Implementing fuzzy containment via rough inclusions: rough

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Its basic notion is rough inclusion, meaning a function (x; y) on pairs of objects .... Any X satisfying G(i) is called a set of objects in ?; if, in addition, X satis es G(ii ...
Implementing fuzzy containment via rough inclusions: rough mereological approach to distributed problem solving Lech Polkowski Institute of Mathematics Warsaw University of Technology Pl. Politechniki 1 00-650 Warszawa Poland [email protected]

Andrzej Skowron Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa Poland [email protected]

Abstract We propose a uni ed formal treatment of design, analysis, synthesis and control in distributed systems of intelligent agents. Our approach goes back to rough set the+ory and we propose rough mereology as a foundational basis for our approach. Rough mereology is a theory of the relation of being a part in a degree which can be regarded as a particular implementation of a general idea of fuzzy containment. We include basic connections between the two theories. Topic category: Qualitative and approximate - reasoning modeling, Intelligent information systems Key words: qualitative and approximate- reasoning modeling, intelligent information systems, rough mereology, distributed systems of intelligent agents, cooperative: design, analysis, synthesis, and control.

1 Introduction We propose a new methodology for reasoning about complex objects in distributed systems. The tasks covered by reasoning systems include design, synthesis, analysis and control of complex objects [1], [4], [7], [8], [10-12]. A complex object, perceived by us as a solution to a requirement issued to the agents in the system, is represented by means of a scheme over the universe of agents and the inventory of atomic parts. We propose rough mereology as a principle on which cooperative agent systems acting under uncertainty can be constructed and their properties analyzed. Rough mereology stems from rough set theory [5]. Its basic notion is rough inclusion, meaning a function (x; y) on pairs of objects which returns the degree in which the object x is a part of the object y. Two basic features of rough mereology viz. (i) collections of objects are regarded as objects (ii) the notions of a part in a degree and an element in a degree are equivalent, allow for regarding rough mereology as a particular implementation of the general idea of partial (fuzzy) containment. The results presented here re ect the main lines of our research [7-12]: 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. The need for principles on which distributed systems of agents supporting uncertain reasoning can be constructed is stressed as the main problem of distributed AI [1]. We present here problems of synthesis and control of complex systems as exemplary cases of applications of our approach. Problems of analysis and design are treated e.g. in [12].

2 Preliminaries Rough set theory. An information system [5] is a triple A=(U; A; V ) where U is a set called the universe of objects and A is a set of attributes; any attribute a 2 A is a mapping a : U ?! V on U: Descriptors over A and V are expressions of the form (a; v) where a 2 A and v 2 a(V ) . C (A; V ) is the set of all boolean combinations of descriptors. For x 2 U and B  A; we let InfB (x) = f(a; a(x)) : a 2 B g. We say that objects x; y 2 U are B -indiscernible when InfB (x) = InfB (y); the B -indiscernibility relation is IND(B ) = f(x; y) 2 U  U : InfB (x) = InfB (y)g. We denote by the symbol [x]B the equivalence class of IND(B ) which contains x.

X;B (x) = kXk[\x[]xB]kB k where kZ k denotes the cardinality of a set Z . We de ne the standard rough inclusion U : exp(U )  exp(U ) ?! [0; 1] by letting

U (X; Y ) = kXkX\Yk k in case X 6= ; and U (;; Y ) = 1:

Rough Mereology. Rough mereology [7] o ers the general formalism for the treatment of partial containment. Rough mereology can be regarded as a far - reaching generalization of mereology of Lesniewski [3]: it replaces the relation of being a (proper) part with a hierarchy of relations of being a part in a degree. The basic notion is the notion of a rough inclusion. A real function (X; Y ) on a universe of objects U with values in the interval [0; 1] is a rough inclusion [7] when it satis es the following conditions: (A) (X; X ) = 1 for any X (meaning normalization); (B) (X; Y ) = 1 implies that (Z; Y )  (Z; X ) for any triple X; Y; Z (meaning monotonicity); (C) (X; Y ) = 1 and (Y; X ) = 1 imply (X; Z )  (Y; Z ) (meaning monotonicity); (D) there is N such that (N; X ) = 1 for any X . An object N satisfying (D) is a -null object. 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: (E) 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. (F) is an inference rule: it is applied to infer the relation of being a part from the relation of being a subpart. (G) 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 satis es the above two conditions then (X; Y ) = 1. Any X satisfying G(i) is called a set of objects in ?; if, in addition, X satis es G(ii,iii), X is called the class of objects in ?: These notions allow for representations of collections of objects as objects. We interpret the formula: (x; y) = r as the statement: x is a part of y in degree r. A model of rough inclusion is the standard rough inclusion U on a universe U which satis es axioms (A)-(G). We now outline the way in which mereology of Lesniewski [3] is related to rough mereology. Mereology of Lesniewski. The mereological theory of Lesniewski formalizes the relation of (being) a (proper, exact) part. We de ne 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 de ned as an object X which satis es the condition (G)(i) , respectively the conditions (G)(i)-(iii), with a collection of objects ?. It turns out that the relation part() induced by  satis es axioms of mereology of Lesniewski [14] on non-null objects of the universe: the relation part() is a non-re exive and transitive relation on U ; the formula x part() y reads: x is a (proper) part of y. The formula x = class() fx1 ; x2 ; :::; xk g is interpreted as

degree r. In this way  induces a fuzzy memebership function. Relations to fuzzy containment. Fuzzy containment is de ned in a fuzzy universe U endowed with fuzzy membership functions X ; Y by the formula [2]:

(X  Y ) = inf Z fI (X (Z ); Y (Z ))g for a many - valued implication I . We quote a result in [9] which shows that rough inclusions generate a class of fuzzy containments stable under residual implications of the form ?! > where > is a continuous t-norm [2] viz.: for any rough inclusion ? !  on U , the function (X; Y ) = inf Z f > ((Z; X ); (Z; Y ))g is also a rough inclusion. Rough mereological connectives. Propagating rough inclusions can be e ected by means of rough mereological connectives. An n-rough mereological connective f (n-rmc, in short) is f : [0; 1]m ! [0; 1] such that [1; :::; 1] 2 f ?1(1). Examples of 2-rmc's are f (x; y) = min(x; y) (Zadeh), f (x; y) = xy (Menger). Rough inclusions from information systems. Rough inclusions can be generated from the information system A; for instance, for a given partition P = fA1 ; : : : ; Ak g of the set A P of attributes into non-empty sets A1 ; : : : ; Ak , and a given set W = fw1 ; : : : ; wk g of weights, wi 2 [0; 1] for i = 1; 2; : : :; k and ki=1 wi = 1 we let

o;P;W (x; y) =

Xk w  kINDi(x; y)k i=1

i

kAi k

where INDi (x; y) = fa 2 Ai : a(x) = a(y)g. We call o;P;W a pre-rough inclusion. The function o;P;W is rough-invariant i.e. if InfB (x) = InfB (x0 ) and InfB (y) = InfB (y0 ) then o;P;W (x; y) = o;P;W (x0 ; y0 ). o;P;W can be extended to a rough inclusion on the set exp(U ) [9] e.g. via the formula: (X; Y ) = >f? fo;P;W (x; y) : y 2 Y g : x 2 X g where > is a t-norm and ? is a t-conorm.

3 Synthesis of approximate reasoning scheme In this section we are concerned with synthesis spaces. Our formalization can be put in a nutshell as follows: synthesis agents are equipped with their local rough inclusions; local classes of synthesis objects at synthesis agents form design objects on which the designer acts by means of its rough inclusion. The designer decompositions of classes into simpler classes form links among classes (=design agents) establishing a design scheme. The design scheme is then re ected into synthesis agents. Synthesis agents. We start with the set Ag of synthesis agents and the set Inv of inventory objects. Any synthesis agent ag is assigned a label lab(ag) = fU (ag); A(ag); St(ag); L(ag); o (ag); F (ag)g where: U (ag) is the universe of objects at ag, A(ag) = fU (ag); A(ag)g is the information system of ag, 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 constructed as formulas in C (A(ag); V ); o (ag)  U (ag)  U (ag)  [0; 1] is a pre-rough inclusion at ag generated from A(ag); F (ag) is a set of rm connectives at ag . Mereology DS : Synthesis agents classify their objects by means of pre - rough inclusions. The D S ? mereology DS (the design - synthesis mereology) is a family f(ag) : ag 2 Agg where (ag) is a xed extension of o (ag) for any ag 2 Ag. Mereology DS de nes design objects (categories of real objects): a design object x is a class((ag))? where ?  U (ag): Thus, design objects are classes of synthesis objects: classes relevant for design are those where objects in ? are -similar. On these classes the mereology of designer acts decomposing them into simpler classes. We denote the set of classes by U (Des). Design agents. Requirements. The designer operates on the universe U (Des) by means of a rough inclusion (Des); any design agent dag is equipped with an information system Adag =(Udag ; Adag ). The table Adag describes objects (possibly complex) from a universe U (Des) in the language of attributes in Adag . A (designer) requirement (dag) at dag is a formula in C (Adag ; V ); the symbol will denote a requirement at some design agent. The symbol x j=d will denote that x satis es : Communication design - synthesis. The communication between synthesis spaces and design spaces is provided by mappings (ag; dag) : for x 2 U (ag); the value (ag; dag)(x) 2 U (Des) is a design object . Let us observe that while (ag; dag)(x) is unique, the object x may belong to more than one design object. This causes the ambiguity in communication

(similarity) relation Des sat:

Satisfactory satis ability 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 an object whose category satis es a requirement 0 = 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 ; 0 2 Des req; Des sat 0 will read "any object satisfying 0 (resp. ) satis es (resp. 0 ) in degree satisfactory to the designer ". We denote by the symbol [ ] the tolerance class Des sat( ): We denote by the symbol _ the disjunction of formulas in [ ] i.e. _ is _f 0 : 0 2 [ ]g: Clearly, if x j=d _ then x satis es in the satisfactory degree.

Mereological compatibility of requirements. We assume that (Des) is compatible with Des sat i.e. if x = class(ag ) fx ; x ; :::; xk g, xi j=d i , x j=d , yi j=d _i , y = class(ag )fy ; y ; ::; yk g then y j=d _ : The compatibility condition means 1

2

1

2

that the designer schemes are designed as insensitive to local communication ambiguities. Approximate logic of synthesis. Consider a synthesis agent ag.The symbol bag will denote the variable which runs over objects in Uag : A valuation vX where X is a set of synthesis agents is a function which assigns to any bag for ag 2 X an x denotes vfagg with vfagg (bag ) = x. element vX (bag ) 2 Uag : The symbol vag We now de ne approximate speci cations at ag as formulas of the form < st(ag); (ag); "(ag) > where st(ag) 2 St(ag); (ag) 2 L(ag); "(ag) 2 [0; 1] : We say that v = vfagg satis es a formula =< st(ag); (ag); "(ag) >, symbolically v j= , in case x j= < st(ag ); (ag ); "(ag ) >. (ag)(v(bag ); st(ag))  " and st(ag) j= (ag). We write x j=< st(ag); (ag); "(ag) > i vag The designer language Linkd: We de ne a language Linkd. For a string dag = dag1dag2:: :dagk dag we have dag 2 Linkd i there exist x; x1 ; x2 ; ::; xk ; x such that x class((Des)) fx1 ; x2 ; ::; xk g where xi 2 Udagi for i  k, x 2 Udag . Constructibility relation. For dag = dag1dag2: ::dagk dag 2 Link, we de ne the dag-constructibility relation (dag)  U (dag1 )  U (dag2 )  :::  U (dagk )  U (dag) by letting (x1 ; x2 ; :::; xk ; x) 2 (dag) i x class((Des))fx1 ; x2 ; ::; xk g.

The synthesis language Link: The agents ag1; ag2; ::; agk ; ago in Ag form the string ag =ag1 ag2:::agk ago 2 Link if and only if there exist design agents dag1 ; dag2 ; : ::; dagk ; dago such that dag = dag1 dag2 :::dagk dago 2 Linkd and there exist objects x1 2 U (ag1 ); :::; xk 2 U (agk ); x 2 U (ago ) such that ((ag1 ; dag1 )(x1 ); ::; (agk ; dagk )(xk ); (ago ; dago )(x)) 2 (dag): We write (ag) =dag:

Elementary constructions. If ag =ag ag ::agk ago 2 Link, then 1

2

c = (ag; f< st(agi ); (agi ); "(agi ) > : i = 0; 1; ::; kg) will be called an elementary construction. We write: Ag(c)=fago ; ag1; ::; agk g, Root(c)=ago ; Leaf (c) = Ag(c) ? fagog: Constructions. For elementary constructions c, c0 with Ag(c) \ Ag(c0) = fagg where ag = Root(c) 2 Leaf (c0); we de ne the ag-composition c ?ag c0 of c and c0 with Root(c ?ag c0 ) = Root(c0 ), Leaf (c ?ag c0 ) = (Leaf (c) ? fagg) [ (Leaf (c0), Ag(c ?ag c0 ) = Ag(c) [ Ag(c0 ). A construction is any expression C obtained from a set of elementary constructions by applying the composition operation a nite number of times. (C; ; ")?schemes. For an elementary construction c = c(ag) as above, we de ne a (c; ; ") ? scheme as (ag; f< st(agi ); (agi ); "(agi ) > i = 1; ::; kg[ < st(ago ); (ago); "(ago ); f (ago) > where (ago ) = ; "(ago) = " and f (ago) 2 F (ago) satis es the condition: then o(ag)(x; st(ago))  f ("(ag1); "(ag2); ::; "(agk ))  "(ago)

if (agi )(xi ; st(agi ))  "(agi) for i = 1; 2; ::; k

i

i

i i

i

i

i

i

o

and " = "(ago). The (C; ; ") -scheme c de nes a function Fc called the output function of c given by Fc (vLeaf (C ) ) = x where x 2 U (ago ) is the unique object produced by C from vLeaf (C ) . Proposition 4.1. For any valuation vX on the set X of leaf agents ag(1); : :; ag(m) of the (C; ; ")?scheme with ago = Root(C ) such that

v(bag(i) ) j= < st(ag(i)); (ag(i)); "(ag(i)) > for i = 1; 2; :::; m, we have Fc (vX ) j=< st(ago); (ago ); "(ago ) > :

Negotiation (top-down) of a scheme for satisfying a requirement. Consider now a designer goal _(dago) . The realization of _(dago ) by synthesis agents consists in negotiations among agents in order to nd a (C; ; ")-scheme c such that (ago ; dago )(Fc (vLeaf C ) j=d _ (dago ) where (ago ) =dago : Theorem 4.2 (the sucient criterium of correctness ) ( )

Assume that a (C; ; ")-scheme realizes a designer goal _(dago ): Then for any valuation vX on the set X of leaf agents ag(1); ::; ag(m) of the (C; ; ")?scheme with ago = Root(C ) such that v(bag(i) ) j= < st(ag(i)); (ag(i)); "(ag(i)) > i = 1; 2; :::; m,

(ago ; dago)((Fc (vX )) j=d _(dago ): 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.

4 Mereological controllers The approximate speci cation (; ") can be regarded as an invariant to be kept over the universe of global states (complex objects) of the distributed system. A mereological controller generalizes the notion of a fuzzy controller [4]. The basic problems.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 speci cation (; "). We assume the leaf agents of the (C; ; ") -scheme c are partitioned into two disjoint sets, namely the set Un 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(bag ) for any ag 2 Control(c) and a value "0 such that Fc (v) j=< st(agc ); ; "0 >. Output: A new valuation v0 such that v0(bag ) = v0(bag ) for ag 2 Un control(c) and Fc (v0 ) j= (st(agc); ; "o) where "o = supf : Fc (w) j= (st(agc ); ; ) for some w such that w(bag ) = v(bag ) for ag 2 Un control(c)g. These requirements can hardly be satis ed directly. A relaxation of (OCP) is (CP) r-CONTROL PROBLEM

ag

ag

c

o

c

o

for some given threshold r. We will describe now the basic idea on which our controllers of complex dynamic objects represented by distributed systems of intelligent agents are built. The main component of the controller are  - incremental rules. -rules have the form: ((ag)) ("(agi1 ); :::; "(agir )) = h("(ag); ?"(ag); "(ag1); :::; "(agk )) where agi1 ; :::; agir are all controllable children of ag (i.e. children of ag having descendents in Control(c)), h : Rk+2 ! Rr and R is the set of reals. Approximations to the function h are extracted from experimental data. The meaning of (ag) is : if x0 j=< st(ag); (ag); "0 (ag) > for x 2 U (ag) where "0 (ag) = "(ag) + "(ag) then if the controllable children agi1 ; :::; agir of ag will issue objects yi1 ; :::; yir with yij j=< st(agij ); (agij ); "(agij ) + "(agij ) > for j = 1; :::; r where ("(agi1 ); :::; "(agir )) = h("(ag); ?"(ag); "(ag1); :::; "(agk )) 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 "(agi1 )  0; :::; "(agir )  0. The above semantics covers the case when -rules allow to compensate in one step the in uence of a noise. Other cases will be treated elsewhere. ?rules can be composed in obvious sense. (ag?ag0 ) denotes the composition of (ag) and (ag ) over ag ? ag0 : The variable (c) will run over compositions of ?rules over c. Theorem 5.1 (the suciency criterium of correctness of the controller) Let Fc (v) j=< st(agc ); ; " > where v is the valuation of leaf agents of the (C; ; ") -scheme c and let Fc (v0 ) j=< st(agc ); ; "0 > where v0 is a valuation of leaf agents of c such that v0 (bag ) = v(bag ) for ag 2 Control(c), "0 < ". If f"new (ag)g is a new assignment to agents de ned by a composition (c) of some -rules such that "new (ag) = "(ag) for ag 2 Un control(c), "new (agc ) = " and fxag : ag 2 Control(c)g is the set of control parameters (inventory objects) satisfying xag j=< st(ag); (ag); "new (ag) > for ag 2 Control(c) then for the object xnew = Fc (v1 ) constructed over the valuation v1 of leaf agents in c such that v1 (bag ) = v0 (bag ) for ag 2 Un control(c) and v1 (bag ) = xag for ag 2 Control(c) it holds that xnew j=< st(agc ); ; " >. 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. One can hardly expect that classical methods of control theory can be successfully applied for such complex systems. Some applicational aspects of this approach can be found in [7], [11], [12]. We are currently working on applications to knowledge discovery in large data - bases via decomposition into smaller tables and synthesis of local ndings. 0

0

5 References [1] Amarel, S., PANEL on AI and Design, Proceedings of IJCAI-91; Twelfth International Conference on Arti cial Intelligence, Sydney, Australia, Vol.1, 563-565, 1991. [2]Dubois, D., Prade H. and Yager R.R., Readings in Fuzzy Sets for Intelligent Systems, Morgan Kaufmann, San Mateo, 1993. [3] Lesniewski, S., Foundations of the general theory of sets , in Stanislaw Lesniewski, Collected Works ( Surma,S. J., Srzednicki, J. T., Barnett, D. I. and Rickey, V.F., Eds.), Kluwer, Dordrecht 1992, pp.128-173. [4] Mamdani E.H., and Assilian S., An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man - Machine Studies, 7, 1-13, 1975. [5]Pawlak, Z., Rough sets: Theoretical Aspects of Reasoning about Data , Kluwer, Dordrecht, 1991. [6] Pawlak, Z., and Skowron, A., Rough membership functions, in Advances in The Dempster - Shafer Theory of Evidence ( Yager,R.R., Fedrizzi, M. and Kacprzyk, J., Eds.), Wiley, New York, 1994, pp. 251-271

Third International Workshop on Rough Sets and Soft Computing, San Jose State University, CA, 1994, pp.78-85. [9] Polkowski, L., Skowron, A., Rough mereology: A new paradigm for approximate reasoning, submitted to International Journal of Approximate Reasoning. [10] Polkowski, L., Skowron, A., Rough mereology and analytical morphology: New developments in rough set theory, Proceedings of WOCFAI-95; Second World Conference on Fundamentals of Arti cial Intelligence, Paris, 1995, Angkor, Paris, 343-354, 1995. [11] Polkowski, L., Skowron, A., Rough mereological approach to knowledge-based distributed AI, Proceedings of WCES-3; Third World Congress on Expert Systems, February 5-9, 1996, Seoul, Korea. [12] Skowron, A., Polkowski, L., Rough mereological foundations for design, analysis, synthesis and control in distributed systems, Proc. Second Intern. Conference on Information Sciences, Wrightsville Beach, NC, September 1995.