We present an outline of a process by which operational software requirements specifications can be written for Formal Concept Analysis (FCA). The Z notation is used to specify the FCA model and the formal operations on it. We posit a novel approach whereby key features of Z and FCA can be integrated and put to work in contemporary software development, thus promoting operational specification as a useful application of conceptual structures.</p

Andrews, S.

Polovina, S.

English

Sheffield Hallam University Research Archive

Operational specification for FCA using ZANDREWS, S. and POLOVINA, S.Available from Sheffield Hallam University Research Archive (SHURA) at:http://shura.shu.ac.uk/21/This document is the author deposited version.  You are advised to consult the publisher's version if you wish to cite from it.Published versionANDREWS, S. and POLOVINA, S. (2008). Operational specification for FCA using Z. In: 16th International Conference on Conceptual Structures, Toulouse, France, July 2008.Repository use policyCopyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Users may download and/or print one copy of any article(s) in SHURA to facilitate their private study or for non-commercial research. You may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain.Sheffield Hallam University Research Archivehttp://shura.shu.ac.uk Operational Specification for FCA using ZSimon Andrews and Simon PolovinaFaculty of Arts, Computing, Engineering and SciencesSheffield Hallam University, Sheffield, UK{s.andrews, s.polovina}@shu.ac.ukAbstract. We present an outline of a process by which operationalsoftware requirements specifications can be written for Formal ConceptAnalysis (FCA). The Z notation is used to specify the FCA model andthe formal operations on it. We posit a novel approach whereby keyfeatures of Z and FCA can be integrated and put to work in contempo-rary software development, thus promoting operational specification asa useful application of conceptual structures.1 IntroductionThe Z notation is a method of formally specifying software systems [1, 2]. It isa mature method with tool support [3] and an ISO standard1. Its strength isin providing a rigorous approach to software development. Formal methods ofsoftware engineering allow system requirements to be unambiguously specified.The mathematical specifications produced can be formally verified and toolsexist to aid with proof and type checking. Being based on typed set theory andfirst order predicate logic, Z is in a position to be exploited as a method ofspecification of systems modeled using FCA.An issue with formal methods has been the amount of effort required toproduce a mathematical specification of the software system being developed.Having a ’ready made’ mathematical model provided by FCA would allow formalmethods to have a new outlet. Whilst FCA can already be used to aid in theunderstanding and implementation of software systems (see next Section), Z canprovide the method and structure by which FCA can be properly integrated intoa development life cycle.Work linking FCA and Z has been undertaken [4] that uses FCA as a meansby which Z specifications can be explored and visualised. However, it does notappear that the link has been established in the other direction, i.e. that an FCAmodel can be taken as a starting point for functional requirements specificationin Z. We are interested in specifying functional system requirements as opera-tions on the FCA data model, thus allowing the strengths of FCA and Z to becombined. Work on algorithms based on FCA has been carried out, for exampleby Carpineto and Romano [5], but here we are suggesting a formal approach to1 Information Technology - Z Formal Specification Notation - Syntax, Type Systemand Semantics, ISO/IEC 13568:2002the abstract specification of system requirements that can assist in transformingthe conceptual model into an implementation.2 FCA in Software DevelopmentFCA has been used in a number of ways for software development; for modelingthe data structure of software applications, such as ICE [6], DVDSleuth [7] andHierMail [8], and as the basis for specialised application building environmentssuch as ToscanJ [9] and Galicia [10]. However there appears to be little workconcerning the use of FCA as part of a general software engineering life cycle.Tilley et al [11] have conducted a survey of FCA support for software engi-neering activities which found that the majority of reported work was concernedwith object-oriented re-engineering of existing/legacy systems and class iden-tification tasks. They found little that related to a wider software engineeringcontext or to particular life cycle phases.One piece of work that does relate FCA directly to phases of the software lifecycle has been carried out by Hesse and Tilley [12]; They discuss how FCA ap-plies to requirements engineering and analysis. By taking a use-case approach,relating information objects to functional processes, they show that a hierar-chical program structure can be produced. They suggest that FCA can play acentral role in the software engineering process as a form of concept-based soft-ware development. The approach of this paper embodies their idea, with FCAproviding the information structure and Z providing the process specification(Figure 1).Fig. 1. FCA and Z in the Software Life Cycle3 From FCA to ZIn FCA a formal context consists of a set of objects, G , a set of attributes, M ,and a relation between G and M , I ⊆ G ×M . A formal concept is a pair (A,B) where A ⊆ G and B ⊆ M . Every object in A has every attribute in B . Forevery object in G that is not in A, there is an attribute in B that that objectdoes not have. For every attribute in M that is not in B there is an object in Athat does not have that attribute. A is called the extent of the concept and B iscalled the intent of the concept. If g ∈ A and m ∈ B then (g ,m) ∈ I , or gIm.In Z, information structures are declared based upon a typed set theory. Toapply this in FCA, G becomes such a type, namely the universal set of objects ofinterest. Similarly, M becomes the universal set of attributes that the objects ofinterest may have. The notation g : G declares an object g of type G and m : Mdeclares an attribute m of type M . Sets can be declared using the powersetnotation, P, and relations declared by placing an appropriate arrow betweenrelated types.3.1 Formal Context as a System StateUsing the Z notation, the formal context and concepts can be specified as statevariables in a state schema (Figure 2), declaring the relation I , along with aconcept function, S , which maps extents to intents. S is declared as an injection;an intent has one and only one extent, an extent has one and only one intent.Thelower section of the schema (the schema predicate) logically describes how I andS are related. A : PG declares that A is a set of objects. B is the intent of A. |ContextAndConceptsI : G ↔ MS : PG 7֌ PM∀A : PG; B : PM | (A,B) ∈ S •∀ g : G; m : M • g ∈ A ∧ m ∈ B ⇔ gIm ∧∀ g : G | g /∈ A • ∃m : M | m ∈ B ∧ ¬ gIm ∧∀m : M | m /∈ B • ∃ g : G | g ∈ A ∧ ¬ gImFig. 2. State Schema specifying a Formal Context and its Conceptscan be read as ’such that’ and • can be read as ’then’.Although a proof is not attempted here, the predicate appears, by inspection,to satisfy Wille’s conditions for deriving concepts so that A = B I and B = AI[13].3.2 Query OperationsIn Z, a query postfix, ?, is used to indicate an input to an operation and anexclamation postfix, !, is used to indicate an output from an operation. Thesymbol Ξ indicates that the operation does not change the value of the statevariables.In Z, if R is a binary relation between X and Y , then the domain of R(domR) is the set of all members of X which are related to at least one memberof Y by R. The range of R (ranR) is the set of all members of Y to which atleast one member of X is related by R.By making use of the concept function, S , and the fact that it is injective,operations to output the intent of an extent and to output the extent of anintent, are easily specified. Figure 3 specifies the latter in an operation schemacalled FindExtent.A strength of the Z notation is its notion of preconditions and postcondi-tions. Preconditions are statements that must be true for the operation to besuccessful and postconditions specify the result of the operation. In FindExtent,the precondition B? ∈ ranS states that the input set of attributes must be inthe range of S . The postcondition A! = S∼(B?) obtains the extent by invertingS and supplying it with the intent.FindIntent is not specified here as it is, essentially, a mirror of FindExtent,with the input being a set of objects and the output being the corresponding setof attributes, B ! = S (A?).FindExtentΞContextAndConceptsB? : PMA! : PGB? ∈ ranSA! = S∼(B?)Fig. 3. An operation to find the extent of an intentA query operation that outputs an object’s attributes, called FindAttributesis shown in Figure 4. The set of attributes is obtained by taking the relationalimage of I through a set containing the object of interest. Again, the operationFindObjects (for an attribute of interest) is similar and is not specified here.FindAttributesΞContextAndConceptsg? : GB ! : PMg? ∈ dom IB ! = I (| {g?} |)Fig. 4. An operation to find an object’s attributesOperation schemas to find object concepts and attribute concepts can bespecified according toWille’s definitions, γg := ({g}II , {g}I ) and γm := ({m}II , {m}I ),by piping together the corresponding object/attribute, extent/intent queries us-ing a chevron notation, >>. The output from the schema preceding the chevronsbecomes the input for the schema that follows them:FindObjectConcept =̂ FindAttributes >> FindExtent ,FindAttributeConcept =̂ FindObjects >> FindIntent .In each case, we are interested in the outputs of both of the piped schemas, sothat γg = (A!,B !?) and γm = (A!?,B !). The postfix !? indicates that somethingis first an output and then an input.3.3 Update OperationsA strength of the Z notation is its notion of before and after states, i.e. a cleardistinction is made between the value of state variables before an operation iscarried out and their values after the operation is carried out. A state variabledecorated with an apostrophe indicates that is in the after state. The symbol ∆indicates that an operation changes the state.An operation to add a new object to the context can be specified by declaringthe object and the object’s attributes as inputs. The operation schema AddObjectis shown in Figure 5. It is a precondition that the attributes currently exist inthe context.In Z, −⊲ subtracts elements from a range and ⊲ restricts a range. These areused in the postcondition involving S to take into account the possibility thatthe attributes of the new object are an existing intent. The relevant concept isupdated by adding the new object to the corresponding extent.AddObject∆ContextAndConceptsg? : GB? : PMg? /∈ dom IB? ⊆ ran II ′ = I ∪ {m : M | m ∈ B ! • g? 7→ m }S ′ = (S −⊲ {B?}) ∪ {⋃(domS ⊲ {B?}) ∪ {g?} 7→ B?}Fig. 5. An operation to add a new objectA similar operation to add a new attribute can be specified, but is not givenhere. Other useful operations that can be specified include those to remove anobject from the context, remove an attribute from the context, remove an at-tribute from an object, remove an object from an attribute and to add an existingattribute to an existing object. It also is possible that other notions in FCA, suchas the superconcept/subconcept relationship and attribute/object implications,will lend themselves to operational specification in Z.4 A User Profile ExampleConsider a user profile system where users belong to groups and groups areassociated with services. The contexts for this system areusergroupContext : USER ↔ GROUPgroupserviceContext : GROUP ↔ SERVICEThe complete state schema UserProfileSystem is not given for the sake ofbrevity. The concept functions are also omitted (in practice, where concepts areexplicitly required, it may be more pragmatic to specify an axiom to obtain themfrom the context, rather than include them explicitly in the system state).An operation is required to form a new group from all users who have accessto a particular set of services. The preconditions are that the group must notalready exist and that there must be at least one user who has access to the setof services (this also ensures that the services exist). The requirement is specifiedin Figure 6.FormGroup∆UserProfileSystemnewgroup? : GROUPservices? : PSERVICEnewgroup? /∈ dom groupserviceContextusergroupcontext o9groupservicecontext ⊲ services? = ∅∃ user : USER | services? ⊆ran({user}⊳ usergroupContext o9groupserviceContext)usergroupContext ′ = usergroupContext ∪ {user : USER | services? ⊆ran({user}⊳ usergroupContext o9groupserviceContext) • user 7→ newgroup?}groupserviceContext ′ = groupserviceContext ∪{service : SERVICE | service ∈ services? • newgroup? 7→ service}Fig. 6. An operation to form a new group in the user profile systemRelational composition is carried out using o9, here to form the relation be-tween users and services. ⊳ is domain restriction. Set comprehension is used inthe postconditions in the form {... • x 7→ y }. The mapping x 7→ y defines theform of the elements of the comprehended set.The above example shows how formal contexts, arising from FCA, can beused in the formal specification of system requirements. The operation schemaFormGroup is an unambiguous specification that can be translated into a pro-gram design.5 ConclusionThe work presented here paves a way by which an FCA model can be specifiedas a Z state schema and that operations on the model (system requirements)can then be specified as Z operation schemas. The strengths of the conceptualmodel are thus combined with the strengths of structured formal methods. TheZ notation is well understood as part of the software life cycle; it has strengthsin the way functional system requirements are structured as schemas and inits notions of pre and post conditions and before and after states. Z has anindustry standard and is supported by a variety of software engineering tools.We therefore envisage FCA systems that are specified using Z as opening up FCAto the comprehensive tools and support that are available for Z and vice versa,thereby promoting operational requirements specification as a useful applicationof conceptual structures.References1. J. M. Spivey. The Z Notation: A Reference Manual. Prentice-Hall, 1989.2. J. B. Wordsworth. Software Development with Z: A Practical Approach to FormalMethods in Software Engineering. Addison-Wesley, 1992.3. http://vl.zuser.org/#tools (Accessed 21 April 2008).4. T. Tilley. Towards an FCA based tool for visualising formal specifications. In B.Ganter and A. de Moor, editors, Using Conceptual Structures: Contributions toICCS 2003, pages 227-240. Shaker-Verlag, Germay, 2003.5. C. Carpineto and G. Romano. Concept Data Analysis: Theory and Application.Wiley, 2004.6. Y. Qian and L. Feijs. Step-wise Concept Lattice Navigation. In B. Ganter and A.de Moor, editors, Using Conceptual Structures: Contributions to ICCS 2003, pages255-267. Shaker-Verlag, Germay, 2003.7. J. Ducrou. DVDSleuth: A Case Study in Applied Formal Concept Analysis for Nav-igating Web Catalogs. In U. Priss, S. Polovina and R. Hill, editors, ConceptualStructures: Knowledge Architectures for Smart Applications, ICCS 2007 proceed-ings, pages 496-500, Springer, 2007.8. R. Cole, P. Eklund and G. Stumme. Document Retrieval for Email Search andDiscovery using Formal Concept Analysis. Applied Artificial Intelligence, Volume17, Number 3, 2003.9. P. Becker and J. Correia. The ToscanaJ Suite for Implementing Conceptual Infor-mation Systems. In B. Ganter, G. Stumme and R. Wille, editors, Formal ConceptAnalysis: Foundations and Applications, pages 324-348. Springer-Verlag, Germany,2005.10. P. Valtchev, D. Grosser, C. Roume and M. Hacene. Galicia: an open platformfor lattices. In B. Ganter and A. de Moor, editors, Using Conceptual Structures:Contributions to ICCS 2003, pages 241-254. Shaker-Verlag, Germay, 2003.11. T. Tilley, R. Cole, P. Becker and P. Eklund. A Survey of Formal Concept AnalysisSupport for Software Engineering Activities. In B. Ganter, G. Stumme and R. Wille,editors, Formal Concept Analysis: Foundations and Applications, pages 250-271.Springer-Verlag, Germany, 2005.12. W. Hesse and T. Tilley. Formal Concept Analysis Used for Software Analysis andModelling. In B. Ganter, G. Stumme and R. Wille, editors, Formal Concept Analy-sis: Foundations and Applications, pages 288-303. Springer-Verlag, Germany, 2005.13. R. Wille. Formal Concept Analysis as Mathematical Theory of Concepts and Con-cept Hierarchies. In B. Ganter, G. Stumme and R. Wille, editors, Formal Con-cept Analysis: Foundations and Applications, pages 1-33. Springer-Verlag, Germany,2005.

A Survey of Formal Concept Analysis Support for Software Engineering Activities. In

Concept Data Analysis: Theory and Application.

Document Retrieval for Email Search and Discovery using Formal Concept Analysis.

DVDSleuth: A Case Study in Applied Formal Concept Analysis for Navigating Web Catalogs. In

Formal Concept Analysis as Mathematical Theory of Concepts and Concept Hierarchies.

Formal Concept Analysis Used for Software Analysis and Modelling.

Galicia: an open platform for lattices.

Software Development with Z: A Practical Approach to Formal Methods in Software Engineering.

Step-wise Concept Lattice Navigation.

The ToscanaJ Suite for Implementing Conceptual Information Systems.

The Z Notation: A Reference Manual.

Towards an FCA based tool for visualising formal specifications.

Operational specification for FCA using Z

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.178.5124

Operational specification for FCA using Z

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