2 Design. Design is the process of developing plans, or schemes of actions which are patterns for making a product. In our approach, the designer task

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1 INFORMATION SCIENCES xx, 1{xx (1998) 1 Rough Mereological Foundations for Design, Analysis, Synthesis, and Control in Distributed Systems ANDRZEJ SKOWRON Institute of Mathematics Warsaw University Banacha 2, Warsaw, Poland skowron@mimuw.edu and LECH POLKOWSKI Institute of Mathematics Warsaw University of Technology Pl. Politechniki 1, Warsaw, Poland polk@mimuw.edu.pl ABSTRACT We propose a unied formal treatment of problems of design, analysis, synthesis and control in distributed systems of intelligent agents. Our approach is rooted in rough set theory and we propose rough mereology as a foundational basis for our approach. Key words: approximate reasoning, rough mereology, distributed systems of intelligent agents, cooperative design, analysis, synthesis, and control. 1. INTRODUCTION We propose a new methodology called rough mereology for reasoning about complex objects in distributed systems. The tasks covered by reasoning systems include design, synthesis, analysis as well as control of complex objects [1], [2], [4], [5], [6], [7], [8], [10], [11], [12], [24], [26] - [28]. Our formal models of design, analysis, synthesis and control are constructed over xed universes of available intelligent and cooperating design, respectively, synthesis, agents. A complex object, perceived by us as a solution to a requirement issued to the agents in the universe, is represented by means of a scheme over the universe of agents over which it can be constructed. We propose rough mereology as a principle on which cooperative agent systems acting under uncertainty can be constructed and their properties analyzed. c Elsevier Science Inc., Avenue of the Americas, New York, NY /94/$6.00

2 2 Design. Design is the process of developing plans, or schemes of actions which are patterns for making a product. In our approach, the designer task is to organize a set of agents in the universe into a hierarchical structure. In this structure, any agent is provided with a requirement in the language of the designer specifying features of the class of complex objects expected to be issued by this agent. The hierarchical structure is composed of elementary teams of agents. The designer makes use of the knowledge of operations which yield complex objects from simpler parts to establish potential links among the agents which lead to the formation of elementary teams of agents. The designed hierarchical structure is a pattern to follow in the process of synthesis. For any node in the designed structure there should exist an agent or a scheme of agents in the synthesis scheme which is able to issue a complex object satisfying the requirement attached to the node in a satisfactory degree. The requirement attached to the node may not be fully understandable by the corresponding agents in the synthesis scheme due to possibly dierent languages used by the designer and the agents; therefore, requirements of the designer are perceived by agents as vague statements and they are rendered in the language of agents as approximate specications. Approximate specications are constructed as pairs consisting of a specication in the language of an agent and an uncertainty coecient which expresses the degree in which the specication is satised. The objects which satisfy approximate specications at synthesis agents are assumed to be accepted by the designer as satisfying its requirements in the corresponding nodes in the satisfactory degree. This agreement is secured in the process of negotiation between the designer and the synthesis agents leading to the formation of the synthesis scheme. Synthesis. The process of synthesis over the given universe of agents of a complex object which satises a given requirement in the satisfactory degree consists in constructing a synthesis scheme of agents. In a synthesis scheme agents corresponding to the nodes of the chosen design scheme are equipped with operations, approximate formulas and standards. Objects satisfying approximate formulas project onto designer objects which satisfy the designer requirement at the corresponding node of the design scheme. Agents in a synthesis scheme negotiate connectives for propagating uncertainty coecients; a proper choice of connectives leads to a scheme which is able to synthesize objects which satisfy the given designer requirement in the satisfactory degree. Analysis. Analysis of a complex object x is rst concerned with the taxonomy problem of the existence of a decomposition scheme of this object into simpler parts. In our approach, the object x is matched against the designer knowledge and classied possibly into the universe of a design

3 agent which is able to decompose it by means of its mereology. The design process results in a design scheme for x. The next problem of analysis of x is the constructibility problem of the existence of a synthesis scheme for x which would correspond to the proposed design scheme and satisfy the correctness criterium. Finally, one can consider the problem of sensitivity of the scheme to the parameter change and problems concerning complexity of the synthesis scheme. Control. In many applications we are in need of controlling the behavior of complex highly non-linear objects. This approach requires a reformulation of the traditional language of control theory as the traditional tools are not sucient. We model controlled complex objects as distributed systems of controllable local teams of agents. The main task of control is to regulate the behavior of a synthesis scheme of agents in order to preserve as an invariant of the scheme a given approximate specication attached to the root (output) agent; in the process of regulation some parameters of the synthesis scheme may change, in particular, approximate specications attached to the remaining agents may undergo changes. The structure of the controlling device (the controller) is described. The controller is designed on the basis of local rules expressing the relationship among changes in uncertainty coecients of agents linked into an elementary synthesis scheme; these local rules are composed in order to obtain requirements for changes in control parameters. These local rules are extracted from data tables of agents. The results presented here reect the main lines of our research: studies of various methods of reasoning under uncertainty, quantitative frameworks for adaptive decision making in distributed systems, logics for supporting approximate reasoning about complex objects in distributed systems. This paper is a consequence of our research over the last two years in this area [13], [19] - [29]. Let us mention that the need for principles on which distributed systems of agents supporting uncertain reasoning can be constructed is stressed as the main problem of distributed articial intelligence [1]. The order of things in our paper is as follows. Section 2 brings our analysis of design -synthesis interface and in Section 3 we discuss the design process. In Section 4 we discuss the aspects of synthesis of complex objects. The control and analysis aspects are discussed in Section 5. We assume that the reader is familiar with ideas and results of rough set theory including tolerance information systems [16], [17], [24], [25] and with rough mereology and its applications [13], [19] - [25]. Our notation follows [19], [25] which we include in Appendix. 3

4 4 2. DESIGN-SYNTHESIS COMMUNICATION In this section we are concerned with communication between the design and synthesis spaces. The basic result of this section is the formalization of vagueness caused by the fact that design and synthesis spaces use distinct languages and in consequence categories of objects used by designer can be given various interpretations by synthesis agents. Design agents. Requirements. The designer operates on the set Des Ag of design agents; any design agent dag is equipped with an information system A dag =(U dag ; A dag ) and a rough inclusion dag. The table A dag describes objects (possibly complex) from a universe U dag in the language of attributes in A dag. The variable b dag runs over objects in U dag : The rough inclusion dag [21] is a binary relation on objects which allows dag to express some objects as parts in a degree of some other objects and to construct a similarity (tolerance) relation between objects. A valuation v X where X is a set of design agents is a function which assigns to any b dag for dag 2 X an element v X (b dag ) 2 U dag : A (designer) requirement (ag) at ag is a formula in C(A dag ; V ) [26], [28]; the symbol will denote a requirement at some design agent. The symbol x j= d d will denote that x satises : Synthesis agents. Any synthesis agent ag is assigned a label lab(ag) = fu(ag); (ag); (ag; (ag)), D(ag); St(ag); L(ag); o (ag); F (ag)g where U(ag) is the universe of objects at ag; (ag) is the design agent associated to ag; (ag; (ag)) : U(ag) 7?! U (ag) ; D(ag) = fu(ag); A(ag); d(ag)g is the decision system [16], [25] of ag; A(ag) is the set of conditional attributes of ag. The value d(ag)(x) is a designer requirement satised by (ag; (ag))(x); St(ag) U(ag) is the set of standard objects (standards) at ag; L(ag) is a set of unary predicates at ag (specifying properties of objects in U(ag)). Predicates of L(ag) are formulas from C(A(ag); V ) [25]; o (ag) U(ag) U(ag) [0; 1] is a pre-rough inclusion at ag [21]. It is a function on pairs of objects which can be extended [21] to collections of objects; F (ag) is a set of mereological connectives at ag [21]-[24].

5 The meaning of standards is that they are some distinguished objects in universes of agents whose compositions are standards also; hence standards form skeletons for which the exact relations hold. Our assumption is that when we can assemble a complex standard from a set of simpler standards then replacing these simpler standards with objects suciently close to them leads by the same synthesis operation to a complex object satisfactorily close to the complex standard. The closeness of objects is measured by uncertainty coecients and mereological connectives are used to propagate uncertainty coecients from the children agents to the parent agent. Mereology DS : Synthesis agents classify their objects by means of pre - rough inclusions; any pre - rough inclusion o (ag) can be extended [21] to a rough inclusion (ag) on the set 2 U(ag) of subsets of U(ag). The D S?mereology DS (the design - synthesis mereology) is a family f(ag) : ag 2 Agg where (ag) is a xed extension [21] of o (ag) for any ag 2 Ag. Mereology DS denes design objects (categories of real objects): a design object x is a class((ag))? [14], [19] where? U(ag): Thus, design objects are classes of synthesis objects. On these classes the mereologies of design agents act decomposing them into simpler classes. One possibility for constructing such mereology for any agent ag can be based on decision tables: we use conditional attributes to describe objects and the decision attribute describes classes of objects. This idea is used in this work. The communication between synthesis spaces and design spaces is provided by mappings (ag; (ag)) : for x 2 U(ag); the value (ag; (ag))(x) 2 U((ag)) is a design object. Let us observe that while (ag; (ag))(x) is unique, the object x may belong to more than one design object. This causes the ambiguity in communication among design agents and synthesis agents: while a synthesis agent ag classies x as cat(x) = (ag; (ag))(x), the design agent (ag) can regard x as an element of a category cat 0 (x) distinct from cat(x). However, categories cat(x); cat 0 (x) containing x should be regarded as similar. We express this similarity on the higher level of requirements by means of a tolerance relation Des sat on descriptions of classes of objects. Satisfactory satisability of requirements. The requirements of the designer specify classes of ideal objects but the ultimate purpose of design is a real object whose category would satisfy. It can happen that 0 an object whose category satises a requirement 6= is accepted as satisfying in a degree satisfactory to the designer: We denote by Des req the set of designer requirements; the vagueness of designer requirements will be formalized by means of a tolerance relation Des sat on the set Des req; for ; 2 Des req; Des sat will read "any object satisfying (resp. ) satises (resp. 0 ) in degree satisfactory to the designer ". We denote by the symbol [ ] the tolerance class Des sat( ): We denote by the 5

6 6 symbol _ the disjunction of formulas in [ ] i.e. _ is _f 0 : 0 2 [ ]g: Clearly, if x j= _ d then x satises in the satisfactory degree. One can regard _ as a formalization of the partial knowledge of the designer about the properties of objects: not knowing as a rule the exact description of objects, the designer works with a class of descriptions plausible from its point of view. Mereological compatibility of requirements. We assume that dag is compatible with Des sat i.e. if x = class( dag ) fx 1 ; x 2 ; :::; x k g, x i j= d _ i, x j= d, y i j= d i, y = class( dag ) fy 1 ; y 2 ; ::; y k g then y j= _ d : The compatibility condition means that the designer schemes are designed as insensitive to local communication ambiguities. Approximate logic of synthesis. The symbol b ag will denote the variable which runs over objects in U ag : A valuation v X where X is a set of synthesis agents is a function which assigns to any b ag for ag 2 X an element v X (b ag ) 2 U ag : The symbol vag x denotes v fagg with v fagg (b ag ) = x. We recall [19], [21] that a function like E (ag) (x; y) = min((ag)(x; y); (ag)(y; x)) can be regarded as a measure of similarity between objects. We now dene syntax of a logic of approximate formulas L [13]. The atomic formulas of L are of the form < st(ag); (ag); "(ag) > where st(ag) 2 St(ag); (ag) 2 L(ag); "(ag) 2 [0; 1] for ag 2 Ag: The formulas of L are built from atomic formulas by means of classical propositional connectives :; _; ^; );, and of a modal unary propositional connective 3 (cf. Sect.4 for semantics of 3). The semantics of L is dened as follows. For ag = Root(C), v 2 V C, we say that v satises a formula =< st(ag); (ag); "(ag) >, symbolically v j=, in case E (ag) (v(b ag ); st(ag)) " and v(b ag ) 2 (ag) D(ag) (see Appendix). We let v j= ^ in case v j= and v j= ; v j= : in case non(v j= ): For a formula < st(ag); (ag); "(ag) > of L, we write x j=< st(ag); (ag); "(ag) > for vag x j= < st(ag); (ag), "(ag) >. Decision rules of synthesis agents. From the triple (D(ag); L(ag); (ag)); the agent ag generates by the standard techniques [24], [25] the decision rules of the form The meaning of this rule is: < st(ag); (ag); "(ag) >=) _. (< st(ag); (ag); "(ag) >=) _ ) is true

7 7 i ((ag; (ag))(st(ag)) j= d ) ^ [ x j= < st(ag); (ag); "(ag) > ) (ag; (ag))(x) j= d _ ]. The interface between the designer agent (ag) and the synthesis agent ag is constructed by choosing a standard object st(ag) 2 U ag, an uncertainty coecient "(ag) and a formula (ag) satisfying the following properties: (i) the projection of st(ag) on the designer level satises the speci- cation if the standard object st(ag) satises the formula (ag) ; (ii) if x 2 U ag and x is close to st(ag) in degree at least "(ag) and the standard st(ag) is satisfying (ag) then the projection of x on the designer level satises _. 3. DESIGN Local decomposition schemes. By means of the rough inclusion dag the agent dag induces on the universe U dag the mereological relation part dag in the sense of Lesniewski [14]. The relation part dag establishes a local decomposition scheme of some complex objects in the universe U dag into simpler objects i.e. the relation x class( dag ) fx 1 ; x 2 ; ::; x k g means that x is designed (built) from parts x 1 ; x 2 ; ::; x k : In this way the designer establishes a hierarchy fpart dag g of local decomposition schemes on the set of possible complex objects. Local decomposition schemes can be composed in the sense that whenever the relation part dag expresses an object x as built from parts x 1 ; x 2 ; ::; x k and the relation part dag 0 expresses, say, x 1 as built from parts x 11 ; x 12 ; ::; x 1m then the composition part dag part dag 0 expresses x as built from parts x 11 ; x 12 ; :::; x 1m ; x 2 ; ::; x k : The designer language Link d : The designer task is now to establish for a given complex object x, specied by a requirement, a scheme of design agents (for simplicity we assume this scheme to be a tree) such that the object x can be decomposed over the scheme by means of a composition of local decomposition schemes into primitive objects which belong to the universes of leaf agents of the scheme. We dene a language Link d Des Ag + where Des Ag + is the set of all nite nonempty strings over Des Ag. For a string dag = dag 1 dag 2 :::dag k dag, where dag is the root agent and dag 1 ; dag 2 ; ::; dag k are leaf agents, we have dag = dag 1 dag 2 :::dag k dag 2 Link d i there exist x; x 1 ; x 2 ; ::; x k ; x such that x class(part dag ) fx 1 ; x 2 ; ::; x k g where x i 2 U dagi for i k, x 2 U dag. We let set(dag) = fdag 1 ; dag 2 :::; dag k ; dagg. For L Link d, we dene Des Ag(L) = [fset(dag) : dag 2 Lg and we denote by a relation on

8 8 Des Ag(L) dened by: dag dag 0 if and only if there exists dag 2 L such that dag; dag 0 2 set(dag) and dag 0 is the root of dag. A set L Link d is a construction support in case (Des Ag(L); ) is a tree. The symbol L will denote a construction support. Constructibility mapping and design operations. For dag = dag 1 dag 2 :::dag k dag 2 L, we dene the dag-constructibility relation (dag) U(dag 1 ) U(dag 2 ) ::: U(dag k ) U(dag) by letting (x 1 ; x 2 ; :::; x k ; x) 2 (dag) i x class(part dag )fx 1 ; x 2 ; ::; x k g. The constructibility mapping con(dag); associated with the relation (dag); is dened by con(dag)(x 1 ; x 2 ; :::; x k ) = x i (x 1 ; x 2 ; :::; x k ; x) 2 (dag): The existence of the mapping con(dag) follows from the uniqueness of classes in mereology generated by the relation part dag : The constructibility mapping con(dag) is in general a many - to - one mapping. We select from con(dag) one - to - one mappings called design operations de- ned as follows: a design operation o(dag) associated with dag is a one - to - one partial mapping such that if o(dag)(x 1 ; x 2 ; :::; x k ) = x then con(dag)(x 1 ; x 2 ; :::; x k ) = x and range o(dag) = range con(dag). We denote by the symbol O(dag) the collection of design operations associated with dag. We will write o(dag) instead of o(dag), where dag = root(dag). We observe that the relation (dag); the mapping con(dag) and design operations o(dag) are compatible with requirements with respect to the tolerance Des sat viz. the conditions (x 1 ; x 2 ; :::; x k ; x) 2 (dag); x i j= d i; x j= d ; y i j= d _ i and (y 1 ; y 2 ; :::; y k ; y) 2 (dag) imply y j= _ d and similar conditions hold for con(dag) and o(dag). Design schemes. Now, given a requirement satised by a standard object along with its Des s at - tolerance class _, the designer initiates the communication and negotiation process among design agents. This process results eventually in a design scheme which is able to design an object satisfying in the satisfactory degree. We describe this initializing process in the following steps. Step 1. The designer selects a design agent dag o such that there exists an object x o 2 U dago which satises the requirement _ and there exists dag o = dag o;1 dag o;2 :::dag o;k dag o 2 Link d such that the condition

9 9 con(dag o )(x 1 ; x 2 ; :::; x k ) = x holds for some vector argument (x 1 ; x 2 ; :::; x k ): Step 2. Agents dag o ; dag o;1 ; dag o;2 ; ::; dag o;k negotiate the choice of _ requirements o;1 ; o;2 ; :.:; and a vector argument o;k (x 1; x 2 ; :::; x k ) such _ that any x i satises the requirement o;i ; this choice determines the design operation o(dag o ) with the property that o(dag o )(x 1 ; x 2 ; ::; x k ) = x: The negotiated condition is the following: if (y 1 ; y 2 ; ::; y k ) 2 domain o(dag o ) _ and any y i satises the requirement o;i then o(dag o )(y 1 ; y 2 ; ::; y k ) satises the requirement _ : Factors involved in negotiations and inuencing them may be e.g. the complexity of o(dag), the cost of assembling via o(dag), the accessibility of x 1 ; ::; x k etc. Next, the designer repeats steps 1 and 2 with the agents dag o;1, dag o;2,..., dag o;k and the respective requirements o;1,..., o;k. This leads to choices of decompositions for objects x 1, x 2,..., x k : dag i = dag i;1 dag i;2 :::dag i;ki dag o;i _ 2 Link, requirements i;1 ; i;2 ; :::; i;k i, objects x i;1 ; x i;2 ; :::; x i;ki and the design operation o(dag i ) are chosen for any x i which satisfy the counterparts of conditions in steps 1 and 2. The negotiation process continues with new requirements. The process stops when all branches terminate with strings in Link d whose all leaf agents can satisfy the chosen requirements with inventory objects. We observe that the result of negotiations for any non-inventory agent dag in L can be described by means of the pair lab(dag) = ( _ (dag); o(dag)); the singleton lab(dag) = ( _ (dag)) summarizes the negotiation outcome for any inventory agent dag in L. We will call the tuple lab(dag) the design label of the agent dag. The construction support L along with the set flab(dag)g of design labels of its agents will be called the design scheme and denoted by D s = (L; flab(dag) : dag 2 Lg). The agent dag o will be called the root agent of D s; dag(1),..., dag(m) are leaf agents of D s. We write D s(dag o ; dag 1 ; ::; dag m ) when dag o, dag 1,..., dag m are all strings used to construct D s. D s(dag i ) is called an elementary design scheme. The operation term of D s, denoted by the symbol T (D s) will be de- ned by induction on the number of strings from Link in L: (i) if L = fdag o g then we let T (D s) = o(dag o )(b dago; 1 ; b dago;2 ; :::; b dago;k ); (ii) if L = fdag o ; dag 1 ; :::; dag k g, an agent dag is a leaf agent of D s, dag k+1 = dag 1 dag 2 :::dag n dag, where dag i 62 set(l) for i = 1; :::; n;

10 10 T (D s) = o(dag o )(:::(:::(:::; b dag ; :::):::); D s 0 results from D s by attaching ag k+1 to D s at dag then T (D s 0 ) = o(dag o )(:::(:::(:::; o(dag)(b dag1 ; b dag2 ; :::; b dagn ); :::):::): The term T (D s) represents the composition of operations along the construction support L: This composition is a global operation which assigns to argument vectors of objects at leaf agents of D s the value which is an object in the universe of the root agent of D s: The following proposition resumes the upshot of the negotiation process. Proposition 3.1. Let v X be any valuation on the set X of leaf agents dag(1); ::; dag(m) of the design scheme D s where the object v(b dag(i) ) satises _ (dag(i)) and lab(dag(i)) = ( _ (dag(i))) for i = 1; 2; :::; m. Then the uniquely dened object T (D s)(v(b dag(1) ); v(b dag(2) ); :::; v(b dag(k) )) 2 U dago satises _ (dag o ) where lab(dag o ) = ( _ (dag o ); o(dag o )). 2 The projection Dg =( _ (dag(1)); _ (dag(2)); ::; _ (dag(n)); _ (dag o )) of the design scheme D s onto the requirement space will be called a designer goal; the designer goal Dg expresses the existence of a design scheme which can design an object satisfying _ (dag o ) from objects satisfying _ (dag(i)) for i = 1; 2; ::; n: We now proceed from the design stage to the synthesis stage. In the synthesis stage the synthesis agents are organized into a synthesis scheme modelled on a given design scheme. 4. SYNTHESIS (OF APPROXIMATE REASONING SCHEME) We begin with a set Ag of synthesis agents (called simply agents in what follows) and a set I of primitive objects (the inventory). Elements of lab(ag) constitute the basic knowledge of a synthesis agent ag; other elements are worked out in negotiations among synthesis agents and in their interactions with the design agents. We now describe this process. The synthesis language Link. We recall that the designer creates a language Link d Des Ag + ; the language Link d is a pattern according to which a language Link Ag + is dened in the process of communication among synthesis agents. To this end, the agents ag 1 ; ag 2 ; :::; ag k, ag o form the string ag =ag 1 ag 2 :::ag k ag o 2 Link if and only if there exist design

11 agents dag 1 ; dag 2,..., dag k, dag o such that dag = dag 1 dag 2 :::dag k dag o 2 Link d, and dag i = (ag i ) for each i, and there exist objects x 1 2 U(ag 1 ),..., x k 2 U(ag k ), x 2 U(ag o ) such that ((ag 1 ; dag 1 )(x 1 ); :::; (ag k ; dag k )(x k ); (ag o ; dag o )(x)) 2 (dag): For ag =ag 1 ag 2 :::ag k ag o 2 Link, an operation o(ag) is a partial oneto-one mapping with domain o(ag) U(ag 1 ) U(ag 2 ) ::: U(ag k ) and range o(ag) U(ag o ) such that if o(ag o )(x 1 ; x 2 ; :::; x k ) = x then o d (dag)((ag 1 ; dag 1 )(x 1 ); :::; (ag k ; dag k )(x k )) = (ag o ; dag o )(x) for some unique design operation o d (dag); let us observe that to a design operation o d (dag) there may correspond more than one operation o(ag). We extend the mapping to strings by dag = (ag) and o d (dag) = (o(ag)). Elementary constructions. We dene an elementary construction c: if ag =ag 1 ag 2 :::ag k ag o 2 Link, then an expression c = (ag; p sign(ag 1 ); p sign(ag 2 ); :::; p sign(ag k ); p sign(ag o )) will be called an elementary construction with pre-signatures p sign(ag o ) = (st(ag o ); (ag o ); "(ag o ); o(ag o )) and p sign(ag i ) = (st(ag i ); (ag i ); "(ag i )) if there exists a D s(dag) with dag = dag 1 dag 2 :::dag k dag o = (ag), o d (dag o ) = (o(ag)); lab(dag o ) = ( _ (dag o ); o d (dag o )); lab(dag i )=( _ (dag i )) such that < st(ag i ), (ag i ), "(ag i ) > =) _ (ag i ) is true for i = 0; 1; 2; ::; k: We will say that c projects onto D s. To stress the relationship between ag and c we will write c(ag) instead of c and ag(c) instead of ag: We let ag o = Root(c); fag 1 ; ag 2 ; :::; ag k g = Leaf(c); fag 1 ; ag 2 ; :::; ag k ; agg = Ag(c): Constructions. For elementary constructions c, c 0 with Ag(c) \ Ag(c 0 ) =fagg where ag = Root(c) 2 Leaf(c 0 ), we dene the ag-composition c? ag c 0 of c and c 0 with Root(c? ag c 0 ) = Root(c 0 ), Leaf(c? ag c 0 ) = (Leaf(c)?fagg)[ (Leaf(c 0 ), Ag(c? ag c 0 ) = Ag(c) [ Ag(c 0 ). The composition c? ag c 0 is called a construction if there exists a D s(dag o ; dag 1 ) such that c projects onto dag o and c 0 projects onto dag 1 ; we say then that c? ag c 0 projects onto D s(dag o ; dag 1 ): A construction is any expression C obtained from a set 11

12 12 of elementary constructions by applying the composition operation a nite number of times. By V C we denote the set of partial valuations v Ag(C). By T (C) we denote the unique term composed of operations o(ag) such that (T (C)) = T (D s) where C projects onto D s and is a natural extension of to composition of operations. (C; ; ")?schemes. For an elementary construction c = c(ag) as above, with p sign(ag) = (st(ag); (ag); "(ag); o(ag)), we dene a (c; ; ")? scheme as a pair (c(ag); sign(ag)) where sign(ag) = p sign(ag) [ ff(ag)g, = (ag), " = "(ag) and f(ag) 2 F (ag) satises the condition: (Unc(c)) if o (ag i )(x i ; st(ag i )) "(ag i ) for i = 1; 2; ::; k then o (ag)(o(ag)(x 1 ; x 2 ; ::; x k ); st(ag)) f("(ag 1 ); "(ag 2 ); ::; "(ag k )) "(ag): The construction c is said to be the support of the (c; ; ")? scheme: A construction C composed of elementary constructions c 1 ; ::; c m ; c o with Root(C) = Root(c o ) = ag o is the support of a (C; ; ")?scheme when each c i is the support of a (c i ; i ; " i )?scheme, where i = (Root(c i )), " i = "(Root(c i )), = (ag o ) and " = "(ag o ). We have Proposition 4.1. Let v X be any valuation on the set X of leaf agents ag(1); :::; ag(m) of the (C; ; ")?scheme with ag o = Root(C) such that v(b ag(i) ) satises < st(ag(i)); (ag(i)); "(ag(i)) > for i = 1; 2; :::; m where p sign(ag(i)) = (st(ag(i)); (ag(i)); "(ag(i)). Then the uniquely dened object x 2 U(ag o ) such that satises x = T (C)(v X (b ag(1) ); v X (b ag(2) ); :::; v X (b ag(m) )) < st(ag o ); (ag o ); "(ag o ) > where p sign(ag o ) = (st(ag o ); (ag o ); "(ag o ); o(ag o )). Projections of (C; ; ")-schemes onto design schemes. We say that a (C; ; ")? scheme projects onto a design scheme D s when the support C projects onto D s. Negotiation (top-down) of a scheme for satisfying a requirement. Consider now a designer goal Dg = ( _ (dag 1 ); _ (dag 2 ); :::; _ (dag k ); _ (dag o )) realized by a design scheme D s(dag o ; dag 1 ; :::; dag m ). The realization by synthesis agents of the designer goal Dg must begin with nding a (C; ; ")- scheme S such that S projects onto D s. We describe this process. 2

13 Stage 1. A string ag o = ag 1 ag 2 ::ag k ag o 2 Ag is chosen such that a construction c(ag o ) projects onto D s(dag o ); let the uncertainty coecients of agents be " 0 (ag 1 ); :::; " 0 (ag k ), " 0 (ag o ): Stage 2. Agents ag 1 ; ag 2 ; ::; ag k ; ag o negotiate a connective f(ag o ) 2 F (ag o ) and uncertainty bounds "(ag 1 ) " 0 (ag 1 ); ::; "(ag k ) " 0 (ag k )::; "(ag o ) " 0 (ag o ) such that (Unc(ag o )) is satised with f(ag o ) and "(ag 1 ); ::; "(ag k ); "(ag o ). Stages 1, 2 give, when successful, a (c(ag o ); ; ")?scheme which projects onto D s(dag o ): Stages 3 and following. Agents ag 1 ; ag 2 ; ::; ag k repeat stages 1,2 with new coecients "(ag i ), and the process is continued until a succesful result is reached. The successful result of negotiations means that a (C; (ag o ); "(ag o )) - scheme is constructed which projects onto D s(dag o ; dag 1 ; :::; dag m ): We state a theorem which follows from Propositions 3.1 and 4.1. Theorem 4.1. (The sucient criterium of correctness) Assume that a (C; ; ")-scheme projects onto a design scheme D s realizing a designer goal Dg = ( _ (dag 1 ); _ (dag 2 ); ::; _ (dag k ); _ (dag o )). Let v X be any valuation on the set X of leaf agents ag(1); ::; ag(m) of the (C; ; ") -scheme with ag o = Root(C) such that v(b ag(i) ) satises where < st(ag(i)); (ag(i)); "(ag(i)) > p sign(ag(i)) = (st(ag(i)); (ag(i)); "(ag(i); o(ag o )) for i = 1; 2; :::; m: Then the uniquely dened object x 2 U(ag o ) such that satises the condition x = T (C)(v X (b ag(1) ); v X (b ag(2) ); :::; v X (b ag(m) )) (ag o ; dag o )(x) j= d _ (dag o ) 13 i.e. x satises the designer requirement in the satisfactory degree. We extend the satisability relation j= to the connective 3 by 2 if there exists C such that v X j= 3 < st(ag o ); (ag o ); "(ag o ) >

14 14 where x j= < st(ag o ); (ag o ); "(ag o ) > x = T (C)(v X (b ag(1) ); v X (b ag(2) ); :::; v X (b ag(m) )) and ag(1), ag(2),...,ag(m) are leaf agents of C. The connective 3 expresses the existence of a scheme which can satisfy (; ") over a given input v. In particular, Theorem 4.1 gives a sucient condition for nding such C. Let us emphasize the fact that the functions f(ag), called mereological connectives above, are expected to be extracted from experiments with samples of objects.the above property allows for an easy to justify correctness criterium of a given (C; ; ")-scheme provided that all parameters in this scheme have been chosen properly. The searching process for these parameters and synthesis of an uncertainty propagation scheme satisfying the formulated conditions constitutes the main and not easy part of design and synthesis. Bottom-up communication. The bottom-up communication process is started by the leaf agents of C; the leaf agents select a valuation v X compatible with C and any leaf agent ag sends to its parent agent ag o the object v X (b ag ) and the value E (ag) (v X (b ag ); st(ag)): This process is repeated with non-leaf agents: any non-leaf agent ag applies the operation o(ag) to the objects x 1 ; x 2 ; :::; x k sent by its children agents, synthesizes the object x = o(ag)(x 1 ; x 2 ; :::; x k ) and nds the value E (ag) (x; st(ag)): This communication process ends at the root agent of C with the object x C. In the case when the assembling process proceeds correctly, the object x C satises the requirement in the satisfactory degree; the correctness of the assembling process is checked by means of the negotiated connectives f(ag) and the found values E (ag) (x; st(ag)): 5. MEREOLOGICAL CONTROLLER The approximate specication (; ") can be regarded as the invariant to be kept over the universe of global states (complex objects) of the distributed system. A mereological controller generalizes the notion of a multirelay fuzzy controller [9], [15], [20], [27]. The basic problems. Now we can formulate some basic problems related to control in terms of properties of a construction C. The control problems can be divided into several classes depending on the model of controlled object. In this work we deal with the simplest case. In this case, the model of a controlled object is the (C; ; ") -scheme c which can be treated as a model of the unperturbed by noise controlled object whose states are satisfying the approximate specication (; ").

15 The (C; ; ") -scheme c denes a function F c called the output function of the (C; ; ") -scheme c given by F c (v) = x i x = T (C)(v(b ag1 ); v(b ag2 ); :::; v(b agr )) where ag 1 ; :::; ag r are leaf agents of C: If ag c is the root agent of C then any object from the set F c (U(ag 1 ) ::: U(ag r )) \ fx 2 U(ag) : x j= (st(ag c ); ; ")g is called the (; ")-invariant object of c. We assume the leaf agents of the (C; ; ") -scheme c are partitioned into two disjoint sets, namely the set U n control(c) of uncontrollable (noise) agents and the set Control(c) of controllable agents. We present now two examples of a control problem for a given (C; ; ") -scheme. (OCP) OPTIMAL CONTROL PROBLEM: Input: (C; ; ") -scheme c; information about actual valuation v of leaf agents i.e. the values v(b ag ) for any ag 2 Control(c) and a value " 0 such that F c (v) j= (st(ag c ); ; " 0 ). Output: A new valuation v 0 such that v 0 (b ag ) = v(b ag ) for any ag 2 Un control(c) and F c (v 0 ) j= (st(ag c ); ; " o ) where " o = supf : F c (w) j= (st(ag c ); ; ) for some w such that w(b ag ) = v(b ag ) for ag 2 Un control(c)g. These requirements on the output can be too hard to be satised directly. One can search for changes in a given uncertainty scheme which allow for the construction of an object not necessarily the closest to a given specication but satisfying a given specication in a degree higher than the sum of a given threshold and the degree dened by the current object. In this way we obtain (CP) r-control PROBLEM Input: (C; ; ") -scheme c; information about actual valuation v of leaf agents (i.e. the values v(b ag ) for any ag 2 Control(c)) and a value " 0 such that F c (v) j= (st(ag c ); ; " 0 ): Output: A new valuation v 0 such that v 0 (b ag ) = v(b ag ) for ag 2 Un control(c) and F c (v 0 ) j= (st(ag c ); ; " o ) where " o > " 0 + r for some given threshold r. The controller. We will describe now the basic idea on which our controllers of complex dynamic objects represented by distributed systems of 15

16 16 intelligent agents are built. The main component of controllers are -rules describing how local changes (i.e. changes at agents of a given distributed system) 4"(ag) of uncertainty coecients "(ag) can be compensated by local changes 4"(ag 1 ); :::; 4"(ag k ) of uncertainty coecients at children ag 1 ; :::; ag k of ag for agents in a given scheme. By composing -rules one can calculate the necessary changes in uncertainty coecients for all agents ag 2 Control(c) and in this way to predict possible changes of controllable parameters (i.e. elementary objects being values of b ag for ag 2 Control(c)). The -rules have the following structure: ("(ag i 1 ); :::; "(ag i r )) = h("(ag);?"(ag); "(ag 1 ); :::; "(ag k )) where ag ; :::; ag i 1 i r are all controllable children of ag (i.e. children of ag having descendents in Control(c)), h : R k+2! R r and R is the set of reals. The description of the function h is extracted from experimental data. In the process of extracting h from data rough set and boolean reasoning methods [3], [25] can be applied. The semantics of the -rule for ag = ag 1 ag 2 :::ag k ag o 2 Link in c is de- ned by uncertainty coecients "(ag); "(ag 1 ); :::; "(ag k ) attached to agents in c in the following way: if an object x 0 issued by the agent ag is satisfying x 0 j= (st(ag); (ag); " 0 (ag)) where " 0 (ag) = "(ag) + "(ag) then if the controllable children ag ; :::; ag i 1 i r of ag will issue objects y i1 ; :::; y ir satisfying y ij j= (st(ag ij ); (ag ij ); "(ag ij ) + "(ag ij )) for j = 1; :::; r where ("(ag i 1 ); :::; "(ag i r )) = h("(ag);?"(ag); "(ag 1 ); :::; "(ag k )) then the agent ag will construct an object y such that y j= (st(ag); (ag); ") where " "(ag). In the above formula we assume "(ag) 0 and "(ag i 1 ) 0; :::; "(ag i r ) 0: The above semantics covers the case when -rules allow to compensate in one step the inuence of a noise. Other cases will be treated in our next paper. If ag 1 ag 2 :::ag k ag o, ag 0 1 ag0 2 :::ag0 r ag0 o 2 Link, fagi 1 ; :::; ag i r g; fag j1 ; :::; ag js g are disjoint sets of controllable children of ag o, ag 0 o, respectively, then two -rules ("(ag i 1 ); :::; "(ag i r )) = h 1 ("(ag o ); "(ag o ); "(ag 1 ); :::; "(ag k )) ("(ag j1 ); :::; "(ag js )) = h 2 ("(ag 0 o); "(ag 0 o); "(ag j1 ); :::; "(ag js ))

17 can be composed at ago 0 if ago 0 = ag it for some t. The result of the composition is dened by ("(ag i 1 ); :::; "(ag i t?1 ); "(ag j1 ); :::; "(ag js ); "(ag it+1 ); :::; "(ag ir )). Hence the composition of -rules leads to the distribution of changes of uncertainty coecients among agents. It is easy to observe that the composition of the -rules leads to a new labelling f" new (ag)g where for all non-root controllable agents of c we have " new (ag) = " new (ag)+ " new (ag) and "(ag) is obtained as the result of negotiations among agents about possible changes dened by compositions of -rules: For the root agent of c and all non-controllable agents in c we assume " new (ag) = "(ag). We obtain the following proposition establishing a basic property related to the correctness of mereological controllers. Proposition 5.1. (The suciency criterium of correctness of the controller) Let F c (v) j= (st(ag c ); ; "(ag c )) where v is the valuation of leaf agents of the (C; ; ") - scheme c and let F c (v 0 ) j= (st(ag c ); ; " 0 (ag c )) where v 0 is a valuation of leaf agents of c such that v 0 (b ag ) = v(b ag ) for ag 2 Control(c), " 0 (ag c ) < "(ag c ). Let f" new (ag)g be a new labelling of agents dened by composition of some -rules such that " new (ag) = "(ag) for ag 2 Un control(c), " new (ag c ) = "(ag c ) = " and fx ag : ag 2 Control(c)g the set of control parameters (inventory objects) satisfying x ag j= (st(ag); (ag); " new (ag)) for ag 2 Control(c): Then x new j= (st(ag c ); ; "(ag c )) holds for the object x new = F c (v 1 ) constructed over the valuation v 1 of leaf agents in c such that v 1 (b ag ) = v 0 (b ag ) for ag 2 Un control(c) and v 1 (b ag ) = x ag for ag 2 Control(c): The above approach can be treated as a rst step towards modelling complex distributed dynamical systems. We expect that it can be extended to solve control problem for complex dynamical systems i.e. dynamical systems which are distributed, highly nonlinear, with vague concepts involved in their description. It is hardly expected that the classical methods of control theory can be successfully applied for such complex systems. 17 2

18 18 6. ANALYSIS The process of analysis is split into the design and synthesis spaces. Given an object x, its analysis can be realized in the following sequence of steps. Step 1. An agent ag 2 Ag such that x 2 U(ag) is chosen Step 2. An agent (ag) such that (ag; (ag))(x) is dened choses its decomposition. Step 3. The decomposition continues, resulting in the analysis tree for (ag; (ag))(x): Step 4. A design scheme for (ag; (ag))(x) is designed. Step 5. A (C; ; ")-scheme for synthesis of x is found. This constitutes the analysis of constructibility of x. Further analysis includes e.g. the analysis of stability as well as robustness of (C; ; "). 7. CONCLUSIONS We expect that our approach is oering some important tools for solving practical tasks related to computer-aided manufacturing [4], [5], [6], [12], [30], computer-aided design [4], [10], logistics [12], [30], adaptive control of complex systems [9], [11], business re-engineering [2], cooperative/ distributed problem solving including planning, dynamic task assignment, automated fabrication and approximate reasoning [18], [30], [32] in early stages of design processes. In the full version of the paper we will also present a general logical framework [13], [21] for approximate reasoning based on rough mereology which seems to be universal basis for several existing approaches dealing with uncertainty (cf. [18]) like Dempster-Shafer approach, Bayesian-based reasoning, belief networks, fuzzy logics, neural network logic etc. Our notion of a complex object includes, among others, proofs understood as schemes constructed in order to support within our knowledge assertions/hypothesis about reality described by our knowledge incompletely. This work has been supported by grant #8-T11C01011 from the State Committee for Scientic Research (KBN). 8. APPENDIX: BASIC NOTIONS Information systems and decision tables. The formalization of vagueness within the framework of rough set theory is based on the assumption that objects are perceived by means of the information about

19 them encompassed in a set of available features or attributes [16], [19], [25]; this informal idea leads to the notion of an information system. An information system is a triple A=(U; A; V ) where U is a nite set called the universe of objects and A is a nite set of attributes; any attribute a 2 A is a mapping a : U?! V on the universe U: We denote by the symbol V a the range a(u) of the attribute a; the set V a is called the value set of a. Descriptors over A and V are expressions of the form (a; v) where a 2 A and v 2 V a. By the symbol C(A; V ) we denote the set of all boolean combinations of descriptors. For 2 C(A; V ); by A we denote the meaning of in the decision table A, i.e. the set of all objects in U with the property, dened inductively as follows: (i) if is of the form (a; v) then A = fx 2 U : a(x) = vg; ( ^ 0 ) A = A \ A 0 ; ( _ 0 ) A = A [ A 0. For an object x 2 U we dene for a set B A the information vector Inf B (x) = f(a; a(x)) : a 2 Bg. We say that objects x; y 2 U are B-indiscernible when Inf B (x) = Inf B (y); the B-indiscernibility relation IND(B) is dened as follows: IND(B) = f(x; y) 2 U U : Inf B (x) = Inf B (y)g. The relation IND(B) is an equivalence relation and we denote by the symbol [x] B the equivalence class of this relation which contains x. A decision table is any information system of the form A = (U; A [ fdg; V ), where d 62 A is a distinguished attribute called the decision. The elements of A are called conditions. A decision rule of a decision table A = (U; A [ fdg; V ) is any expression of the form =) (d; i) where i 2 V d and 2 C(A; V ). The decision rule =) (d; i) for A is true in A i A ((d; i)) A. Given a set X U, the numerical characterization of a degree in which an object x belongs to X relative to the attribute set B A is provided by the rough membership function X;B [17]. For B A, X U and x 2 U, we let X;B (x) = kx \ [x] Bk k[x] B k where kzk denotes the cardinality of a set Z. We extend the notion of a rough membership function to a standard rough inclusion U on the power set exp(u) of U. To this end, we dene U : exp(u) exp(u)?! [0; 1] by letting U (X; Y ) = kx\y k kxk in case X 6= ; and U (;; Y ) = 1: The reader will nd in [16], [19], [24], [25] a deep discussion of rough settheoretic tools for decision rules generation and for synthesis of adaptive decision systems. Rough Mereology 19

20 20 The basic notion of rough mereology is that of a rough inclusion. A rough inclusion oers the most general formalism for the treatment of partial containment. Rough mereology can be regarded as a far - reaching generalization of mereology of Lesniewski: it replaces the relation of being a (proper) part with a hierarchy of relations of being a part in a degree. Rough inclusions. A real function (X; Y ) on a universe of objects U with values in the interval [0; 1] is a rough inclusion when it satises the following conditions: (A) (X; X) = 1 for any X ; (B) (X; Y ) = 1 implies that (Z; Y ) (Z; X) for any triple X; Y; Z; (C) there is N such that (N; X) = 1 for any X. An object N satisfying (C) is a -null object: such objects are excluded in mereology of Lesniewski. We let X = Y i (X; Y ) = 1 = (Y; X) and X 6= Y i non(x = Y ). We introduce other conditions for rough inclusion: (D) if objects X; Y have the property : if Z 6= N and (Z; X) = 1 then there is T 6= N with (T; Z) = 1 = (T; Y ) then it follows that: (X; Y ) = 1. (D) is an inference rule: it is applied to infer the relation of being a part from the relation of being a subpart. (E) For any collection? of objects there is an object X with the properties: (i) if Z 6= N and (Z; X) = 1 then there are T 6= N; W 2? such that (T; Z) = (T; W ) = (W; X) = 1; (ii) if W 2? then (W; X) = 1; (iii) if Y satises the above two conditions then (X; Y ) = 1. (E) will be applied below to show the existence and uniqueness of classes of objects. A model of rough inclusion is the standard rough inclusion U. We now outline the way in which mereology of Lesniewski [14] follows out of rough mereology. Mereology of Lesniewski. We dene the relation part() on the universe U from a rough inclusion as follows: Xpart()Y i (X; Y ) = 1 and (Y; X) < 1. The objects set()? and, respectively, class()? are dened as an object X which satises the condition (E)(i), respectively the conditions (E)(i)- (iii), with?. It turns out that the relation part() induced by satises all axioms of mereology of Lesniewski [14] on non-null objects of the universe: any rough inclusion introduces a model of mereology on the collection of

21 non--null objects of the universe U [14]. In particular, the relation part() is a non-reexive and transitive relation on U ; the formula x part() y reads: x is a (proper) part of y. The formula x = class() fx 1 ; x 2 ; :::; x k g is interpreted as the statement that the object x is composed (designed, synthesized) from parts x 1 ; x 2 ; :::; x k : Many valued logic. It is well known [14] that in mereology the notions of a subset, an element, and an ingredient are all equivalent. Therefore rough mereological containment (X; Y ) can be interpreted as the membership degree of X in Y and therefore rough mereological approach encompasses fuzzy set approach. We mention here only the following instance of this principle: for residual implication?! > and a rough inclusion, the function (X; Y ) = inf Z f?! >((Z; X); (Z; Y ))g is also a rough inclusion [21]. Hence rough inclusions encompass fuzzy containment functions [9], [31]. Rough mereological connectives. Propagating rough inclusions can be eected by means of rough mereological connectives. An n-rough mereological connective f is a map f : [0; 1] m! [0; 1] such that [1; :::; 1] 2 f?1 (1). Examples of mereological connectives are some connectives of many valued logic e.g. f(x; y) = min(x; y) (Zadeh), f(x; y) = xy (Menger). When a connective f is chosen, we dene the (f; )-closeness function E f; (X; Y ) = f((x; Y ); (Y; X)). In case f = min; the same formula denes the mereological distance function E (X; Y ). Rough inclusions from information systems. Rough inclusions can be generated from the information system A; for instance, for a given partition P = fa 1 ; : : : ; A k g of the set A of attributes into non-empty sets A 1 ; : : : ; A k, and a P given set W = fw 1 ; : : : ; w k g of weights, w i 2 [0; 1] for k i = 1; 2; : : : ; k and i=1 w i = 1 we can let o;p;w (x; y) = kx i=1 w i kind i(x; y)k ka i k where IND i (x; y) = fa 2 A i : a(x) = a(y)g. We call o;p;w a pre-rough inclusion. The function o;p;w is attribute-dependent i.e. if Inf B (x) = Inf B (x 0 ) and Inf B (y) = Inf B (y 0 ) then o;p;w (x; y) = o;p;w (x 0 ; y 0 ). o;p;w can be extended to a rough inclusion on the set 2 U [21]. Tolerance relations. A relation X X is a tolerance relation on X if (i) is reexive: xx for any x 2 X (ii) is symmetric: xy implies yx for any pair x; y of elements of X. By (x) we denote the tolerance class of x i.e. the set (x) =fy : yxg: For a chosen threshold k 2 [0; 1], we dene a tolerance F;;k via X f;;k Y i E f; (X; Y ) k. The family f f;;k : k 2 [0; 1]g will be applied in the rest of our work as a measure of 21

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