcations of “modular components”(called units)of the system under development.The units may be parametrized,and then we can identify them with local construc-tions;non-parametrized units are modeled as algebras.Another essential part of an architectural speciﬁcation is a prescription of how the units,once developed,。

A Stakeholder Framework for Analyzing and Evaluating Cororate Social Performance

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conceptualframework会计概念框架

nd limitations of a conceptual framework
Financial statements - IASC Making Corporate Reports Valuable - ICAS Statement of Principles - ASB
Statement of principles
objectives and users of financial statements attributes of financial statements that enable them to fulfil
identification of a set of (ideally mutually exclusive and collectively exhaustive) criteria to be used in choosing between alternative solutions to standard setting problems in assessing the quality and utility of financial reports
nature, function and limits of financial
statements.”
FASB
“A basic structure for organising one’s
thinking about what one is trying to do and
how to go about it”
Some conceptual frameworks
FASB conceptual framework project The Corporate report - ASC Guidelines for financial reporting standards - The

Granger Causality Basic Theory and Application to Neuroscience

Steven L. Bressler
Center for Complex Systems and Brain Sciences, Florida Atlantic University, USA
Preprint submitted to Elsevier Science
Hale Waihona Puke 7 February 2008
2
1
Introduction
In neuroscience, as in many other ﬁelds of science and engineering, signals of interest are often collected in the form of multiple simultaneous time series. To evaluate the statistical interdependence among these signals, one calculates cross correlation functions in the time domain and ordinary coherence functions in the spectral domain. However, in many situations of interest, symmetric 1 measures like ordinary coherence are not completely satisfactory, and further dissection of the interaction patterns among the recorded signals is required to parcel out eﬀective functional connectivity in complex networks. Recent work has begun to consider the causal inﬂuence one neural time series can exert on another. The basic idea can be traced back to Wiener [1] who conceived the notion that, if the prediction of one time series could be improved by incorporating the knowledge of a second one, then the second series is said to have a causal inﬂuence on the ﬁrst. Wiener’s idea lacks the machinery for practical implementation. Granger later formalized the prediction idea in the context of linear regression models [2]. Speciﬁcally, if the variance of the autoregressive prediction error of the ﬁrst time series at the present time is reduced by inclusion of past measurements from the second time series, then the second time series is said to have a causal inﬂuence on the ﬁrst one. The roles of the two time series can be reversed to address the question of causal inﬂuence in the opposite direction. From this deﬁnition it is clear that the ﬂow of time plays a vital role in allowing inferences to be made about directional causal inﬂuences from time series data. The interaction discovered in this way may be reciprocal or it may be unidirectional. Two additional developments of Granger’s causality idea are important. First, for three or more simultaneous time series, the causal relation between any two of the series may be direct, may be mediated by a third one, or may be a combination of both. This situation can be addressed by the technique of conditional Granger causality. Second, natural time series, including ones from economics and neurobiology, contain oscillatory aspects in speciﬁc frequency bands. It is thus desirable to have a spectral representation of causal inﬂuence. Major progress in this direction has been made by Geweke [3,4] who found a novel time series decomposition technique that expresses the time domain Granger causality in terms of its frequency content. In this article we review the essential mathematical elements of Granger causality with special emphasis on its spectral decomposition. We then discuss practical issues concerning how to estimate such measures from time series data. Simulations are used to illustrate the theoretical concepts. Finally, we apply the technique to analyze the dynamics of a large-scale sensorimotor network in the cerebral cortex durHere by symmetric we mean that, when A is coherent with B, B is equally coherent with A.

Quantum Groups and Noncommutative Geometry

a r X i v :h e p -t h /0006167v 1 21 J u n 2000QUANTUM GROUPS AND NONCOMMUTATIVE GEOMETRYShahn MajidSchool of Mathematical Sciences,Queen Mary and Westﬁeld College University of London,Mile End Rd,London E14NS,UK November,1999Abstract Quantum groups emerged in the latter quarter of the 20th century as,on the one hand,a deep and natural generalisation of symmetry groups for certain integrable systems,and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain.Just as the last century saw the birth of classical geometry,so the present century sees at its end the birth of this quantum or noncommutative geometry,both as an elegant mathematical reality and in the form of the ﬁrst theoretical predictions for Planck-scale physics via ongoing astronomical measurements.Noncommutativity of spacetime,in particular,amounts to a postulated new force or physical eﬀect called cogravity.I Introduction Now that quantum groups and their associated quantum geometry have been around for more than a decade,it is surely time to take stock.Where did quantum groups come from,what have they achieved and where are they going?This article,which is addressed to non-specialists (but should also be interesting for experts)tries to answer this on two levels.First of all on the level of quantum groups themselves as mathematical tools and building blocks for physical models.And,equally importantly,quantum groups and their associated noncommutative geometry in terms of their overall signiﬁcance for mathematics and theoretical physics,i.e.,at a more conceptual level.Obviously this latter aspect will be very much my own perspective,which is that of a theoretical physicist who came to quantum groups a decade ago as a tool to unify quantum theory and gravity in an algebraic approach to Planck scale physics.This is in fact only one of the two main origins in physics of quantum groups;the other being integrable systems,which I will try to cover as well.Let me also say that noncommutative geometry has other approaches,notably the one of A.Connes coming out of operator theory.I will say something about this too,although,until recently,this has largely been a somewhat diﬀerent approach.We start with the conceptual signiﬁcance for theoretical physics.It seems clear to me that future generations looking back on the 20th century will regard the discovery of quantum mechanics in the 1920s,i.e.the idea to replace the coordinates x,p of classical mechanics bynoncommuting operators x,p,as one of its greatest achievements in our understanding of Nature, matched in its signiﬁcance only by the uniﬁcation of space and time as a theory of gravity.But whereas the latter was well-founded in the classical geometry of Newton,Gauss,Riemann and Poincar´e,quantum theory was something much more radical and mysterious.Exactly which variables in the classical theory should correspond to operators?They are local coordinates on phase space but how does the global geometry of the classical theory look in the quantum theory,what does it fully correspond to?The problem for most of this century was that the required mathematical structures to which the classical geometry might correspond had not been invented and such questions could not be answered.As I hope to convince the reader,quantum groups and their associated noncommutative geometry have led in the last decades of the20th century to theﬁrst deﬁnitive answers to this kind of question.There has in fact emerged a more or less systematic generalisation of geometry every bit as radical as the step from Euclidean to non-Euclidean,and powerful enough not to break down in the quantum domain.I do doubt very much that what we know today will be theﬁnal formulation,but it is a deﬁnitive step in a right and necessary direction and a turning point in the future development of mathematical and theoretical physics.For example,any attempt to build a theory of quantum gravity with classical starting point a smooth manifold–this includes loop-variable quantum gravity,string theory and quantum cosmology,is necessarily misguided except as some kind of eﬀective approximation:smooth manifolds should come out of the algebraic structure of the quantum theory and not be a starting point for the latter. There is no evidence that the real world is any kind of smooth continuum manifold except as a macroscopic approximation and every reason to think that it is fundamentally not.I therefore doubt that any one of the above could be a‘theory everything’until it becomes an entirely algebraic theory founded in noncommutative geometry of some kind or other.Of course,this is my personal view.At any rate,I do not think that the fundamental importance of noncommutative geometry can be overestimated.First of all,anyone who does quantum theory is doing noncommutative geometry whether wanting to admit it or not,namely noncommutative geometry of the phase space.Less obvious but also true,we will see in Section II that if the position space is curved then the momentum space is by itself intrinsically noncommutative.If one gets this far then it is also natural that the position space or spacetime by itself could be noncommutative,which would correspond to a curved or nonAbelian momentum group.This is one of the bolder predictions coming out of noncommutative geometry.It has the simple physical interpretation as what I call cogravity,i.e.curvature or‘gravity’in momentum space.As such it is independent of i.e. dual to curvature or gravity in spacetime and would appear as a quite diﬀerent and new physical eﬀect.Theoretically cogravity can,for example,be detected as energy-dependence of the speed8185909510000500981200yearpapersFigure 1:Growth of research papers on quantum groupsof light.Moreover,even if cogravity was very weak,of the order of a Planck-scale eﬀect,it could still in principle be detected by astronomical measurements at a cosmological level.Therefore,just in time for the new millennium,we have the possibility of an entirely new physical eﬀect in Nature coming from fresh and conceptually sound new mathematics .Where quantum groups precisely come into this is as follows.Just as Lie groups and their associated homogeneous spaces provided deﬁnitive examples of classical diﬀerential geometry even before Riemann formulated their intrinsic structure as a theory of manifolds,so quantum groups and their associated quantum homogeneous spaces,quantum planes etc.,provide large (i.e.inﬁnite)classes of examples of proven mathematical and physical worth and clear geomet-rical content on which to build and develop noncommutative diﬀerential geometry.They are noncommutative spaces in the sense that they have generators or ‘coordinates’like the non-commuting operators x ,p in quantum mechanics but with a much richer and more geometric algebraic structure than the Heisenberg or CCR algebra.In particular,I do not believe that one can build a theory of noncommutative diﬀerential geometry based on only one example such as the Heisenberg algebra or its variants (however fascinating)such as the much-studied noncommutative torus.One needs many more ‘sample points’in the form of natural and varied examples to obtain a valid general theory.By contrast,if one does a search of BIDS one ﬁnds,see Figure 1,vast numbers of papers in which the rich structure and applications of quantum groups are explored and justiﬁed in their own right (data complied from BIDS:published pa-pers since 1981with title or abstract containing ‘quantum group*’,‘Hopf alg*’,‘noncommutative geom*’,‘braided categ*’,‘braided group*’,‘braided Hopf*’.)This is the signiﬁcance of quantum groups.And of course something like them should be needed in a quantum world where there is no evidence for a classical space such as the underlying set of a Lie group.Finally,it turns out that noncommutative geometry,at least of the type that we shall de-scribe,is in many ways cleaner and more straightforward than the special commutative limit. One simply does not need to assume commutativity in most geometrical constructions,including diﬀerential calculus and gauge theory.The noncommutative version is often less inﬁnite,dif-ferentials are often more regularﬁnite-diﬀerences,etc.And noncommutative geometry(unlike classical geometry)can be specialised without eﬀort to discrete spaces or toﬁnite-dimensional algebras.It is simply a powerful and natural generalisation of geometry as we usually know it. So my overall summary and prediction for the next millennium from this point of view is:•All geometry will be noncommutative(or whatever comes beyond that),with conventional geometry merely a special case.•The discovery of quantum theory,its correspondence principle(and noncommutative ge-ometry is nothing more than the elaboration of that)will be considered one of the century’s greatest achievement in mathematical physics,commensurate with the discovery of clas-sical geometry by Newton some centuries before.•Quantum groups will be viewed as theﬁrst nontrivial class of examples and thereby point-ers to the correct structure of this noncommutative geometry.•Spacetime too(not only phase space)will be known to be noncommutative(cogravity will have been detected).•At some point a future Einstein will combine the then-standard noncommutative geomet-rical ideas with some deep philosophical ideas and explain something really fundamental about our physical reality.In the fun spirit of this article,I will not be above putting down my own thoughts on this last point.These have to do with what I have called for the last decade the Principle of representation-theoretic self-duality[1].In eﬀect,it amounts to extending the ideas of Born reciprocity,Mach’s principle and Fourier theory to the quantum domain.Roughly speaking, quantum gravity should be recast as gravity and cogravity both present and dual to each other and with Einstein’s equation appearing as a self-duality condition.The longer-term philosophical implications are a Kantian or Hegelian view of the nature of physical reality,which I propose in Section V as a new foundation for next millennium.We now turn to another fundamental side of quantum groups,which is at the heart of their other origin in physics,namely as generalised symmetry groups in exactly solvable lattice models. It leads to diverse applications ranging from knot theory to representation theory to Poisson geometry,all areas that quantum groups have revolutionised.What is really going on here in my opinion is not so much the noncommutative geometry of quantum groups themselves as a diﬀerent kind of noncommutativity or braid statistics which certain quantum groups induce onany objects of which they are a symmetry.The latter is what I have called‘noncommutativity of the second kind’or outer noncommutativity since it not so much a noncommutativity of one algebra as a noncommutative modiﬁcation of the exchange law or tensor product of any two independent algebras or systems.It is the notion of independence which is really being deformed here.Recall that the other great‘isation’idea in mathematical physics in this century(after ‘quantisation’)was‘superisation’,where everything is Z2-graded and this grading enters into how two independent systems are interchanged.Physics traditionally has a division into bosonic or force particles and fermionic or matter particles according to this grading and exchange behaviour.So certain quantum groups lead to a generalisation of that as braided geometry[2]or a process of braidiﬁcation.These quantum groups typically have a parameter q and its meaning is a generalisation of the−1for supersymmetry.This in turn leads to a profound generalisation of conventional(including super)mathematics in the form of a new concept of algebra wherin one‘wires up’algebraic operations much as the wiring in a computer,i.e.outputs of one into inputs of another.Only,this time,the under or over crossings are nontrivial(and generally distinct)operations depending on q.These are the so-called‘R-matrices’.Afterwards one has the luxury of both viewing q in this way or expanding it around1in terms of a multiple of Planck’s constant and calling it a formal‘quantisation’–q-deformation actually uniﬁes both ‘isation’processes.For example,Lorentz-invariance,by the time it is q-deformed[3],induces braid statistics even when particles are initially bosonic.In summary,•The notion of symmetry or automorphism group is an artifact of classical geometry and ina quantum world should naturally be generalised to something more like a quantum groupsymmetry.•Quantum symmetry groups induce braid statistics on the systems on which they act.In particular,the notion of bose-fermi statistics or the division into force and matter particles is an artifact of classical geometry.•Quantisation and the departure from bosonic statistics are two limits of the same phe-nomenon of braided geometry.Again,there are plenty of concrete models in solid state physics already known with quantum group symmetry.The symmetry is useful and can be viewed(albeit with hindsight)as the origin of the exact solvability of these models.These two points of view,the noncommutative geometrical and the generalised symmetry, are to date the two main sources of quantum groups.One has correspondingly two mainﬂavours or types of quantum groups which really allowed the theory to take oﬀ.Both were introduced at the mid1980s although the latter have been more extensively studied in terms of applicationsto date.They include the deformationsU q(g)(1)of the enveloping algebra U(g)of every complex semisimple Lie algebra g[4][5].These have as many generators as the usual ones of the Lie algebra but modiﬁed relations and,additionally, a structure called the‘coproduct’.The general class here is that of quasitriangular quantum groups.They arose as generalised symmetries in certain lattice models but are also visible in the continuum limit quantumﬁeld theories(such as the Wess-Zumino-Novikov-Witten model on the Lie group G with Lie algebra g).The coordinate algebras of these quantum groups are further quantum groups C q[G]deforming the commutative algebra of coordinate functions on G.There is again a coproduct,this time expressing the group law or matrix multiplication.Meanwhile, the type coming out of Planck scale physics[6]are the bicrossproduct quantum groupsC[M]◮⊳U(g)(2)associated to the factorisation of a Lie group X into Lie subgroups,X=GM.Here the in-gredients are the conventional enveloping algebra U(g)and the commutative coordinate algebra C[M].The factorisation is encoded in an action and coaction of one on the other to make a semidirect product and coproduct◮⊳.These quantum arose at about the same time but quite independently of the U q(g),as the quantum algebras of observables of certain quantum spaces. Namely it turns out that G acts on the set M(and vice-versa)and the quantisation of those orbits are these quantum groups.This means that they are literally noncommutative phase spaces of honest quantum systems.In particular,every complex semisimple g has an associated complexiﬁcation and its Lie group factorises G C=GG⋆(the classical Iwasawa decomposition) so there is an exampleC[G⋆]◮⊳U(g)(3)built from just the same data as for U q(g).In fact the Iwasawa decomposition can be understood in Poisson-Lie terms with g⋆the classical‘Yang-Baxter dual’of g.In spite of this,there is,even after a decade of development,no direct connection between the two quantum groups:gւց(4)U q(g)←?→C[G⋆]◮⊳U(g).They are both‘exponentiations’of the same classical data but apparently of completely diﬀerent type(this remains a mystery to date.)Figure2:The landscape of noncommutative geometry todayAssociated to these twoﬂavours of quantum groups there are corresponding homogeneous spaces such as quantum spheres,quantum spacetimes,etc.Thus,of theﬁrst type there is a q-Minkowski space introduced in[7]as a q-Lorentz covariant algebra,and independently about a year later in[8]as2×2braided hermitian matrices.It is characterised by[x i,t]=0,[x i,x j]=0.(5) Meanwhile,of the second type there is a noncommutativeλ-Minkowski space with[x i,t]=λx i,[x i,x j]=0(6)which is the one that provides theﬁrst known predictions testable by astronomical measurements (by gamma-ray bursts of cosmological origin[9]).This kind of algebra was proposed as spacetime in[10]and in the4-dimensional case it was shown in[11]to be covariant under a Poincar´e quantum group of bicrossproduct form.These are clearly in sharp contrast.There are of course many more objects than these.q-spheres,q-planes etc.In Section IV we turn to the notion of‘quantum manifold’that is emerging from all these examples.Riemann was able to formulate the notion of Riemannian manifold as a way to capture known examples like spheres and tori but broad enough to formulate general equations for the intrinsic structure of space itself(or after Einstein,space-time).We are at a similar point now and what this ‘quantum groups approach to noncommutative geometry’is is more or less taking shape.It has the same degree of‘ﬂabbiness’as Riemannian geometry(it is not tied to speciﬁc integrable systems etc.)while at the same time it includes the‘zoo’of already known naturally occurring examples,mostly linked to quantum groups.Such things as Ricci tensor and Einstein’s equation are not yet understood from this approach,however,so I would not say it is the last word.This approach is in fairly sharp contrast to‘traditional’noncommutative geometry as it was done before the emergence of quantum groups.That theory was developed by mathematiciansand mathematical physicists also coming from quantum mechanics but being concerned more with topological completions and Hilbert spaces.Certainly a beautiful theory of von-Neumann and C∗algebras emerged as an analogue of point-set topology.Some general methods such as cyclic cohomology were also developed in the1970s,with remarkable applications throughout mathematics[12].However,for concrete examples with actual noncommutative diﬀerential geometry one usually turned either to an actual manifold as input datum or to the Weyl algebra (or noncommutative torus)deﬁned by relationsvu=e2πıθuv.(7)This in turn is basically the usual CCR or Heisenberg algebra[x,p]=ı (8)in exponentiated form.And at an algebraic level(i.e.until one considers the precise C∗-algebra completion)this is basically the usual algebra B(H)of operators on a Hilbert space as in quantum mechanics.Or at roots of unity it is M n(C)the algebra of n×n matrices.So at some level these are all basically one example.Unfortunately many of the tricks one can pull for this kind of example are special to it and not a foundation for noncommutative diﬀerential geometry of the type we need.For example,to do gauge theory Connes and M.Rieﬀel[13] used derivations for two independent vectorﬁelds on the torus.The formulation of‘vector ﬁeld’as a derivation of the coordinate algebra is what I would call the traditional approach to noncommutative geometry.For quantum groups such as C q[G]one simply does not have those derivations(rather,they are in general braided derivations).Similarly,in the traditional approach one deﬁnes a‘vector bundle’as aﬁnitely-generated projective module without any of the infrastructure of diﬀerential geometry such as a principal bundle to which the vector bundle might be associated,etc.All of that could not emerge until quantum groups arrived(one clearly should take a quantum group asﬁber).This is how the quantum groups approach diﬀers from the work of Connes,Rieﬀel,Madore and others.It is also worth noting that string theorists have recently woken up to the need for a noncommutative spacetime but,so far at least,have still considered only this‘traditional’Heisenberg-type algebra.In the last year or two there has been some success in merging these approaches,however;a trend surely to be continued. By now both approaches have a notion of‘noncommutative manifold’which appear somewhat diﬀerent but which have as point of contact the Dirac operator.Preliminaries.A full text on quantum groups is[14].To be self-contained we provide here a quick deﬁter on we will see many examples and various justiﬁcations for this concept. Thus,a quantum group or Hopf algebra is•A unital algebra H,1over theﬁeld C(say)•A coproduct∆:H→H⊗H and counitǫ:H→C forming a coalgebra,with∆,ǫalgebra homomorphisms.•An antipode S:H→H such that·(S⊗id)∆=1ǫ=·(id⊗S)∆.Here a coalgebra is just like an algebra but with the axioms written as maps and arrows on the maps reversed.Thus the coassociativity and counity axioms are(∆⊗id)∆=(id⊗∆)∆,(ǫ⊗id)∆=(id⊗ǫ)∆=id.(9)The antipode plays a role that generalises the concept of group inversion.Other than that the only new mathematical structure that the reader has to contend with is the coproduct∆and its associated counit.There are several ways of thinking about the meaning of this depending on our point of view.If the quantum group is like the enveloping algebra U(g)generated by a Lie algebra g,one should think of∆as providing the rule by which actions extend to tensor products.Thus,U(g)is trivially a Hopf algebra with∆ξ=ξ⊗1+1⊗ξ,∀ξ∈g,(10)which says that when a Lie algebra elementξacts on tensor products it does so byξin the ﬁrst factor and thenξin the second factor.Similarly it says that when a Lie algebra acts on an algebra it does so as a derivation.On the other hand,if the quantum group is like a coordinate algebra C[G]then∆expresses the group multiplication andǫthe group identity element e. Thus,if f∈C[G]the coalgebra is(∆f)(g,h)=f(gh),∀g,h∈Gǫf=f(e)(11)at least for suitable f(or with suitable topological completions).In other words it expresses the group product G×G→G by a map in the other direction in terms of coordinate algebras. From yet another point of view∆simply makes the dual H∗also into an algebra.So a Hopf algebra is basically an algebra such that H∗is also an algebra,in a compatible way,which makes the axioms‘self-dual’.For everyﬁnite-dimensional H there is a dual H∗.Similarly in the inﬁnite-dimensional case.It said that in the Roman empire,‘all roads led to Rome’.It is remarkable that several diﬀerent ideas for generalising groups all led to the same axioms.The axioms themselves wereﬁrst introduced(actually in a super context)by H.Hopf in1947in his study of group cohomology but the subject only came into its own in the mid1980s with the arrival from mathematical physics of the large classes of examples(as above)that are neither like U(g)nor like C[G],i.e.going truly beyond Lie theory or algebraic group theory.Acknowledgements.An announcement of this article appears in a short millennium article[15] and a version more focused on the meaning for Planck scale physics in[16].II Quantum groups and Planck scale physicsThis section covers quantum groups of the bicrossproduct type coming out of Planck-scale physics[6]and their associated noncommutative geometry.These are certainly less well-developed than the more familiar U q(g)in terms of their concrete applications;one does not have inter-esting knot invariants etc.On the other hand,these quantum groups have a clearer physical meaning as models of Planck scale physics and are also technically easier to construct.Therefore they are a good place to start.Obviously if we want to unify quantum theory and geometry then a necessaryﬁrst step is to cast both in the same language,which for us will be that of algebra.We have already mentioned that vectorﬁelds can be thought of classically as derivations of the algebra of functions on the manifold,and if one wants points they can be recovered as maximal ideals in the algebra, etc.This is the more of less standard idea of algebraic geometry dating from the late19th century and early on in the20th.It will certainly need to be modiﬁed before it works in the noncommutative case but it is a starting point.The algebraic structure on the quantum side will need more attention,however.II.A CogravityWe begin with some very general considerations.In fact there are fundamental reasons why one needs noncommutative geometry for any theory that pretends to be a fundamental one.Since gravity and quantum theory both work extremely well in their separate domains,this comment refers mainly to a theory that might hope to unify the two.As a matter of fact I believe that, through noncommutative geometry,this‘holy grail’of theoretical physics may now be in sight.Theﬁrst point is that we usually do not try to apply or extend our geometrical intuition to the quantum domain directly,since the mathematics for that has traditionally not been known. Thus,one usually considers quantisation as the result of a process applied to an underlying classical phase space,with all of the geometrical content there(as a Poisson manifold).But demanding any algebra such that its commutators to lowest order are some given Poisson bracket is clearly an illogical and ill-deﬁned process.It not only does not have a unique answer but also it depends on the coordinates chosen to map over the quantum operators.Almost always one takes the Poisson bracket in a canonical form and the quantisation is the usual CCR or canonical commutation relations algebra.Maybe this is the local picture but what of the global geometry of the classical phase space?Clearly all of these problems are putting the cart before the horse:the real world is to our best knowledge quantum so that should comeﬁrst.We should build models guided by the intrinsic(noncommutative)geometry at the level of noncommutative algebras and only at the end consider classical limits and classical geometry(and Poisson brackets)as emerging from a choice,where possible,of‘classical handles’in the quantum system.In more physical terms,classical observables should come out of quantum theory as some kind of limit and not really be the starting point;in quantum gravity,for example,classical geometry should appear as an idealisation of the expectation value of certain operators in certain states of the system.Likewise in string theory one starts with strings moving in classical spacetime, deﬁnes Lagrangians etc.and tries to quantise.Even in more algebraic approaches,such as axiomatic quantumﬁeld theory,one still assumes an underlying classical spacetime and classical Poincar´e group etc.,on which the operatorﬁelds live.Yet if the real world is quantum then phase space and hence probably spacetime itself should be‘fuzzy’and only approximately modeled by classical geometrical concepts.Why then should one take classical geometrical concepts inside the functional integral except other than as an eﬀective theory or approximate model tailored to the desired classical geometry that we hope to come out.This can be useful but it cannot possibly be the fundamental‘theory of everything’if it is built in such an illogical manner.There is simply no evidence for the assumption of nice smooth manifolds other than now-discredited classical mechanics.And in certain domains such as,but not only,in Planck scale physics or quantum gravity,it will certainly be unjustiﬁed even as an approximation.Next let us observe that any quantum system which contains a nonAbelian global symmetry group is already crying out for noncommutative geometry.This is in addition to the more obvious position-momentum noncommutativity of quantisation.The point is that if our quantum system has a nonAbelian Lie algebra symmetry,which is usually the case when the classical system does, then from among the quantum observables we should be able to realise the generators of this Lie algebra.That is,the algebra of observables A should contain the algebra generated by the Lie algebra,A⊇U(g).(12)Typically,A might be the semidirect product of a smaller part with external symmetry g by the action of U(g)(which means that in the bigger algebra the action of g is implemented by the commutator).This may soundﬁne but if the algebra A is supposed to be the quantum analogue of the‘functions on phase space’,then for part of it we should regard U(g)‘up side down’not as an enveloping algebra but as a noncommutative space with g the noncommutative coordinates.In other words,if we want to elucidate the geometrical content of the quantum algebra of observables then part of that will be to understand in what sense U(g)is a coordinate algebra,U(g)=C[?].(13)Here?cannot be an ordinary space because its supposed coordinate algebra U(g)is noncom-mutative.。

A CONCEPTUAL FRAMEWORK

Johan van Niekerk1,Rossouw von Solms21,2Centre for Information Security Studies,Nelson Mandela Metropolitan University,South Africa1johanvn@nmmu.ac.za,+27415043048,PO Box77000,Port Elizabeth,60002rossouw@nmmu.ac.za,+27415043669,PO Box77000,Port Elizabeth,6000ABSTRACTThe importance of establishing an information security culture in an organization has become a well established idea.The aim of such a culture is to address the various human factors that can affect an organization’s overall information security efforts.However,understanding both the various el-ements of an information security culture,as well as the relationships between these elements,can still be problematic.Schein’s deﬁnition of a corporate culture is often used to aid understanding of an information security culture.This paper brieﬂy introduces Schein’s model.It then incorporates the important role knowledge plays in information security into this deﬁnition.Finally,a conceptual framework to aid understanding of the interactions between the various elements of such a culture, is presented.This framework is explained by means of illustrative examples,and it is suggested that this conceptual framework can be a useful aid to understanding information security culture. KEYWORDSInformation Security,Information Security Culture,Corporate Culture,Organizational Learning, Schein’s Model.1INTRODUCTIONToday,most organizations need information systems to survive and rmation has become a valuable asset to modern organizations.It is therefore imperative for modern organizations to take the protection of their information resources seriously.This protection of information resources,also known as information security,consist of many processes.Some of these processes are,to a large extent,dependent on human co-operated behavior.Employees,whether intentionally or through neg-ligence,often due to a lack of knowledge,are the greatest threat to information security(Mitnick& Simon,2002,p.3).Without an adequate level of user cooperation and knowledge,many security techniques are liable to be misused or misinterpreted by users.This may result in even an adequate security measure becoming inadequate(Siponen,2001).An organization’s information security strat-egy should thus comprehensively address this”human factor”.It is important to note that there are two dimensions to this”human factor”in information security,namely knowledge,and cooperation, or behavior.These dimensions are to a large extend interrelated to each other.Organizations cannot protect the integrity,conﬁdentiality,and availability of information in today’s highly networked systems environment without ensuring that each person involved shares the security vision of the organization,understands his/her roles and responsibilities,and is adequately trained to perform them(ISO/IEC TR13335-1,2004,p.14).In order to assist in ensuring information security,individual users thus needs knowledge regarding their speciﬁc role in the security process. This knowledge can be provided via education,training and awareness campaigns.Once these users have sufﬁcient knowledge about their roles in the security process,there is still no guarantee that they will adhere to their required security roles.It is possible that users understand their roles correctly but still don’t adhere to a security policy because it conﬂicts with their beliefs and values(Schlienger&Teufel,2003).It is therefore imperative to also ensure that the users have the correct attitude,and thus the desired behavior,towards information security.In order to ensure the desired user behavior,it is necessary to cultivate an organizational sub-culture of information security (V on Solms,2000;Schlienger&Teufel,2003).Such a culture should support all business activities in such a way that information security becomes a natural aspect in the daily activities of every employee (Schlienger&Teufel,2003).Education of employees plays a very important role in the establishment of such a culture.It is paramount that the people are educated to want to be more secure in their day to day operation(Nosworthy,2000).Such a change of attitude is of utmost importance,because a change in attitude automatically leads to a subsequent behavioral change(Nosworthy,2000).Through the establishment of an information security culture,the employees can become a security asset, instead of being a risk(V on Solms,2000).Many recent studies have shown that the establishment of an information security culture in the organization is in fact necessary for effective information security(Eloff&V on Solms,2000; V on Solms,2000).However,such a culture must be supported by adequate knowledge regarding information security(Van Niekerk&V on Solms,2005).Without adequate knowledge,users who want to behave securely,might still apply a security control incorrectly.Conversely,a user who has adequate knowledge,but believes that secure behavior is unnecessary in his/her speciﬁc role,might still behave in an insecure way.Due to this co-dependence between the knowledge dimension of the human factor in information security,and the behavioral dimension,it would be beneﬁcial to deal with both these dimensions holistically.It would thus make sense to have a single conceptual framework that can be used to reason about both the knowledge,and the behavioral aspects of this human factor in information security.This paper will brieﬂy adapt the”generic”deﬁnition of corporate culture to the speciﬁc needs of an information security culture.This adapted deﬁnition will then be used toprovide a conceptual framework for examining the various aspects of the human factors in information security.In examining this adapted deﬁnition,it is important to realize that knowledge,and the underly-ing educational programs needed to impart such knowledge,is often seen as part of corporate culture. It is not the intent of this paper to dispute this view.In fact,this paper supports the view that knowl-edge and education will always play a role towards ensuring speciﬁc behavior patterns.However,this paper does attempt to highlight the fact that the knowledge”dimension”is of particular importance in an information security culture,and that security knowledge plays a very speciﬁc enabling role in information security.The additional knowledge”dimension”this paper will present,represents the knowledge needed to effectively implement,or use,the security measures if the desired attitude can be assumed.The knowledge that form an underlying part of any corporate culture is still assumed to be present.In that respect,an information security culture is assumed to be the same as a”normal”corporate culture.2RESEARCH PARADIGM AND RATIONALEThe work in this paper is based on qualitative,or phenomenological-,research methods,as described in Creswell(1998).This paper should thus be seen as”an inquiry process of understanding based on distinct methodological traditions of inquiry that explore a social or human problem”(Creswell,1998, p.15).The research presented here does not attempt to deﬁne new knowledge,but rather to provide a more in-depth understanding of the phenomenon described as”information security culture”.As far as could be determined,the speciﬁc conceptual model,as well as the underlying interactions between the various levels of information security culture,as presented in this paper,has never been published before.It is the authors’belief that the use of this conceptual model could improve the understanding of the concept of information security culture.Since the concept of organizational culture has been largely”borrowed”by information security researchers from the humanities,it was deemedﬁtting to also”borrow”the research paradigm,used in this paper from the humanities.The model for corporate culture as presented in Schein(1999)has become widely accepted amongst information security researchers(Schlienger&Teufel,2003).However,this model describes corporate culture in general,and not information security culture speciﬁcally.In order to ensure a rigorous research approach,even concepts with a seemingly obvious meaning will be revisited in this paper.The description of these concepts in the presented information security framework is deemed necessary because there might exist differences between the ontologies commonly adhered to by information security specialists and researchers from the management sciences.The aim of this paper is thus to present an holistic,conceptual model of information security culture,for information security practitioners and students.This model aims to clarify,at a conceptual level,the interactions between various elements comprising such an information security culture.The model also attempt to clearly deﬁne,in an information security context,concepts such as the strength and the stability(or predictability)of an information security culture.The model presented in this paper is intended to clarify,and improve,the understanding of exiting concepts.It is hoped that this model will be of use to other information security researchers when examining the human factors in information security.Before the speciﬁc concept of an information security culture is examined,this paper willﬁrst explore the existing deﬁnition of corporate culture.3CORPORATE CULTUREEvery organization has a particular culture,comprising an omnipresent set of assumptions that is often difﬁcult to fathom,and that directs the activities within the organization(Smit&Cronj´e,1992, p.382).Such a culture could be deﬁned as;the beliefs and values shared by people in an organization (Smit&Cronj´e,1992,p.382).Beliefs and values,however,are both concepts that can be difﬁcult to quantify.It is therefor often tempting to think of culture as just”the way we do things around here”(Schein,1999,p.15),or that”something”that makes an organization more successful than others (Smit&Cronj´e,1992,p.383).However,oversimplifying the concept of culture is the biggest danger to understanding it(Schein,1999,p.15).A better way to think about culture is to examine the different”levels”at which culture exists (Schein,1999,p.15).This way of thinking about corporate culture is already widely accepted in information security(Schlienger&Teufel,2003).In order to clarify these levels of culture,each of the levels will be brieﬂy examined:•Level One:Artifacts.Artifacts are what you can observe,see,hear,and feel,in an orga-nization(Schein,1999,p.15).Artifacts would include visible organizational structures and processes.At the level of artifacts,culture is very clear and has an immediate emotional im-pact,which could be positive or negative,on the observer(Schein,1999,p.16).Observing the artifacts alone,however,does not explain why the members of the organization behave as they do(Schein,1999,p.16).In order to understand the reasons for the behavior patterns of organization members it is necessary to examine”deeper”levels of culture(Schein,1999,p.16),such as the organization’s espoused values.•Level Two:Espoused Values.An organization’s espoused values are the”reasons”an organi-zational insider would give for the observed artifacts(Schein,1999,p.17),for example;that the organization believes in team work,that everyone in the organization’s view is important in the decision making process,etc.Espoused values generally consist of the organization’s ofﬁcial viewpoints,such as mission-or vision-statements,strategy documents,and any other documents that describe the organization’s values,principles,ethics,and visions(Schein,1999, p.17).However,it is possible for two organizations to have very different observable artifacts and yet share very similar espoused values(Schein,1999,pp.18-19).This is because there is an even deeper level of thought and perception that drives the overt,or observable,behavior (Schein,1999,p.19).The espoused values are values which the organization wants to live up to.The interpretation,and application,of these espoused values in the day to running of the organization depends on the shared tacit assumptions between the employees of that organiza-tion.•Level Three:Shared Tacit Assumptions.The shared tacit assumptions in an organization develop in any successful organization.Often these assumptions are formed in the organiza-tion’s early years,because certain strategies have proven to be successful(Schein,1999,p.19).If strategies based on speciﬁc beliefs and values continue to be successful,these beliefs and values gradually come to be shared and taken for granted.The beliefs and values become tacit assumptions about the nature of the world and how to succeed in it(Schein,1999,p.19).These values,beliefs,and assumptions that have become shared and taken for granted in an organi-zation,form the essence of that organization’s culture.Beliefs,in this sense,refer to a group of people’s convictions about the world and how it works,whilst values refer to a community’s basic assumptions about what ideals are worth pursuing(Smit&Cronj´e,1992,p.383).It is important to remember that the shared tacit assumptions resulted from a joint learning process.The corporate culture of any organization,is a result of all three the above levels.At its most basic,and most difﬁcult to quantify,level,the members of the organization share certain beliefs and values.These shared tacit assumptions act as a kind of”ﬁlter”,which affects how individuals will carry out their normal day-to-day activities.It also inﬂuences how these individuals interpret the or-ganization’s policies,and how they implement its procedures.These policies and procedures form part of the organization’s espoused values.The espoused values can be seen as the”visible”contri-bution of the organization’s management towards the organization’s culture.To a degree,espousedvalues provide cultural direction.The interpretation of this”direction”,however,is extremely de-pendant on the underlying shared tacit assumptions.These three levels of corporate culture could be seen to correspond closely to the behavioral aspects of the”human factor”in information security. As mentioned earlier,this”human factor”in information security consist of two dimensions,namely knowledge and behavior,which are very inter-related.Due to the co-dependency between these two dimensions it is not possible to ignore the impact a lack of information security related knowledge would have on an organizational sub-culture of information security.4INFORMATION SECURITY CULTUREIn”normal”deﬁnitions of organizational culture,the relevant job-related knowledge are generally ignored,because it can be assumed that the average employee would have the needed knowledge to do his/her job.In the case of information security,the required knowledge is not necessarily needed to perform the employee’s normal job functions.Knowledge of information security is generally only needed when it is necessary to perform the normal job functions in a way that is consistent with good information security practices.It can not be assumed that the average employee has the necessary knowledge to perform his/her job in a secure manner.If an organization is trying to foster a sub-culture of information security,all activities would have to be performed in a way that is consistent with good information security practice.Having adequate knowledge regarding information security is a prerequisite to performing any normal activity in a secure rmation security knowl-edge,or a lack thereof,could therefor be seen as a fourth level to an information security culture that will affect each of the other three layers.For example:Artifacts:Artifacts are what actually happens in the organization.Without the necessary skills and proﬁciencies,it would be impossible to perform security related tasks correctly.Thus,for the day-to-day task to happen in a secure way,the users would have to have sufﬁcient knowledge of how to perform their tasks securely.Espoused Values:To create the policy document,the person,or team,responsible for the drafting of the policy must know what to include in such a policy in order to adequately address the organization’s security needs.Shared Tacit Assumptions:This layer consists of the beliefs and values of employees.If such a be-lief should conﬂict with one of the espoused values,knowing why a speciﬁc control is needed, might play a vital role in ensuring compliance(Schlienger&Teufel,2003).It should be clear that in an information security culture,knowledge underpins and supports all three the”normal”levels of corporate culture.Without adequate knowledge,information secu-rity cannot be ensured.The co-dependency between the three”normal”levels of an organization’s information security culture,and knowledge,the”fourth level”,implies that each of these four lev-els will have an impact on how”secure”,or desirable,the overall information security culture will be.Theﬁrst part of the model presented in this paper is thus an adaptation of Schein’s model.This adaptation incorporates the underlying need for information security related knowledge into Schein’s model.Knowledge are added as a fourth level of culture that is speciﬁc to an information security culture.This adaptation is necessary because in an information security culture the requisite knowl-edge cannot be assumed to be present.Figure1,provides a graphical exposition of this adaptation. In this presented conceptual model,knowledge is dealt with as an additional level to culture,as op-posed to viewing knowledge as a sub-component of each of the original three levels.This is done solely because modeling knowledge as an additional level makes it easier to clearly show the effect knowledge,or a lack thereof,would have on the overall information security culture.Figure1:Levels of Culture(adapted from Schein,1999,p.16)5INFORMATION SECURITY CULTURE:A CONCEPTUAL FRAMEWORKThe overall effect of an organization’s information security culture can be seen as an accumulation of the effects of each of the culture’s underlying levels.Each of these levels can either positively or negatively inﬂuence the overall information security culture.In orderto clearly demonstrate theinteractions between these four levels,and their effects on the overall security efforts,it is necessary toﬁrst provide a basic reference framework.5.1Basic Elements and Terminology of the Conceptual FrameworkThe basic elements of this framework are depicted in Figure2.The elements in Figure2can be described as follows:•BL:Minimum Acceptable Base Line-This line indicates what would be an acceptable min-imum security baseline.In other words,a culture whose net effect would meet the minimum requirements for some industry standard.•SL:Nett Security Level-This line indicates the actual nett effect of the culture on the overallsecurity effort.This line can be seen as the cumulative effect of the four underlying levels of the culture.The nett security level(SL)can either be more secure(to the right),less secure(to the left),or just as secure(overlapping)as the minimum acceptable baseline(BL).•AF:Artifacts-This node represents the relative strength of the artifact level(AF)of the cul-ture.If this node is to the left of the minimum acceptable baseline(BL),it indicates that the measurable artifacts are not as secure as they should be.A node to the right of the baseline (BL)would indicate artifacts that are even more secure than the acceptable minimum.A node exactly on the baseline(BL)would indicate artifacts that are just as secure as required by this baseline.•EV:Espoused Values-This node represents the relative strength of the organization’s espoused value level(EV).The various policies and procedures comprising this level could be more,less, or just as comprehensive than those recommended as the minimum acceptable baseline.•SA:Shared Tacit Assumptions-This node represents the relative strength of the organization’s shared tacit assumption level(SA).The underlying beliefs or values of the employees could be either more,less,or just as in favor of good secure practices as required by the minimum acceptable baseline.•KN:Knowledge-This node represents how much knowledge the organization’s employees have regarding information security.Employees can be more knowledgeable than a certain minimum level needed to perform their jobs securely,they could be less knowledgeable,or they could have exactly the minimum requisite level of knowledge.As mentioned above,the horizontal alignment of the nodes representing the various cultural levels,AF,EV,SA and KN,in comparison to the minimum acceptable baseline,should be interpreted as an indication of the relative strength of each level.In a similar fashion,the horizontal alignment of the nodes in comparison to the same horizontal alignment of the other levels should be interpreted as an indication of how stable,or predictable,the culture is.The nett security level line(SL)is an indication of the average strength of the culture,or the nett combined effect of all four the levels.The culture depicted in Figure2should thus,ﬁrstly,be interpreted as a strong,or secure culture.All four levels in Figure2has a strength greater than the baseline,which also results in a nett security level that is positive,or greater than the baseline.Secondly,all four levels are perfectly aligned with each other.This results in a culture that should be completely stable,or predictable.The culture depicted in Figure2could thus be said to be the ideal culture in terms of information security since it is both strong and stable.The terms strong,and stable,as used above,should not be confused as being indicative of how pervasive or resistent to change the culture might be.According to Schein(1999,pp.25-26), corporate culture is always strong in the sense of affecting every single aspect of daily life in an organization at a more than superﬁcial level.Culture is also always stable,in the sense that it resists attempts at changing it.In that sense,culture is one of the most stable facets in an organization (Schein,1999,p.26).When referring to an information security culture,the term strong,as used in this paper,should be interpreted as a desirable culture that is conductive to information security.The term stable,as used in the same context,should be interpreted as an indication of how predictable the resulting artifacts,or nett security level of the culture would be for any speciﬁc scenario.All of the factors mentioned above would contribute to the overall desirability of an information security culture.How strong,and stable an organization’s information security culture is,would depend on the interaction between the various levels of culture.Figure 3:Possible interactions between the various levels of an Information Security Culture.(BL =Minimum Acceptable Baseline,SL =Nett Security Level,AF=Artifacts,EV =Espoused Values,SA=Shared Tacit Assumptions,KN=Knowledge)5.2Interpreting the Conceptual FrameworkEach of the underlying cultural levels will contribute towards the overall strength and stability of such a culture.For example,if an organization has espoused values that are in line with recommended best practices for security,this would make the overall security better.Conversely,should the espoused values fail to address all relevant security related issues,the overall security would be weaker.The combination of the espoused values,and the ”ﬁltering effect”of the shared tacit assump-tions and the user knowledge,on these espoused values,results in the visible,and measurable arti-facts .From a security viewpoint,the artifact level is a very good indication of the overall security of the organization’s information,since this level reﬂects what actually happens in the day to day op-erations.Fig.3shows a few possible effects interactions between the various levels of culture could have on the overall state of the organization’s information security.The examples in Fig.3assumes that the desirability of the various levels can be quantiﬁed and normalized to the same scale.In other words,it is assumed that,for example,the desirability of the relevant espoused values can be measured and expressed as a value that indicates the contribution of this level towards the overall security.It is also assumed that the other levels can be expressed in the same way,and that the scale of such measurements can be normalized in such a way that these values will indicate the relative desirability of that level when compared to the other levels.The line marked SL (Security Level )represents the nett effect of the interactions between various levels of the culture.The four examples in Fig.3can be interpreted as follows:•A:”Neutral”and Stable .The desirability of the various levels of culture are ”neutral”,oraverage.In other words the strength of each level neither exceeds,nor falls short,of the mini-mum acceptable baseline standards.Since all the levels have the same level of desirability,the various levels will neither negate nor reinforce the effects of other levels on the overall security.The effects of such a culture would thus be predictable and stable.•B:Insecure and ”Mostly Stable”.Both the espoused values and the shared tacit assumptionsin this culture are of sufﬁcient strength to meet the minimum acceptable baseline standard.However,in this culture,the employees do not have the requisite level of information security related knowledge.It is thus possible for the measurable artifacts to fall short of the minimum acceptable baseline.For example,either the policy dealing with a speciﬁc control might be lacking because the person(s)responsible for creating the policy lack the necessary knowledge,or the knowledge needed to implement this control in day-to-day operations might be lack-ing amongst the responsible employees.In both such cases,the resulting artifacts might be weaker than expected.This misalignment between the various levels also means that it would be difﬁcult to predict the exact relative strength of the overall security level.In this case one could probably assume the culture will be mostly predictable,hence stable,because the lack of knowledge would probably not apply equally to all controls.•C:Insecure and Unstable.The various levels contributing to the culture are not aligned.This would mean that the nett effects of the culture might be unpredictable,due to the opposing forces at play in this culture.The espoused values are very desirable,but the users lack the requisite knowledge and do not have the desired beliefs and values,resulting in a measurable artifact level that is not secure.For any speciﬁc security control,a user may,or may not,have the requisite knowledge to fulﬁll his/her role in the implementation of that speciﬁc control.That same user could also agree with the relevant espoused value,or could have beliefs that are contrary to that espoused value.It would thus be very difﬁcult to predict the nett security level of this culture.Such a culture would not be a desirable culture.•D:Secure and Unstable.The various levels contributing to the culture are not aligned.The espoused values are desirable,and the users have adequate knowledge.The high level of user knowledge in this case somewhat negates the fact that the users do not have the desired beliefs and values,resulting in an overall culture that is more secure than the minimum acceptable baseline.However,this culture should be considered not desirable,because its effects cannot always be predicted.It might be possible for the users to behave insecurely with regards to a speciﬁc security control because the speciﬁc control conﬂicts with their beliefs(Schlienger& Teufel,2003).The above examples only reﬂect a few possible scenarios.It should however be clear that the nett effect of any information security culture can be inﬂuenced,either positively,or negatively,by how”secure”the underlying levels of such a culture is.In such a model it might also be possible to deduce the relative state of one or more of the cultural levels.For example,if the organization has good espoused values,but the measurable artifacts indicate bad security,it might be inferred that the employees lack either the required knowledge or the desired attitude.6CONCLUSIONThis paper suggested that,for an effective information security culture,the requisite information se-curity knowledge amongst an organization’s users could be seen as a fourth layer to Schein’s(Schein, 1999)model for corporate culture.The various interactions between the layers of such an information security culture were then presented conceptually.The conceptual model presented showed that the nett overall effect that an information security culture would have on the organization’s information security efforts would depend on the relative desirability,or strength,of each underlying level in such a culture.Furthermore,the alignment of the strengths of the individual underlying culture levels relative to the other levels,would to a large extend determine how predictable,hence stable,the effects of such a culture would be.The ideal culture would thus be one where all four underlying levels are stronger than the minimum acceptable baseline,and are also perfectly aligned relative to each other.The example in Figure2would be such an ideal culture.The assumption made when presenting the example,namely that the desirability of the various levels can be quantiﬁed and normalized to the same scale,should by no means be taken as an assertion made by this paper.The aim of the paper was not to present such metrics and normalization processes but rather to show,at a certain level of abstraction,how this conceptual model could be used to reason about information security culture.It should,however,be possible to quantify and normalize the。

The Laplacian of a uniform hypergraph

The Laplacian of a Uniform Hypergraph∗Shenglong Hu†,Liqun Qi‡February5,2013AbstractIn this paper,we investigate the Laplacian,i.e.,the normalized Laplacian tensor of a k-uniform hypergraph.We show that the real parts of all the eigenvalues of theLaplacian are in the interval[0,2],and the real part is zero(respectively two)if andonly if the eigenvalue is zero(respectively two).All the H+-eigenvalues of the Laplacianand all the smallest H+-eigenvalues of its sub-tensors are characterized through thespectral radii of some nonnegative tensors.All the H+-eigenvalues of the Laplacianthat are less than one are completely characterized by the spectral components of thehypergraph and vice verse.The smallest H-eigenvalue,which is also an H+-eigenvalue,of the Laplacian is zero.When k is even,necessary and suﬃcient conditions for thelargest H-eigenvalue of the Laplacian being two are given.If k is odd,then its largest H-eigenvalue is always strictly less than two.The largest H+-eigenvalue of the Laplacianfor a hypergraph having at least one edge is one;and its nonnegative eigenvectors arein one to one correspondence with theﬂower hearts of the hypergraph.The secondsmallest H+-eigenvalue of the Laplacian is positive if and only if the hypergraph isconnected.The number of connected components of a hypergraph is determined bythe H+-geometric multiplicity of the zero H+-eigenvalue of the Lapalacian.Key words:Tensor,eigenvalue,hypergraph,LaplacianMSC(2010):05C65;15A181IntroductionIn this paper,we establish some basic facts on the spectrum of the normalized Laplacian tensor of a uniform hypergraph.It is an analogue of the spectrum of the normalized Lapla-cian matrix of a graph[6].This work is derived by the recently rapid developments on both ∗To appear in:Journal of Combinatorial Optimization.†Email:Tim.Hu@connect.polyu.hk.Department of Applied Mathematics,The Hong Kong Polytechnic University,Hung Hom,Kowloon,Hong Kong.‡Email:maqilq@.hk.Department of Applied Mathematics,The Hong Kong Polytechnic Uni-versity,Hung Hom,Kowloon,Hong Kong.This author’s work was supported by the Hong Kong Research Grant Council(Grant No.PolyU501909,502510,502111and501212).1the spectral hypergraph theory [7,16,19–21,23,27,29,30,33–35]and the spectral theory of tensors [4,5,11,13–15,17,19–22,24–26,28,31,32,36].The study of the Laplacian tensor for a uniform hypergraph was initiated by Hu and Qi [16].The Laplacian tensor introduced there is based on the discretization of the higher order Laplace-Beltrami operator.Following this,Li,Qi and Yu proposed another deﬁnition of the Laplacian tensor [19].Later,Xie and Chang introduced the signless Laplacian tensor for a uniform hypergraph [33,34].All of these Laplacian tensors are in the spirit of the scheme of sums of powers.In formalism,they are not as simple as their matrix counterparts which can be written as D −A or D +A with A the adjacency matrix and D the diagonal matrix of degrees of a graph.Also,this approach only works for even-order hypergraphs.Very recently,Qi [27]proposed a simple deﬁnition D −A for the Laplacian tensor and D +A for the signless Laplacian tensor.Here A =(a i 1...i k )is the adjacency tensor of a k -uniform hypergraph and D =(d i 1...i k )the diagonal tensor with its diagonal elements being the degrees of the vertices.This is a natural generalization of the deﬁnition for D −A and D +A in spectral graph theory [3].The elements of the adjacency tensor,the Laplacian tensor and the signless Laplacian tensors are rational numbers.Some results were derived in [27].More results are expected along this simple and natural approach.On the other hand,there is another approach in spectral graph theory for the Laplacian of a graph [6].Suppose that G is a graph without isolated vertices.Let the degree of vertex i be d i .The Laplacian,or the normalized Laplacian matrix,of G is deﬁned as L =I −¯A ,where I is the identity matrix,¯A =(¯a ij )is the normalized adjacency matrix,¯a ij =1√d i d j ,if vertices i and j are connected,and ¯a ij =0otherwise.This approach involves irrational numbers in general.However,it is seen that λis an eigenvalue of the Laplacian L if and only if 1−λis an eigenvalue of the normalized adjacency matrix ¯A.A comprehensive theory was developed based upon this by Chung [6].In this paper,we will investigate the normalized Laplacian tensor approach.A formal deﬁnition of the normalized Laplacian tensor and the normalized adjacency tensor will be given in Deﬁnition 2.7.In the sequel,the normalized Laplacian tensor is simply called the Laplacian as in [6],and the normalized adjacency tensor simply as the adjacency tensor.In this paper,hypergraphs refer to k -uniform hypergraphs on n vertices.For a positive integer n ,we use the convention[n ]:={1,...,n }.Let G =(V,E )be a k -uniform hypergraph with vertex set V =[n ]and edge set E ,and d i be the degree of the vertex i .If k =2,then G is a graph.For a graph,let λ0≤λ1≤···≤λn −1be the eigenvalues of L in increasing order.The following results are fundamental in spectral graph theory [6,Section 1.3].(i)λ0=0andi ∈[n −1]λi ≤n with equality holding if and only if G has no isolated vertices.(ii)0≤λi ≤2for all i ∈[n −1],and λn −1=2if and only if a connected component of Gis bipartite and nontrivial.2(iii)The spectrum of a graph is the union of the spectra of its connected components. (iv)λi=0andλi+1>0if and only if G has exactly i+1connected components.I.Ourﬁrst major work is to show that the above results can be generalized to the Laplacian L of a uniform hypergraph.Let c(n,k):=n(k−1)n−1.For a k-th order n-dimensional tensor,there are exactly c(n,k)eigenvalues(with algebraic multiplicity)[13,24]. Letσ(L)be the spectrum of L(the set of eigenvalues,which is also called the spectrum of G).Then,we have the followings.(i)(Corollary3.2)The smallest H-eigenvalue of L is zero.(Proposition3.1)m(λ)λ≤c(n,k)with equality holding if and only if G has noλ∈σ(L)isolated vertices.Here m(λ)is the algebraic multiplicity ofλfor allλ∈σ(L).(ii)(Theorem3.1)For allλ∈σ(L),0≤Re(λ)with equality holding if and only ifλ=0;and Re(λ)≤2with equality holding if and only ifλ=2.(Corollary6.2)When k is odd,we have that Re(λ)<2for allλ∈σ(L).(Theorem6.2/Corollary6.5)When k is even,necessary and suﬃcient conditions are given for2being an eigenvalue/H-eigenvalue of L.(Corollary6.6)When k is even and G is k-partite,2is an eigenvalue of L.(iii)(Theorem3.1together with Lemmas2.1and3.3)Viewed as sets,the spectrum of G is the union of the spectra of its connected components.Viewed as multisets,an eigenvalue of a connected component with algebraic multiplic-ity w contributes to G as an eigenvalue with algebraic multiplicity w(k−1)n−s.Here s is the number of vertices of the connected component.(iv)(Corollaries3.2and4.1)Let all the H+-eigenvalues of L be ordered in increasing order asµ0≤µ1≤···≤µn(G)−1.Here n(G)is the number of H+-eigenvalues of L(with H+-geometric multiplicity),see Deﬁnition4.1.Thenµn(G)−1≤1with equality holding if and only if|E|>0.µ0=0;andµi−2=0andµi−1>0if and only if log2i is a positive integer and G has exactly log2i connected components.Thus,µ1>0if and only if G is connected.On top of these properties,we also show that the spectral radius of the adjacency tensor of a hypergraph with|E|>0is equal to one(Lemma3.2).The linear subspace generated by the nonnegative H-eigenvectors of the smallest H-eigenvalue of the Laplacian has dimension exactly the number of the connected components of the hypergraph(Lemma3.4).Equalities that the eigenvalues of the Laplacian should satisfy are given in Proposition3.1.The only two H+-eigenvalues of the Laplacian of a complete hypergraph are zero and one(Corollary 4.2).We give the H+-geometric multiplicities of the H+-eigenvalues zero and one of the Laplacian respectively in Lemma4.4and Proposition4.2.We show that when k is odd and G is connected,the H-eigenvector of L corresponding to the H-eigenvalue zero is unique3(Corollary6.4).The spectrum of the adjacency tensor is invariant under multiplication by any s-th root of unity,here s is the primitive index of the adjacency tensor(Corollary6.3). In particular,the spectrum of the adjacency tensor of a k-partite hypergraph is invariant under multiplication by any k-th root of unity(Corollary6.6).II.Our second major work is that we study the smallest H+-eigenvalues of the sub-tensors of the Laplacian.We give variational characterizations for these H+-eigenvalues(Lemma 5.1),and show that an H+-eigenvalue of the Laplacian is the smallest H+-eigenvalue of some sub-tensor of the Laplacian(Theorem4.1and(8)).Bounds for these H+-eigenvalues based on the degrees of the vertices and the second smallest H+-eigenvalue of the Laplacian are given respectively in Propositions5.1and5.2.We discuss the relations between these H+-eigenvalues and the edge connectivity(Proposition5.3)and the edge expansion(Proposition 5.5)of the hypergraph.III.Our third major work is that we introduce the concept of spectral components of a hypergraph and investigate their intrinsic roles in the structure of the spectrum of the hypergraph.We simply interpret the idea of the spectral componentﬁrst.Let G=(V,E)be a k-uniform hypergraph and S⊂V be nonempty and proper.The set of edges E(S,S c):={e∈E|e∩S=∅,e∩S c=∅}is the edge cut with respect to S.Unlike the graph counterpart,the number of intersections e∩S c may vary for diﬀerent e∈E(S,S c). We say that E(S,S c)cuts S c with depth at least r≥1if|e∩S c|≥r for every e∈E(S,S c).A subset of V whose edge cut cuts its complement with depth at least two is closely related to an H+-eigenvalue of the Laplacian.These sets are spectral components(Deﬁnition2.5). With edge cuts of depth at least r,we deﬁne r-th depth edge expansion which generalizes the edge expansion for graphs(Deﬁnition5.1).Aﬂower heart of a hypergraph is also introduced (Deﬁnition2.6),which is related to the largest H+-eigenvalue of the Laplacian.We show that the spectral components characterize completely the H+-eigenvalues of the Laplacian that are less than one and vice verse,and theﬂower hearts are in one to one correspondence with the nonnegative eigenvectors of the H+-eigenvalue one(Theorem4.1). In general,the set of the H+-eigenvalues of the Laplacian is strictly contained in the set of the smallest H+-eigenvalues of its sub-tensors(Theorem4.1and Proposition4.1).We introduce H+-geometric multiplicity of an H+-eigenvalue.The second smallest H+-eigenvalue of the Laplacian is discussed,and a lower bound for it is given in Proposition5.2.Bounds are given for the r-th depth edge expansion based on the second smallest H+-eigenvalue of L for a connected hypergraph(Proposition5.4and Corollary5.5).For a connected hypergraph, necessary and suﬃcient conditions for the second smallest H+-eigenvalue of L being the largest H+-eigenvalue(i.e.,one)are given in Proposition4.3.The rest of this paper begins with some preliminaries in the next section.In Section2.1, the eigenvalues of tensors and some related concepts are reviewed.Some basic facts about the spectral theory of symmetric nonnegative tensors are presented in Section2.2.Some new observations are given.Some basic deﬁnitions on uniform hypergraphs are given in Section2.3.The spectral components and theﬂower hearts of a hypergraph are introduced.In Section3.1,some facts about the spectrum of the adjacency tensor are discussed.4Then some properties on the spectrum of the Laplacian are investigated in Section3.2. We characterize all the H+-eigenvalues of the Laplacian through the spectral components and theﬂower hearts of the hypergraph in Section4.1.In Section4.2,the H+-geometric multiplicity is introduced,and the second smallest H+-eigenvalue is explored.The smallest H+-eigenvalues of the sub-tensors of the Laplacian are discussed in Section 5.The variational characterizations of these eigenvalues are given in Section5.1.Then their connections to the edge connectivity and the edge expansion are discussed in Section5.2 and Section5.3respectively.The eigenvectors of the eigenvalues on the spectral circle of the adjacency tensor are characterized in Section6.1.It gives necessary and suﬃcient conditions under which the largest H-eigenvalue of the Laplacian is two.In Section6.2,we reformulate the above conditions in the language of linear algebra over modules and give necessary and suﬃcient conditions under which the eigenvector of an eigenvalue on the spectral circle of the adjacency tensor is unique.Someﬁnal remarks are made in the last section.2PreliminariesSome preliminaries on the eigenvalues and eigenvectors of tensors,the spectral theory of symmetric nonnegative tensors and basic concepts of uniform hypergraphs are presented in this section.2.1Eigenvalues of TensorsIn this subsection,some basic facts about eigenvalues and eigenvectors of tensors are re-viewed.For comprehensive references,see[13,24–26]and references therein.Let C(R)be theﬁeld of complex(real)numbers and C n(R n)the n-dimensional complex(real)space.The nonnegative orthant of R n is denoted by R n+,the interior of R n+is denotedby R n++.For integers k≥3and n≥2,a real tensor T=(t i1...i k)of order k and dimension nrefers to a multiway array(also called hypermatrix)with entries t i1...i k such that t i1...i k∈Rfor all i j∈[n]and j∈[k].Tensors are always referred to k-th order real tensors in this paper,and the dimensions will be clear from the content.Given a vector x∈C n,deﬁnean n-dimensional vector T x k−1with its i-th element beingi2,...,i k∈[n]t ii2...i kx i2···x ikfor alli∈[n].Let I be the identity tensor of appropriate dimension,e.g.,i i1...i k =1if and only ifi1=···=i k∈[n],and zero otherwise when the dimension is n.The following deﬁnitions are introduced by Qi[24,27].Deﬁnition2.1Let T be a k-th order n-dimensional real tensor.For someλ∈C,if polynomial system(λI−T)x k−1=0has a solution x∈C n\{0},thenλis called an eigenvalue of the tensor T and x an eigenvector of T associated withλ.If an eigenvalue5λhas an eigenvector x ∈R n ,then λis called an H-eigenvalue and x an H-eigenvector.Ifx ∈R n +(R n ++),then λis called an H +-(H++-)eigenvalue.It is easy to see that an H-eigenvalue is real.We denote by σ(T )the set of all eigenval-ues of the tensor T .It is called the spectrum of T .We denoted by ρ(T )the maximum module of the eigenvalues of T .It is called the spectral radius of T .In the sequel,unlessstated otherwise,an eigenvector x would always refer to its normalization x k √ i ∈[n ]|x i|k.This convention does not introduce any ambiguities,since the eigenvector deﬁning equations are homogeneous.Hence,when x ∈R n +,we always refer to x satisfying n i =1x k i =1.The algebraic multiplicity of an eigenvalue is deﬁned as the multiplicity of this eigenvalue as a root of the characteristic polynomial χT (λ).To give the deﬁnition of the characteristic polynomial,the determinant or the resultant theory is needed.For the determinant theory of a tensor,see [13].For the resultant theory of polynomial equations,see [8,9].Deﬁnition 2.2Let T be a k -th order n -dimensional real tensor and λbe an indeterminate variable.The determinant Det (λI −T )of λI −T ,which is a polynomial in C [λ]and denoted by χT (λ),is called the characteristic polynomial of the tensor T .It is shown that σ(T )equals the set of roots of χT (λ),see [13,Theorem 2.3].If λis a root of χT (λ)of multiplicity s ,then we call s the algebraic multiplicity of the eigenvalue λ.Let c (n,k )=n (k −1)n −1.By [13,Theorem 2.3],χT (λ)is a monic polynomial of degree c (n,k ).Deﬁnition 2.3Let T be a k -th order n -dimensional real tensor and s ∈[n ].The k -th order s -dimensional tensor U with entries u i 1...i k =t j i 1...j i k for all i 1,...,i k ∈[s ]is called the sub-tensor of T associated to the subset S :={j 1,...,j s }.We usually denoted U as T (S ).For a subset S ⊆[n ],we denoted by |S |its cardinality.For x ∈C n ,x (S )is deﬁned as an |S |-dimensional sub-vector of x with its entries being x i for i ∈S ,and sup(x ):={i ∈[n ]|x i =0}is its support .The following lemma follows from [13,Theorem 4.2].Lemma 2.1Let T be a k -th order n -dimensional real tensor such that there exists an integer s ∈[n −1]satisfying t i 1i 2...i k ≡0for every i 1∈{s +1,...,n }and all indices i 2,...,i k such that {i 2,...,i k }∩{1,...,s }=∅.Denote by U and V the sub-tensors of T associated to [s ]and {s +1,...,n },respectively.Then it holds thatσ(T )=σ(U )∪σ(V ).Moreover,if λ∈σ(U )is an eigenvalue of the tensor U with algebraic multiplicity r ,then it is an eigenvalue of the tensor T with algebraic multiplicity r (k −1)n −s ,and if λ∈σ(V )is an eigenvalue of the tensor V with algebraic multiplicity p ,then it is an eigenvalue of the tensor T with algebraic multiplicity p (k −1)s .62.2Symmetric Nonnegative TensorsThe spectral theory of nonnegative tensors is a useful tool to investigate the spectrum of a uniform hypergraph[7,23,27,33–35].A tensor is called nonnegative,if all of its entriesare nonnegative.A tensor T is called symmetric,if tτ(i1)...τ(i k)=t i1...i kfor all permutationsτon(i1,...,i k)and all i1,...,i k∈[n].In this subsection,we present some basic facts about symmetric nonnegative tensors which will be used extensively in the sequel.For comprehensive references on this topic,see[4,5,11,14,22,28,31,32]and references therein.By[23,Lemma3.1],hypergraphs are related to weakly irreducible nonnegative tensors. Essentially,weakly irreducible nonnegative tensors are introduced in[11].In this paper,we adopt the following deﬁnition[14,Deﬁnition2.2].For the deﬁnition of reducibility for a nonnegative matrix,see[12,Chapter8].Deﬁnition2.4Suppose that T is a nonnegative tensor of order k and dimension n.We call an n×n nonnegative matrix R(T)the representation of T,if the(i,j)-th element of R(T)is deﬁned to be the summation of t ii2...i kwith indices{i2,...,i k} j.We say that the tensor T is weakly reducible if its representation R(T)is a reducible matrix.If T is not weakly reducible,then it is called weakly irreducible.For convenience,a one dimensional tensor,i.e.,a scalar,is regarded to be weakly irreducible.We summarize the Perron-Frobenius theorem for nonnegative tensors which will be used in this paper in the next lemma.For comprehensive references on this theory,see[4,5,11, 14,28,31,32]and references therein.Lemma2.2Let T be a nonnegative tensor.Then we have the followings.(i)ρ(T)is an H+-eigenvalue of T.(ii)If T is weakly irreducible,thenρ(T)is the unique H++-eigenvalue of T.Proof.The conclusion(i)follows from[32,Theorem2.3].The conclusion(ii)follows from[11,Theorem4.1].2 The next lemma is useful.Lemma2.3Let B and C be two nonnegative tensors,and B≥C in the sense of compo-nentwise.If B is weakly irreducible and B=C,thenρ(B)>ρ(C).Thus,if x∈R n+is aneigenvector of B corresponding toρ(B),then x∈R n++is positive.Proof.By[31,Theorem3.6],ρ(B)≥ρ(C)and the equality holding implies that|C|=B. Since C is nonnegative and B=C,we must have the strict inequality.7The second conclusion follows immediately from the ﬁrst one and the weak irreducibility of B .For another proof,see [31,Lemma 3.5].2Note that the second conclusion of Lemma 2.3is equivalent to that ρ(S )<ρ(B )for any sub-tensor S of B other than the trivial case S =B .By [14,Theorem 5.3],without the weakly irreducible hypothesis,it is easy to construct an example such that the strict inequality in Lemma 2.3fails.For general nonnegative tensors which are weakly reducible,there is a characterization on their spectral radii based on partitions,see [14,Theorems 5.2amd 5.3].As remarked before [14,Theorem 5.4],such partitions can result in diagonal block representations for symmetric nonnegative tensors.Recently,Qi proved that for a symmetric nonnegative tensor T ,it holds that [28,Theorem 2]ρ(T )=max {T x k :=x T (T x k −1)|x ∈R n +, i ∈[n ]x k i =1}.(1)We summarize the above results in the next theorem with some new observations.Theorem 2.1Let T be a symmetric nonnegative tensor of order k and dimension n .Then,there exists a pairwise disjoint partition {S 1,...,S r }of the set [n ]such that every tensor T (S j )is weakly irreducible.Moreover,we have the followings.(i)For any x ∈C n ,T x k =j ∈[r ]T (S j )x (S j )k ,and ρ(T )=max j ∈[r ]ρ(T (S j )).(ii)λis an eigenvalue of T with eigenvector x if and only if λis an eigenvalue of T (S i )with eigenvector x (S i )k √ j ∈S i |x j|k whenever x (S i )=0.(iii)ρ(T )=max {T x k |x ∈R n +,i ∈[n ]x k i =1}.Furthermore,x ∈R n +is an eigenvector ofT corresponding to ρ(T )if and only if it is an optimal solution of the maximization problem (1).Proof.(i)By [14,Theorem 5.2],there exists a pairwise disjoint partition {S 1,...,S r }of the set [n ]such that every tensor T (S j )is weakly irreducible.Moreover,by the proof for [14,Theorem 5.2]and the fact that T is symmetric,{T (S j ),j ∈[r ]}encode all the possible nonzero entries of the tensor T .After a reordering of the index set,if necessary,we get a diagonal block representation of the tensor T .Thus,T x k = j ∈[r ]T (S j )x (S j )k follows for every x ∈C n .The spectral radii characterization is [14,Theorem 5.3].(ii)follows from the partition immediately.(iii)Suppose that x ∈R n +is an eigenvector of T corresponding to ρ(T ),then ρ(T )=x T (T x k −1).Hence,x is an optimal solution of (1).8On the other side,suppose that x is an optimal solution of (1).Then,by (i),we haveρ(T )=T x k =T (S 1)x (S 1)k +···+T (S r )x (S r )k .Whenever x (S i )=0,we must have ρ(T )( j ∈S i (x (S i ))k j )=T (S i )x (S i )k ,since ρ(T )( j ∈S i(y (S i ))k j )≥T (S i )y (S i )k for any y ∈R n +by (1).Hence,ρ(T (S i ))=ρ(T ).By Lemma 2.3,(1)and the weak irreducibility of T (S i ),we must have that x (S i )is a positive vector whenever x (S i )=0.Otherwise,ρ([T (S i )](sup(x (S i ))))=ρ(T (S i ))with sup(x (S i ))being the support of x (S i ),which is a contradiction.Thus,max {T (S i )z k |z ∈R |S i |+,i ∈S iz k i =1}has an optimal solution x (S i )in R |S i |++.By optimization theory [2],we must have that(T (S i )−ρ(T )I )x (S i )k −1=0.Then,by (ii),x is an eigenvector of T .22.3Uniform HypergraphsIn this subsection,we present some preliminary concepts of uniform hypergraphs which will be used in this paper.Please refer to [1,3,6]for comprehensive references.In this paper,unless stated otherwise,a hypergraph means an undirected simple k -uniform hypergraph G with vertex set V ,which is labeled as [n ]={1,...,n },and edge set E .By k -uniformity,we mean that for every edge e ∈E ,the cardinality |e |of e is equal to k .Throughout this paper,k ≥3and n ≥k .For a subset S ⊂[n ],we denoted by E S the set of edges {e ∈E |S ∩e =∅}.For a vertex i ∈V ,we simplify E {i }as E i .It is the set of edges containing the vertex i ,i.e.,E i :={e ∈E |i ∈e }.The cardinality |E i |of the set E i is deﬁned as the degree of the vertex i ,which is denoted by d i .Then we have that k |E |= i ∈[n ]d i .If d i =0,then we say that the vertex i is isolated .Two diﬀerent vertices i and j are connected to each other (or the pair i and j is connected),if there is a sequence of edges (e 1,...,e m )such that i ∈e 1,j ∈e m and e r ∩e r +1=∅for all r ∈[m −1].A hypergraph is called connected ,if every pair of diﬀerent vertices of G is connected.A set S ⊆V is a connected component of G ,if every two vertices of S are connected and there is no vertices in V \S that are connected to any vertex in S .For the convenience,an isolated vertex is regarded as a connected component as well.Then,it is easy to see that for every hypergraph G ,there is a partition of V as pairwise disjoint subsets V =V 1∪...∪V s such that every V i is a connected component of G .Let S ⊆V ,the hypergraph with vertex set S and edge set {e ∈E |e ⊆S }is called the sub-hypergraph of G induced by S .We will denoted it by G S .In the sequel,unless stated otherwise,all the notations introduced above are reserved for the speciﬁc meanings.Here are some convention.For a subset S ⊆[n ],S c denotes the complement of S in [n ].For a nonempty subset S ⊆[n ]and x ∈C n ,we denoted by x S the monomial i ∈S x i .Let G =(V,E )be a k -uniform hypergraph.Let S ⊂V be a nonempty proper subset.Then,the edge set is partitioned into three pairwise disjoint parts:E (S ):={e ∈E |e ⊆S },9E(S c)and E(S,S c):={e∈E|e∩S=∅,e∩S c=∅}.E(S,S c)is called the edge cut of G with respect to S.When G is a usual graph(i.e.,k=2),for every edge in an edge cut E(S,S c)whenever it is nonempty,it contains exactly one vertex from S and the other one from S c.When G is a k-uniform hypergraph with k≥3,the situation is much more complicated.We will say that an edge in E(S,S c)cuts S with depth at least r(1≤r<k)if there are at least r vertices in this edge belonging to S.If every edge in the edge cut E(S,S c)cuts S with depth at least r,then we say that E(S,S c)cuts S with depth at least r.Deﬁnition2.5Let G=(V,E)be a k-uniform hypergraph.A nonempty subset B⊆V is called a spectral component of the hypergraph G if either the edge cut E(B,B c)is empty or E(B,B c)cuts B c with depth at least two.It is easy to see that any nonempty subset B⊂V satisfying|B|≤k−2is a spectral component.Suppose that G has connected components{V1,...,V r},it is easy to see that B⊂V is a spectral component of G if and only if B∩V i,whenever nonempty,is a spectral component of G Vi.We will establish the correspondence between the H+-eigenvalues that are less than one with the spectral components of the hypergraph,see Theorem4.1.Deﬁnition2.6Let G=(V,E)be a k-uniform hypergraph.A nonempty proper subset B⊆V is called aﬂower heart if B c is a spectral component and E(B c)=∅.If B is aﬂower heart of G,then G likes aﬂower with edges in E(B,B c)as leafs.It is easy to see that any proper subset B⊂V satisfying|B|≥n−k+2is aﬂower heart. There is a similar characterization between theﬂower hearts of G and these of its connected components.Theorem4.1will show that theﬂower hearts of a hypergraph correspond to its largest H+-eigenvalue.We here give the deﬁnition of the normalized Laplacian tensor of a uniform hypergraph.Deﬁnition2.7Let G be a k-uniform hypergraph with vertex set[n]={1,...,n}and edge set E.The normalized adjacency tensor A,which is a k-th order n-dimension symmetric nonnegative tensor,is deﬁned asa i1i2...i k :=1(k−1)!j∈[k]1k√i jif{i1,i2...,i k}∈E,0otherwise.The normalized Laplacian tensor L,which is a k-th order n-dimensional symmetric tensor,is deﬁned asL:=J−A,where J is a k-th order n-dimensional diagonal tensor with the i-th diagonal element j i...i=1 whenever d i>0,and zero otherwise.10When G has no isolated points,we have that L =I −A .The spectrum of L is called the spectrum of the hypergraph G ,and they are referred interchangeably.The current deﬁnition is motivated by the formalism of the normalized Laplacian matrix of a graph investigated extensively by Chung [6].We have a similar explanation for the normalized Laplacian tensor to the Laplacian tensor (i.e.,L =P k ·(D −B )1)as that for the normalized Laplacian matrix to the Laplacian matrix [6].Here P is a diagonal matrixwith its i -th diagonal element being 1k √d iwhen d i >0and zero otherwise.We have already pointed out one of the advantages of this deﬁnition,namely,L =I −A whenever G has no isolated vertices.Such a special structure only happens for regular hypergraphs under the deﬁnition in [27].(A hypergraph is called regular if d i is a constant for all i ∈[n ].)By Deﬁnition 2.1,the eigenvalues of L are exactly a shift of the eigenvalues of −A .Thus,we can establish many results on the spectra of uniform hypergraphs through the spectral theory of nonnegative tensors without the hypothesis of regularity.We note that,by Deﬁnition 2.1,L and D −B do not share the same spectrum unless G is regular.In the sequel,the normalized Laplacian tensor and the normalized adjacency tensor are simply called the Laplacian and the adjacency tensor respectively.By Deﬁnition 2.4,the following lemma can be proved similarly to [23,Lemma 3.1].Lemma 2.4Let G be a k -uniform hypergraph with vertex set V and edge set E .G is connected if and only if A is weakly irreducible.For a hypergraph G =(V,E ),it can be partitioned into connected components V =V 1∪···∪V r for r ≥1.Reorder the indices,if necessary,L can be represented by a block diagonal structure according to V 1,...,V r .By Deﬁnition 2.1,the spectrum of L does not change when reordering the indices.Thus,in the sequel,we assume that L is in the block diagonal structure with its i -th block tensor being the sub-tensor of L associated to V i for i ∈[r ].By Deﬁnition 2.7,it is easy to see that L (V i )is the Laplacian of the sub-hypergraph G V i for all i ∈[r ].Similar convention for the adjacency tensor A is assumed.3The Spectrum of a Uniform HypergraphBasic properties of the eigenvalues of a uniform hypergraph are established in this section.3.1The Adjacency TensorIn this subsection,some basic facts of the eigenvalues of the adjacency tensor are discussed.1The matrix-tensor product is in the sense of [24,Page 1321]:L =(l i 1...i k ):=P k ·(D −A )is a k -th order n -dimensional tensor with its entries being l i 1...i k := j s ∈[n ],s ∈[k ]p i 1j 1···p i k j k (d j 1...j k −a j 1...j k ).11。

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operators. A timed action is a discrete transition, labeled with an action name, a guard, a deadline and a jump. Guards and deadlines are predicates on clocks characterizing respectively, the states at which the action is enabled and the states at which the action becomes urgent (time progress stops). We require that the deadline implies the corresponding guard which guarantees time reactivity. The jumps are functions that specify clock assignments when the action is executed. { The operators are timed extensions of untimed operators. They preserve both time reactivity and activity of components. The latter is the property meaning that if some action can be executed after waiting by some time in a component, then some action of the composed system can be executed after waiting by some (not necessarily the same) time. We propose timed extensions of choice and parallel composition operators that are associative and commutative and are related by an expansion theorem. Choice operators are parameterized by an order relation on actions that is proven to be useful, in particular to de ne parallel composition with maximal progress. { In addition to the usual laws of untimed operators, timed operators satisfy speci c laws re ecting the structure of timed actions and assumptions about their synchronization. We identify di erent synchronization modes that take into account the possibility of waiting of the components and study their properties. The paper is organized as follows. Section 2 presents the basic model, which is essentially automata with clocks, an abstraction of timed automata without the usual restrictions on guards and assignments. Section 3 and section 4 present respectively, basic results on priority choice operators and parallel composition, such as associativity, activity preservation and the expansion theorem. Section 5 presents the algebraic framework.
1 Introduction
Timed formalisms are extensions of untimed ones by adding clocks, real-valued variables that can be tested and modi ed at transitions. Clocks measure the time elapsed at states when some implicitly or explicitly given time progress conditions are satis ed. Timed automata, timed process algebras and timed Petri nets can be considered as timed formalisms. The semantics of timed formalisms can be de ned by means of transition systems that perform time steps or (timeless) transitions. Clearly, such transition systems must satisfy well-timedness requirements related with the possibility for time to progress forever. It is recognized that the compositional description of timed systems that satisfy even weak well-timedness requirements, is a non trivial problem. An inherent di culty is that usually, the semantics of operators compose separately time steps and transitions by preserving urgency: time can progress in a system by some amount if all its components respect their time progress constraints. This leads to very elegant semantics based on a nice \orthogonality principle" between time progress and discrete state changes. Parallel composition and other operators have been de ned according to this principle for timed process algebras and hybrid automata. However, composing independently time steps and transitions may easily introduce timelocks. It is questionable if zation rule for time progress is always appropriate. For instance, if two systems are in states from which they will never synchronize, it may be desirable not to further constrain time progress by the strong synchronization rule. In several papers ( SY96,BS98,BST97]) we have studied compositional description methods that are based on \ exible" composition rules that relax urgency constraints so as to preserve a weak well-timedness property that we call time reactivity. The latter means that if no discrete transition can be executed from a state then time can progress. Contrary to other stronger properties, time reactivity is very easy to satisfy by relating directly time progress conditions and enabling conditions of discrete transitions. We have proposed a simple sub-class of timed automata, called timed automata with deadlines that are time reactive and we have shown how can be de ned choice and parallel composition operators that preserve time reactivity. In this paper, we present a uni ed algebraic framework that encompasses the already presented results and provides laws for choice and parallel composition on timed systems, modulo strong bisimulation. The algebraic framework is characterized by the following.