Granular Problem Solving and Software Engineering

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1 Granular Problem Solving and Software Engineering Haibin Zhu, Senior Member, IEEE Department of Computer Siene and Mathematis, Nipissing University, 100 College Drive, North Bay, Ontario, P1B 8L7, Canada Abstrat Granulation is an important omponent of Granular Computing (GrC) as a problem solving paradigm. Speifiation and regulation of granulation are neessary in helping researhers and pratitioners apply GrC into different appliations. At present, there is insuffiient investigation of this topi. This paper defines onepts and mehanisms of problem solving, investigates the fundamental priniples and proesses of granulation, and demonstrates that the development of software engineering methodologies follows the basi idea of GrC: granulation. With the guide of granulation, this paper also proposes a new method of granulation by rolifiation (the at or the result of designing roles), i.e., roles and agents. The major ontribution is establishing the foundations of granular problem solving and larifying the relationships between granular problem solving and software engineering. Keywords: problem solving, granular omputing, granule, granulation, role, agent. 1. INTRODUCTION Problem solving is an everyday ativity in our lives. People learn to solve problems in early hildhood. Every kind of job involves problem solving. Different jobs require various problem solving strategies, speialties and skills. Granular Computing (GrC) is a general omputing methodology using granules to establish an effiient omputational model for omplex appliations [1] and believed to be a new type of problem solving paradigm [6, 7, 11-17]. Granules and granulation are no doubt the major omponents of GrC. However, there are few broad, formal and aurate definitions and analyses of granules and granulation. This paper proposes formal definitions for granules and granulations based on those of problems and problem solving. These definitions establish effetive guidelines for the appliation of GrC. Software Engineering (SE) has been in existene for more than three deades. It has lead to the development of many onepts, strutures, and methodologies for problem solving. It is now neessary to analyze the relationships between SE and Granular Problem Solving (GrPS). Roles and agents are new onepts and mehanisms in SE. They an improve onventional SE strutures and methodologies. Therefore, they provide a possible new way for granulation to be investigated. The remainder of this paper is arranged as follows: Setion 2 defines problems and problem solving; Setion 3 analyzes the different forms and methods of granulation; Setion 4 demonstrates the major SE methodologies in the sense of GrPS; Setion 5 proposes a new granulation method based on roles and agents; Setion 6 reviews the related work with GrPS; and Setion 7 onludes the paper and disusses possible future work. 2. PROBLEM SOLVING Problem solving involves many fundamental onepts and mehanisms suh as indution, dedution, abstration, lassifiation, and deomposition. Abstration is the first thinking methodology used when attempting to ompose a new onept. Ditionaries [4] define a problem as a question to be onsidered, solved, or answered; any question or matter involving doubt, unertainty, or diffiulty; a question proposed for solution or disussion; or a misgiving, objetion, or omplaint. Whatever the problem, a real-world set of objets is involved. A realworld problem normally onsists of a web of interating and interrelated parts [2]. Definition 1: Objet. Everything in the world is an objet. An objet possesses identity and attributes (its struture). The mark o denotes a speifi objet and O the set of all the objets. Definition 2: Problem. A problem p is a speial questionable objet that needs to be solved. A problem an be defined as a tuple, p::=<v, a>, where, ::= expresses is defined as, v is an operation, and a is an objet alled parameter or argument, a may be null. It an be also expressed as v(a). In natural language, v is a verb and a is a noun. The mark p denotes a speifi problem and P the set of all the problems. Evidently, P O is always true. For example, do(homework) is a problem. Definition 3: Solution. A solution s is a way to solve a problem p or a onrete result objet of the way. The mark s denotes a speifi solution and S the set of all the solutions. Evidently, S O is always true. For example,

2 both learn, understand, and do and the homework done are solutions of problem do(homework). Definition 4: Problem solving. Problem solving for a problem p P an be defined as an ativity to find a map <p, s> where s S. The mark s denotes a speifi solution and S the set of all the problems. In Definition 4, a problem p may be very simple or ompliated. For a speifi problem p, its map <p, s> may or may not exist. Definition 5: Solvable problem. A problem p is solvable if there is one solution s making <p, s> exist. The properties of problems and problem solving: 1) One problem may or may not have a solution. 2) One problem may have many (>1) solutions. 3) A solution may be onrete or abstrat. 4) The set of solutions S is expanding. It means that a problem may have no solution for the time being but may have a solution in a later time. 5) There are many forms of solutions. For example, the problem landonbyacar (themoon) has no urrent solution; the homework done and learn, understand and do are solutions of the problem do(homework), where, the former is a onrete solution but the latter an abstrat one; the problem allonaplane(afriend) had no solution in 1960s but has one now; mathematial problems have solutions in the forms, suh as numbers, formula, equations, tables, and graphs; health problems have solutions in the forms, suh as pills, surgeries, and transplants; and software problems have solutions in the form of software programs. 3. GRANULATION Granules an be taken as the basi entities of knowledge [7]. Granulation of a universe involves the deomposition of the universe into parts, or the grouping of individual elements into lasses, based on available information and knowledge [18]. Granulation is a proess of deomposition from whole to parts [9, 11-17]. Granulation in fat involves both omposition and division. A granule is a group of objets that are drawn together based on some riteria [6, 7]. The definitions in Setion 2 are atually a proess of granulation. If we take granules as sets, granulation may split a set into smaller ones or onsolidate others into a single set. The major aim is to make a problem easier to solve. Definition 6: Granule. A granule p g is a problem, a partial problem, or a ombination of parts of the operation and parameter of a problem. A granule an be defined as p g ::= <v, a>, and the expression v(a) denotes both a granule and a problem, i.e., a problem is also a granule (or initial granule). A final granule is a problem that an be solved in a limited time by a person with relevant knowledge. A final granule does not require splitting in order to be solved. The mark p g denotes a speifi granule, P g the set of all granules. Evidently, P g P is always true. Definition 7: Granulation. Granulation is an iterative proess to find a set of final granules from an initial granule v(a) by division or omposition. The proess stops when every granule is final. From the viewpoint of division, a granule v(a) an be split into three forms of granules, where d is used to express is divided into : note that in the following disussions, m, n, m, n, i, and j are all integers. Division 1: v(a) d {v i (a) m>1, 0 i<m}. It expresses that the operation v is split into m partial operations and eah partial operation v i an form a granule by ating on the parameter a. Division 2: v(a) d {v(a j ) n>1, 0 j<n}. It expresses that the parameter a is split into partial parameters a j and the operation v forms a granule by ating on a partial parameter. Division 3: v(a) d {v j (a j ) m>1, n>1, 0 i<m, 0 j<n }. It expresses that the operation v is split into partial operations v i and the parameter a is split into partial parameters a j and a partial operation v i forms a granule by ating on a partial parameter a j. A ompliated problem may be divided into many simple problems. A solution for one problem may be a part of another problem. For example, (12+35) (34-23) (or (+(12, 35), -(34, 23))) is a problem. It an be divided into several simple problems, suh as 12+35, 34-23, and 47 11, where, 47 is the solution of and 11 is that of The aim of granulation is to make a problem easy to understand and handle. Problem solving does not always require splitting. Sometimes, omposition is required. From the viewpoint of omposition, three forms are as follows orresponding to the forms of divisions respetively, where, is used to express is omposed from : Composition 1: {(v 0, v 1,, v m )(a) m>1, 0 m <m} {v i (a) m>1, 0 i<m}. It expresses that a new operation (v 0, v 1,, v m ) is omposed from m +1 partial operations of the m partial operations v i and the new operation (v 0, v 1,, v m ) an form a granule by ating on the parameter a. Composition 2: {v(a 0, a 1,, a n ) n>1, 0 n <n} {v(a j ) n>1, 0 j<n}. It expresses that a new parameter (a 0, a 1,, a n ) is omposed from n +1 partial parameters of the n partial parameters a j and the operation v an form a granule by ating on the new parameter (a 0, a 1,, a n ). Composition 3: {(v 0, v 1,, v m )(a 0, a 1,, a n ) m>1, 0 m <m, n>1, 0 j<n} {v j (a j ) m>1, n>1, 0 i<m,

3 0 j<n}. It expresses that an new operation (v 0, v 1,, v m ) is omposed from m +1 partial operations of the m partial operations v i and a new parameter (a 0, a 1,, a n ) is omposed from the n +1 partial parameters of the n partial parameters a j and the new operation (v 0, v 1,, v m ) forms a granule by ating on the new parameter (a 0, a 1,, a n ). It seems easy to split or ompose granules based on the different forms of granules. In fat, there are many diffiulties to differentiate among many ambiguous and similar objets. The above disussion only onerns what to do but it is more ompliated to implement how to do. Granulation methods provide proesses to obtain omponent granules from a whole granule or vise versa. For example, build(ar) is a whole problem, it an be divided into partial granules, suh as, build(engine), build(transmission), build(body), build(tire) and build(auxiliary). This proess is Division 2. Granules buy(software), reuse(software), makebyown (software), and borrow(software) an be omposed into build(software). It is Composition 1. Division overlaps with the refinement and Composition the oarsening relationships among granules mentioned in [13, 15]. There are more onrete methods than just division and omposition. Some abstrat guidelines are indistinguishability, similarity, equivalene, proximity and funtionality proposed by Zadeh [19, 20] and Keet [10]. Other useful mehanisms in SE inlude ausality, dedution (attribute assignment, instantiation, and lassifiation), indution or abstration (value extration, struture extration, and meta-struture extration). Method 1: ausality. This is to find a reason granule from a result granule or a result one from a reason one. For example, driveindependently(acar) is a result granule, pass(roadtest) is its reason granule. There may be multiple reasons for one result or multiple results for one reason. They an be ombined by (and) or (or). Method 1 an be both a division and a omposition. Method 2: dedution. It is to form more onrete granules from an abstrat one. It is kind of division. Dedution 1: attribute assignment. This is a proess to reate granules by setting onrete values to the argument part of the granule having strutures. For example, ollet(balloon) is a granule, granules an be reated as, ollet(greanballoon), ollet(redballoon) and ollet(whiteballoon). Dedution 2: instantiation. This is a proess to reate granules by setting all the onrete values to a granule having strutures. For example, drive(ar) is a granule, granules an be reated as, drive(fordcar), drive(hondacar) and drive(bmwcar). Dedution 3: by lassifiation. This is to form more onrete granules having strutures from an abstrat one having strutures. For example, drive(vehile) is an abstrat problem, it an be divided into several more onrete granules, suh as, drive(ar), drive(truk), drive(bus), drive(boat) and even drive(plane). Method 3: abstration (indution). It is to form a more abstrat granule from onrete ones. It is kind of omposition. Abstration 1: attribute extration. This is a proess to reate a granule by extrating a ommon attribute from values. For example, ollet(greenballoon), ollet(redballoon) and ollet(whiteballoon) an be omposed into a granule ollet(olorbolloon). Abstration 2: struture extration. This is a proess to reate a granule by extrating a group of attributes from values. For example, a granule print(student) an be omposed from granules print( John Smith, 22, B.S., 92), print( Ted MDonald, 20, B.A., 87), and print( Mary Thomas, 19, B.B.A., 88), where student is a struture (name, age, major, average grade). Abstration 3: meta-struture extration. This is a proess to form a more abstrat granule having strutures from onrete granules having strutures. For example, granule produe(food) an be omposed by produe(bread), produe(sandwih), produe(soup), and produe(pisa). Abstration is the most ompliated way to solve a problem. It is disovery and reation. The granules to be reated may have never previously existed. Compared with abstration, dedution is a little easier. Abstration is also the key methodology for data mining [17, 18]. Definition 8: sub-problem. Suppose p=v(a) is a problem. Every granule split from v(a) in the forms of v j (a), v(a j ), v i (a j ), v(a 0, a 1,, a n ), (v 0, v 1,.., v m )(a), or (v 0, v 1,.., v m )(a 0, a 1,, a n ) is alled a sub-problem of p. Definition 9: Granular Problem Solving (GrPS). GrPS is problem solving with granulation, i.e., a problem is iteratively granulated into sub-problems until all the subproblems are final granules. Axiom: A problem is solvable if: all its sub-problems are solvable when they are in (and) relations; or one of its sub-problems is solvable when they are in (or) relations. Evidently, dedution and abstration introdue and relations and ausality introdues both and and or relations. This is the axiomati priniple for divide and onquer.

4 4. GRANULATION IN SOFTWARE ENGINEERING As for omputer software, a problem is an abstrat requirement desription. Problems in SE are abstrat, ambiguous, and sophistiated. They are muh more ompliated than information tables [18] and relations in a database system. To find suh a solution is alled SE. Granulation is the inherent method for SE. Granular omputing strategy has been extensively used in software system analysis, espeially in designing lasses in objetoriented analysis [5]. Granulate and onquer may improve divide and onquer, beause granulation involves not only division but also omposition. Chandrasekaran defines a design problem by a set of funtions and a tehnology [3]. The solution to a design problem onsists of a omplete speifiation of a set of omponents and their relations that together desribe an artifat that delivers the funtions and satisfies the onstraints. Inluding a tehnology is onsistent with the property that the solution set S is expanding. Definition 10: Software Problem. A software problem p is a speial problem that is to be solved with a omputer program. The mark p denotes a speifi software problem, P the set of all software problems. Evidently, P P is always true. For example, build(an interative system) is a software problem. Definition 11: Software System. A software system s is a runnable program on a omputer. The mark s denotes a speifi software system and S the set of all software systems. A software system is implemented based on final granules. Evidently, S S is always true. For example, a runnable interative system is a solution of the problem build (an interative system). Based on Definition 11, SE is a proess to design a software system to solve a software problem. It is one kind of problem solving and a proess of granulation. From the above definitions and properties of problem solving, it is understandable why some SE projets are suessful yet others fail. Following the idea of divide (or granulate) and onquer and the granulation proess in Setion 3, the methodologies proposed and applied in SE are atually ombinations of granulation methods. 4.1 Strutural Software Engineering Strutural Software Engineering (SSE) was the earliest method to divide and onquer sine software risis was reported in the software industry. It uses proedures (or funtions, subprograms, and subroutine) to divide a ompliated problem solving into sub-problems that an be handled by a single person. Definition 12: Proedure. It an be expressed as v (a ), v is an operation and a is an objet, i.e., a O. It is atually in the same form of problem. The mark f denotes a speifi proedure and F the set of all the proedures. In SSE, a software problem is a proedure. SSE is in fat granulation from the initial proedure v (a ) that is alled the main program in many programming languages. The software system reated in SSE is implemented in a speial programming language based on the final granules (final proedures). Broadly speaking, SSE applies all the six granulation methods. However, it only onentrates on the operation part v of a problem and the parameter part a is seondary to the operation part, i.e., it mainly uses funtionalities to form granules. SSE is atually to divide the operation v of a problem at first and the parameter a of the problem is then divided to follow the division of the operation. It supports the divisions 1-3 and ompositions 1 and 2. As for the methods of granulation, SSE mainly applies ausality in proedure design, dedution 1 in proedure all, and abstrations 1 and 2 in onstruting data strutures [18]. Granulation by ausality is substantially used in Logi Programming (LP) [23]. In fat, Funtional Programming (FP) [23] uses the same granulation methods as SSE. The only differene is that FP uses lists as its major omputation omponents but SSE uses proedures. 4.2 Objet-Oriented Software Engineering Objet-Oriented Software Engineering (OOSE) is onsidered a better omputing paradigm than SSE aording to fundamental priniples and onepts suh as lasses, instanes, inheritane, polymorphism, and overloading. Classes are the kernel onept of OOSE [23]. Definition 13: lass. A lass is defined as ::= <n, D, F > where, n is the identifiation of the lass; D is attributes for storing the state of an objet; and F is a set of the funtion definitions or implementations (F F). The mark denotes a speifi lass, and C the set of all lasses. After lasses are defined, objets an be redefined based on lasses. To differentiate them from objets in a broad meaning, they are alled instanes. Definition 14: instane. An instane o is defined based on a lass, i.e., o ::= < n,, d > where, n is the identifiation of the objet; is the objet s lass identified by the lass identifiation or name; and d is a set of values respnding to the attributes defined in its lass. The mark o denotes a speifi instane and O the set of all the instanes. OOSE aims to divide the parameter a of a problem at first. The operation v of the problem is then divided following the division of the parameter. It is atually an iterative proess of onstruting a granule using both divisions 1-3 and ompositions 1-3. To granulate with

5 lasses and objets in OOSE, the properties funtionality, similarity, and proximity are applied. As for the methods of granulation, OOSE applies all the dedution and abstration methods. Classes in OOSE are atually a way of granulation by lassifiation, i.e., dedution 3. Dedution 3 in OOSE is to add more internal properties (onnotation) from the original granule to derease the extension (smaller granules). Creating instanes from a lass is granulation by instantiation, i.e., dedution 2. Designing a lass from a group of objets and designing a lass from a group of lasses are granulation by abstration. Abstration is the most diffiult task for software engineers. Fortunately, many objet-oriented programming languages establish extensive libraries of lasses based on their designers efforts on abstration. Classes are built by olleting similar (similarities) attributes, same servies or operations (funtionalities) from a group of objets. At a first glane, a lass supports omposition 3. However, urrent OOSE does not have a definite method to form a lass by ompositions. It is totally up to the experiene, knowledge and wisdom of software engineers. 5. NEW TYPES OF GRANULATION Role-Based Collaboration (RBC) is a omputational thinking methodology or a problem solving paradigm that mainly uses roles as underlying mehanisms to failitate abstration, lassifiation, separation of onern, dynamis, interations and ollaboration [22, 24, 25]. Based on the priniples of RBC [22, 24], Role-Based Software Engineering (RBSE) is a newly proposed methodology to improve software development [26]. It introdues new granulation mehanisms not applied in OOSE in the sense of problem solving: roles, agents, environments and groups. By RBSE, an iterative proess of granulation in division, ompositions, ausality, dedution, and abstration are established. Definition 15: Role. A role is defined as r ::= <n, I, A, A p, A o, R x, O a >, where, n is the identifiation of the role; I ::= < M in, M out > denotes a set of messages, where M in expresses the inoming messages to the relevant agents, and M out expresses a set of outgoing messages or message templates to roles, i.e., M in, M out M; A a set of agents who are urrently playing this role; A p a set of agents who are qualified to play this role; A o a set of agents who used to play this role; R x a set of roles interrelated with it; and O a a set of objets to be aessed by the agents playing this role. The mark r denotes a speifi role and R the set of all the roles. Roles are onsidered as an abstration and deomposition mehanism related to agents. When an agent plays a role, it aepts messages, provides servies and sends out requests related to its role. A role onstitutes a part of an agent s behavior that is obtained by onsidering the interations of that role and hiding all other interations. Definition 16: Agent. An agent is defined as a g ::= < n, a, s, d, r, R p, R o, G a >, where, n is the identifiation of the agent; a a speial lass that desribes the ommon properties of users; s the qualifiations of the agent; d the redit of the agent; r a role that the agent is urrently playing; R p a set of roles that the agent is qualified to play (r a.r p ); R o a set of roles that the agent played before; and G a a set of groups that the agent belongs to [24]. The mark a g denotes a speifi agent and A g the set of all the agents. In RBSE, roles are split into or omposed from (v 0, v 1,..., v m ) and agents are split into or omposed from (a 0, a 1,, a n ) to support omposition 3 of granulation, i.e., an agent playing a role or a role being played by an agent forms a granule. Roles are used to show ommon interfaes among agents in interations. Agents are speial objets being able to play roles. They are different from objets in that they play roles whereas objets do not. Roles are good mehanisms to assist granulation. Compared with other mehanisms, roles and agents an be applied to more ompliated real-world problems suh as system analysis and system design, beause these divide and onquer tasks are muh more ompliated than those in databases. A new method of granulation an be introdued, where rolifiation means the at or the result of designing roles. Method 4: rolifiation. Granulation by rolifiation is to use roles or the ommon interfaes of requests and servies (similarities and funtionalities) and agents or the relevant speial objets to reate new granules. In this method, a granule an be expressed as r(a g ), where r omes from (v 0, v 1,..., v m ) and a g from (a 0, a 1,, a n ). Rolifiation is a granulation method on top of lasses. Roles an be further refined into more operable granulations: urrent roles: a granule an be formed by playing the same urrent role r by a group of agents; potential roles: a granule an be formed by the same potential roles R p attahed by a group of agents; and past roles: a granule an be formed by the same past roles R o attahed by a group of agents. Based on these roles, more relationships among granules an be established. A ase study of suh method an be found in [21]. 6. RELATED WORK Chandrasekaran [3] proposes a task (i.e. software problem) struture for design by analyzing a general lass of methods. He shows that there is no one ideal method for designs and good design problem solving is a result of reursively seleting methods based on a number of riteria, inluding knowledge availability. Yao and Zhong

6 [18] propose several methods to form granules: by equality of attribute values; by equivalene of attribute values; and by similarity of attribute values. Their methods an be applied in data analysis and data mining. Liu [8] presents his formal methods in Artifiial Intelligene (AI) reasoning with granules. He defines the elementary granule of deision rules as a pair (φ, d(φ)) in whih φ is the syntax and d(φ) is the semantis. Han and Dong [5] disuss the arhiteture of granular omputing models, strategies, and appliations. They investigate granular omputing from the perspetives of SE, inluding requirement speifiation and analysis, system analysis and design, algorithm design, strutured programming, software testing, and system deployment. Yao [15] examines the basi priniples and issues of granular omputing, analyzes the tasks of granulation and omputing with granules, disusses the onstrution, interpretation, and representation of granules, as well as priniples and operations of omputing and reasoning with granules. Yao [13] disusses granular omputing theory from the perspetive of information granulation and granular relationships. These relationships are good supplements in granular problem solving. 7. CONCLUSIONS The development of SE methodologies may be based on different forms and methods of granulation. Granulation is one of the foundations of problem solving and SE. Rolifiation is a new type of granulation in GrPS. To improve SE methodologies should be to introdue indistinguishability and proximity, beause urrent SE methodologies have not yet introdued these properties. Roles are also a potential granulation mehanism to be applied in data mining. More ase studies and evidenes are required to ollet to show more merits of roles. ACKNOWLEDGMENTS This researh is in part supported by National Sienes and Engineering Researh Counil of Canada (NSERC: ). Thanks also go to Mike Brewes of Nipissing University for proofreading this artile. REFERENCES [1] Bargiela, A. and Pedryz, W., The roots of granular omputing, Pro. of the IEEE Int l Conf. on Granular Computing, May 2006, pp [2] Capra, F. 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