Rough Mereology Lech Polkowski1 and Andrzej Skowron2 1
Institute of Mathematics, Warsaw University of Technology Pl. Politechniki 1, 00-650 Warsaw, Poland e-mail:
[email protected] 2 Institute of Mathematics, Warsaw University Banacha 2, 02-097 Warsaw, Poland e-mail:
[email protected]
Abstract. We show how to organize a collection of intelligent coop-
erating agents into a rough mereological controller for the purpose of assembling a complex object with desired quality from its elementary parts. This scheme for synthesis of a complex object from its elementary parts provides a framework for e.g. quality control in concurrent engineering as well as cooperative problem-solving.
1 Introduction There are various approaches to the problem of measuring the degree to which elements belong to a given set. Among them are fuzzy set theory [10] and rough set theory [3]. We propose to take as a primitive notion a function (called a rough inclusion ) returning the degree of inclusion of one set into another. We assume [5] some natural conditions motivated by properties of rough membership functions [4] which characterize rough inclusions. Any rough inclusion can be regarded as the family of partial inclusion relations (indexed by degree of inclusion being the value of the rough inclusion); the relation of partial inclusion in degree can also be interpreted as the relation of being a part in degree . It is natural to require that the relation of being a part in degree 1 satis es the axioms of mereology, an alternative set theory proposed by Stanislaw Lesniewski [1] originally as a scheme to avoid the Russellian antinomies in naive set theory of Cantor. Our conditions on a rough inclusion guarantee that this demand is ful lled. We therefore call the theory of rough inclusions rough mereology [5]. There are some motivations for investigating rough mereology, which seem to be important for applications, among them the following. Rough merology o ers a semantics for a higher level language built for the purpose of reasoning about complex structures when it is sucient to deal only with objects and relations of being a part to a degree; in this case we are free from the implementation details of this language in a lower level language which is more speci c and can be based e.g. on standard set theory. This higher level language should be more convenient for applications concerning e.g. assembling of a complex object from elementary parts ( concurrent engineering as well as cooperative problem-solving [9]), in particular for quality control [7],[8], or for
specifying and analysing of a complex structure where the inference engine must take into account the uncertain character of knowledge about objects and complex structures. Rough mereology permits to consider objects which are not constructible from full parts (parts in degree 1) nevertheless built from objects which are their parts to a certain degree. The primitive notion of mereology [1] is the relation of being a part subject to the conditions 1. part(X; Y ) ) non[part(Y; X)] and 2. part(X; Y ) ^ part(Y; Z) ) part(X; Z). The secondary relation of being an ingredient is de ned by 3. ingredient (X; Y ) , (part(X; Y ) _ X = Y ). Consider a function satisfying conditions A-D below on a collection of objects. A) (X; Y ) 2 [0; 1] for any pair X; Y of objects B) (X; X) = 1 for any object X C) [(X; Y ) = 1] ) [(Z; Y ) (Z; X)] for any triple X; Y; Z of objects D) 9N 8X (N; X) = 1. The symbol X = Y will stand for (X; Y ) = 1 ^ (Y; X) = 1. An object N satisfying D will be called a null object . The condition C expresses monotonicity of . We will further require that satis es the conditions E) if 8Z 6= N[((Z; Y ) = 1) ) 9T 6= N((T; Z) = 1 = (T; X)] then (Y; X) = 1 for any pair X; Y of objects F) for any collection U of objects there exists an object X with the following properties (i) 8Z 6= N[(Z; X) = 1 ) 9W 6= N 9T 2 U ((W; Z) = (W; T) = (T; X) = 1] (ii) 8Z 2 U [(Z; X) = 1] (iii) 8Y [(X; Y ) < 1 ) (either (i) or (ii) does not hold when X is replaced by Y )] E expresses the inference rule about being a part from being a subpart and F expresses the existence and uniqueness of the object being the class of objects from a given collection. Functions satisfying A-F are called rough inclusions [5]. Rough inclusions can be interpreted as hierarchies of relations of being a part in a degree. Rough mereology, the theory of rough inclusion contains mereology of Lesniewski [5,6], (cf. [5, Thms. 3.3.1 - 3.10.2]). Example 1. An information system [3] is a triple A = (U; A; V ) where (i) U is a ( nite) set called the universe (ii) A is a set of conditional attributes , any a 2 A is a mapping a : U ?! V with Va = a(U) - the value set of a (iii) V = [fVa : a 2 Ag. Any object x 2 U is described by its information vector Inf (x) = f(a; a(x) : a 2 Ag; objects x and y are indiscernible whenever
2
Inf (x) = Inf (y). We denote by INF(x) the indiscernibility class containing x. The degree in which objects in U are elements of a given subset X U is given by the rough membership function [4] X de ned on U by \Xj X (x) = j INF(x) j INF(x) j We may observe that the above function X is a particular case of a more general function de ned on subsets X; Y U by (X; Y ) = jXjX\Yj j in case X 6= ; and (;; Y ) = 1. One may check that is a rough inclusion on the powerset of U. Given we de ne the relation part by 1. part (X; Y ) i (X; Y ) = 1 ^ (Y; X) < 1 for any pair X; Y of objects and then the relation of being an ingredient is de ned by 2. ingr (X; Y ) i (X; Y ) = 1 for any pair X; Y of objects. We will call a rough mereological n-connective (rmn ? c, in short) any mapping F : [0; 1]n ?! [0; 1] such that F (x1; x2; . . .; xn) = F (x(1) ; x(2); . . .; x(n)) for any permutation of f1; 2; . . .; ng (symmetry) and F (x1 ; x2; . . .; xn) is nondecreasing with respect to each coordinate (monotonicity). The following binary functors of many-valued logic are examples of rm2 ? c0s (of respectively, Menger, Zadeh, Ruspini): F1(x; y) = x y, F2 (x; y) = minfx; yg, F3 = maxfx + y ? 1; 0g. Let us note that selecting a binary rmc F de nes a family f : 2 [0; 1]g of tolerance relations where (X; Y ) i F ((X; Y ), (Y; X)) . The function E(X; Y ) = F ((X; Y ), (Y; X)) will be called the mereological closeness function .
2 Synthesis of complex objects Synthesis (assembling) of complex objects from parts falls into province of coope-
rative product development (concurrent engineering, cooperative problem solving)
[9]. In this many - faceted task one can distinguish few principal aspects. The spatial aspect (a complete process plan ) is represented usually by a hierarchical tree structure (P-graph) ; the nodes of the tree represent process steps along with information about components, connections, tools and operations. The communication aspect involves noti cation of operators at various nodes of changes in the process (e.g. constraints adjusting ) and con ict resolution (negotation) among operators. The quality control aspect [7], [8] deals with routines assuring that the quality of a nal assembly assigned by the assembler agrees with the quality decided by a posteriori testing. The representation aspect deals with problems of modelling assemblies in terms of objects representing components and connections (the connection graph ), representing classes of parts by common information (group technology) , and representing the inventory by means of speci cations de ning classes of parts as well as by descriptions of physical instances of parts. 3
Setting an assembling scheme as above can be divided into two stages: the learning stage and the production stage . We describe now a formal rough mereological model of the production stage. The learning stage will be discussed elsewhere.
3 Rough mereological controllers We describe a system of cooperating intelligent agents acting as a rough mereological controller i.e. it transforms an input (a semantic constraint (speci cation) which may be expressed in natural language) into an output being the nal assembly and its quality value. This system acts on the following principles. 1. The agents are attached to nodes of a tree being a scheme of the assembly process organization. Any agent is provided with an information system. Indiscernibility classes of the system are represented by pattern assemblies. The root agent is provided with a decision system whose decision attribute Q represents the quality value of the nal assembly. The leaf agents have an access to the inventory subsets and have a given strategy for computing values of the so called initial rough inclusion 0 from their information systems. Any non-leaf agent a has given a set Sa of assembling procedures, a set Ca of rough mereological connectives and a mapping a : Sa ?! Ca . 2. The root agent receives an input being a semantic constraint (speci cation) on the nal assembly expressed as a formula in variables representing conditional attributes of the root agent (or in natural language descriptors of these attributes). The root agent decomposes this input into a constraint sent to its children. 3. The top-down communication consists in decomposing the input constraint at each agent into constraints for its children nodes. 4. Any leaf agent a satis es its constraint by applying a best-choice strategy leading to selecting a part X from the inventory. The strategy of a is based on 0 and consists in covering the space of informationvectors INF(X) of inputs by tolerance sets (Yi ) (with respect to appropriately chosen tolerance relation ) of information vectors INF(Yi ) of selected pattern parts Y1 ; . . .; Yk . Choosing a part X may consist e.g. in creating the set V of virtual parts (possibly not accessible in the subinventory of a) satisfying the constraint and selecting an object X from the tolerance set (Yi ) of a pattern Yi (possibly chosen at random) closest to V with respect to E(X; Y ). The agent a calculates from the initial rough inclusion 0 the vector [E(X; Y1); . . .; E(X; Yk )] of rough mereological distances from X to the pattern parts Y1; . . .; Yk . 5. The horizontal communication consists in children nodes of a given parent node negotiating their constraints by means of some strategy and nding a con ict resolution i.e. an assignment of a constraint to any of them. 6. The bottom-up communicationconsists in any child a sending to the parental node the object X assembled at a along with the vector [E(X; Y1); . . .; E(X; Yk )] of rough mereological distances from the assembly X to the pattern assemblies at a. 4
7. Any parent agent a selects a procedure o from Sa , applies o to objects X1 ; X2 ; . . .; Xn sent by its children to assembly X = o(X1 ; X2; . . .; Xn) and applies a (o) to vectors ? ? E Xi ; Y1i ; . . .; E Xi ; Ykii sent by children to calculate the vector [E(X; Y1 ); . . .; E(X; Yk )] where Y1 , Y2 ; . . .; Yk are pattern assemblies at a. 8. The root agent performs Step 7 assembling a nal product X and calculating the vector [E(X; Y1); . . .; E(X; Yk )] and applies to this vector some best t strategy to nd a pattern Yi best- tting X. Then it outputs the pair (X; Q(Yi )). Let us indicate the main points of this approach. a) The rule structure of a rough controller is twofold: (i) the deep structure of the controller is expressed in terms of semantic relations among information systems of agents and their children: it may be expressed either as natural language expressions relating conditional attributes of the agent to conditional attributes of its children or as true formulae in variables of conditional attributes of the agent and its children; (ii) the surface structure of the controller is expressed by best-choice strategies of leaf agents, rules of selecting a (o) and by algoritms of propagating rough mereological closeness E from the children of an agent to the agent. b) Quantization of fuzzy variables in a fuzzy controller [2] into a number of points corresponding to elements of universe of discourse has its counterpart in the rough mereological controller in covering the space of information vectors of inputs at the root agent by tolerance sets of information vectors of some patterns Y1; . . .; Yk . c) The fuzzi cation process known for fuzzy controllers consists in the rough mereological controller in representing the input at the root agent by the vector [E(X; Y1); . . .; E(X; Yk )] of the values of rough mereological closeness of the nal assembly X to the patterns Y1 ; . . .; Yk . This process is realized in a sequence of steps providing the decomposition of the root agent input into inputs for leaf agents, assignment of values of the initial rough inclusion 0 and propagation of rough mereological closeness by algorithms based on functions Fa . d) Control rules of the rough mereological controller are of the form: ( ) if fYb : b a leaf agentg is a set of patterns selected by leaf agents, then the quality value Q(Y ) of the pattern nal assembly Y = class(Yb : b a leaf agent) is q 2 VQ . Let us observe that the control rules have a counterpart at each non-leaf agent a in the form of rules like: if fYb : b a child of ag is a set of patterns selected by children of a, then Y = class(Yb : b a child of a) is a pattern at a. These local control rules are encoded in information systems of a and all b relative to 5
inner constraints (see below). Any non-leaf agent may be therefore regarded as a controller on its own. e) The defuzzi cation process known for fuzzy controllers is realized in the rough mereological controller by assigning to the vector [E(X; Y1); . . .; E(X; Yk )] of c) the quality value Q(X).
4 Example Our example shows the assembling of a complex object (a gentleman of fortune ) from elementary parts (arms, legs, bodies, heads) and quality control by a rough merological controller.
The system of cooperating intelligent agents H ss B sU L s N sR s s L P s A s M s A s The agent R (the root agent) produces the nal assembly, M; P ; N ; U are non-leaf agents submitting auxiliary assemblies and A ; A ; L ; L ; B; H are leaf 2 1
2 1
1
2
1
2
agents responsible for submitting elementary parts from the inventory. We only list leaf agents as other information systems follow by combining pattern objects of leaf agents according to the above scheme relative to deep structure formulae and inner contraints (see below). We also list attribute variables. A1; A2 YA1= left normal arm, YA2 = right normal arm, YA3 = hook-ended arm. YA1 YA2 YA3
cbuc wcut whan hpis hrop mesa poar pobj hcap hsw frsa sa 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 0 1 0 0 1 1
cbuc: xA;1, wcut: xA;2, whan: xA;3, hpis: xA;4 , hrop: xA;5, mesa: xA;6, poar: xA;7, pobj: xA;8, hcap: xA;9, hsw: xA;10, frsa: xA;11, sa: xA;12 L1; L2 YL1 = left normal leg, YL2 = right normal leg, YL3 = strapped wooden leg YL1 YL2 YL3
fobj fpa fsof mesl wou kck fcol fwet frsl sl 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1
fobj: xL;1, fpa: xL;2, fsof: xL;3, mesl: xL;4, wou: xL;5, kck: xL;6, fcol: xL;7, fwet: xL;8, frsl: xL;9, sl: xL;10 6
B
H
pis cut bel kni par co crl crr YB1 1 1 1 1 1 1 1 0 YB2 1 0 1 0 1 1 0 0 YB3 0 0 0 1 0 0 0 0 YB4 1 1 1 0 0 1 0 0
YH 1 YH 2 YH 3 YH 4
hat ker ptc pig shd 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 1 0
pis: xB;1 , cut: xB;2, bel: xB;3 , kni: xB;4 , par: xB;5 , co: xB;6 , crl: xB;7 , crr: xB;8 , hat: xH;1 , ker: xH;2 , ptc: xH;3 , pig: xH;4 , shd: xh;5 The meaning of attributes is as follows: cbuc-carry bucket, wcut-wield cutlass, whan-wield handpike, hpis-hold pistol, hrop-hold rope-end, mesa, mesltap message, poar-pull oar, pobj-push objects, hcap-hold capstan bar, hsw-hold steering wheel, frsa, frsl- t right side, sa, sl- t left side, fobj-feel objects, fpafeel pain, fsof-feel soft/hard, wou-get wounded, kck-kick, fcol-feel cold/warm, fwet-fel wet/dry, pis-brace of pistols, cut-cutlass, bel-buckled belt, kni-sheathed knife, par-parrot, co-coat, crl-crutch by left, crr-crutch by right, hat-hat, kerkerchief cover, ptc-eye-patch, pig-tarry pigtail, shd-blind eye shade. (e.g. YH 3 is kerchief-covered head with tarry pigtail and both eyes normal). The control rules (specifying quality) are: Q (YB1 YL2 YL3YA2 YA1 YH 1 ) = Q(YB1 YL2YL3 YA2YA1 YH 4 ) = LJS Q (YB2 YL2 YL1YA2 YA1 YH 1 ) = CF Q (YB2 YL2 YL1YA2 YA1 YH 4 ) = Q(YB2 YL2YL1 YA2YA3 YH 1 ) = Q (YB2 YL2 YL1YA2 YA3 YH 4 ) = CE Q (YB3 YL2 YL1YA2 YA1 YH 2 ) = D Q (YB3 YL2 YL1YA2 YA1 YH 3 ) = BP Q (YB3 YL2 YL1YA2 YA1 YH 4 ) = BD Q (YB3 YL2 YL1YA2 YA3 YH 4 ) = BB Q (YB4 YL2 YL1YA2 YA1 YH 1 ) = Q(YB4 YL2YL1 YA2YA1 YH 4 ) = A Q is given by its values on pattern nal assemblies.
5 Deep structure Conditional attributes of any agent a are coded as boolean variables. For a variable xa;i, we denote by xa;i the expression :xa;i . An atomic formula will be any expression of the form xa;i or xa;i. A well - formed formula will be any expression belonging to the smallest set of expressions containing all atomic formulae and closed under propositional connectives _ and ^. A well - formed formula will be called proper if there is no variable xa;i such that both xa;i and xa;i are among variables of . The symbol [ ]v denotes the logical value of a well-formed formula under a valuation v on its variables where valuations are induced by objects i.e. for an object X, the valuation v = vX is given by vX (xa;i ) = xa;i(X). A well - formed formula is said to be true if [ ]v = 1 for any object - induced valuation v. We say that an object (assembly) X satis es a well-formed formula if [ ]vX = 1 and we write in this case X j= . 7
The following true formulae render the description of the deep structure on the level of consecutive non-leaf agents. xU;1 , xB;1 ^ xB;2 ^ xB;4 (xU;1 means heavily armed : pistols, cutlass, knife) xU;2 , (xB;1 ^ xB;2 ^ xB;4 _ xB;2 ^ xB;1 ^ xB;4 _ xB;4 ^ xB;1 ^ xB;2 ) xU;3 , (xB;1 ^ xB;2 ^ xB;4 _ xB;2 ^ xB;1 ^ xB;4 _ xB;4 ^ xB;1 ^ xB;2 ) xU;4 , xB;7 ; xU;5 , xB;8 ; xU;6 , xB;5 ; xU;7 , xB;6 xN;1 , xU;1; xN;2 , xU;2; xN;3 , xU;3; xN;4 , xU;4 _ xU;5 xN;5 , xU;6; xN;6 , xU;7 xP;1 , xN;1 ; xP;2 , xN;2 ; xP;3 , xN;3 ; xP;4 , xN;4 , xP;5 , xN;5 _ xN;6 ; xP;6 , xA;11 ^ xA;2 xP;1 , xM;1; xP;2 , xM;2 ; xP;3 , xM;3; xP;4 , xM;4 ; xP;5 , xM;5 xM;6 , xP;6 _ (xA;12 ^ xA;2) xR;1 , xM;1; xR;2 , xM;2; xR;3 , xM;3; xR;4 , xM;4; xR;5 , xM;5 xR;6 , xM;6; xR;7 , xH;1 _ xH;2; xR;8 , xH;4 xR;9 , xM;4 ^ xM;5 ^ xM;6 ^ xH;1 xR;10 , xM;4 ^ xM;5 ^ xM;6 ^ xH;1 ^ xH;3 ^ xH;5 ^ xM;3 xR;11 , xM;5 ^ xM;6 ^ xH;1
6 Logic of constraint negotiation and con ict resolution We distinguish between inner constraints imposed on agents by the designer independently of a particular assembling task and outer constraints issued in the particular assembling process. Inner constraints can be interpreted as meaning postulates in the sense of Carnap and outer constraints express the particular demand on the nal assembly. The constraints may be expressed as natural language phrases and then rendered into formulae in the corresponding variables. For constraints 1 ; 2; . . .; k at a, the negotiation function is f a 1 ; 2 ;...; k = [ 1] ^ [ 2] ^ . . . ^ [ k ] where [ i] is the formula i in which each variable xa;j has been replaced by its deep structure rendering in terms of variables of the children of a. WV Let (x0a;i) be the normal disjunctive formVof f a 1 ; 2 ;...; k . Then any object X such that X j= f a 1 ; 2 ;...; k satis es X j= (x0a;i ) for some proper prime V 0 implicant (xa;i) of f a 1 ; 2 ;...; k . Our negotiation and con ict resolution strategy will therefore consist of the two following steps at a. V Step 1. A proper prime implicant (x0a;i ) of the negotiation function f ac1 ; 2 ;:::; k is selected. This prime implicant is a constraint (con ict) resolution at a. Step 2. Let V(x0a;i) = b1 ^ b2 ^ . . . ^ bk where b1; b2; . . .; bk are the children of a and any bi is a conjunction of variables of bi . Then the constraint bi is sent to bi . In case bi is a leaf agent, the condition bi is resolvable if there exists a part X accessible by the agent bi i.e. (X; Y ) where Y is a 8
pattern part with E(Y ; V ) = minfE(Y; V ) : a pattern Y g and V = fX 0 : X 0 is a virtual part such that X 0 j= b1 g; such X is a constraint resolution at bi . We rst describe an exemplary set of inner constraints: at A1 : xA;12, at A2 : xA;11, at L1 : xL;10, at L2 : xL;9, at B : xB;8 _ xB;7, at U : xB;8 _ xL;2 ^ xL;9; xB;8 _ xL;2 ^ xL;9, at N : xU;4 _ xL;2 ^xL;10; xU;4 _xL;2^xL;10; xU;4 _xU;5, at P : xN;1 ^xN;2 _xA;2 ^xA;11, at M : xP;1 ^ xP;2 ^ xP;6 _ xA;2 ^ xA;12, at R : xM;4 ^ xM;6 _ xH;3 ^ xH;5 ; xM;1 ^ xM;2 _ xH;5 ; xM;5 _ xH;5 ^ (xH;1 _ xH;2), xM;5 _ xH;1 Let the outer constraint at R be (1; R) xR;1 ^ xR;4 ^ xR;5 ^ xR;7 ^ xR;11 The proper prime implicants of the negotiation function f R are (1; R; H) xM;1 ^ xM;4 ^ xM;5 ^ xH;2 ^ xH;1 ^ xH;3 ^ xH;5 (2; R; H) xM;1 ^ xM;4 ^ xM;5 ^ xH;2 ^ xH;1 ^ xH;3 ^ xH;5 (3; R; H) xM;1 ^ xM;4 ^ xM;5 ^ xH;2 ^ xH;3 ^ xH;5 Let the constraints negotiated by M and H be (1; H) xH;2 ^ xH;1 ^ xH;3 ^ xH;5 (1; M) xM;1 ^ xM;4 ^ xM;5 Continuing, we get constraints for other leaf agents: (1; A2) xA;2 ^ xA;11 (1; L1) xL;10 ^ xL;2 (1; L2) xL;2 ^ xL;9 (1; B) xB;1 ^ xB;2 ^ xB;4 ^ xB;5 ^ xB;7 ^ xB;8 The constraint resolution process leads in the case outlined above to the nal assembly X = YL2YB1 YL3YA2 YA1YH 4 . One may check that X is the only positive outcome of the input (1; R). We will present here the bottom - up communication process. In this process any child sends to its parent a two-component messsage, the rst component is an assembly meeting the negotiated solution to constraints and the second component is the vector of values of rough merological closeness between the assembly and any of pattern objects. We will adopt the following strategy for de ning 0 . For any leaf agent, 0(X; Y ) is calculated from the information system describing elementary parts to which the agent has an access by the formula Y)j 0 (X; Y ) = j IND(X; jAj where A is the attribute set and IND(X; Y ) = fa 2 A : a(X) = a(Y )g. We will adopt the strategy that takes the rough mereological connective F (x; y) = minfx; yg; accordingly, we have E(X; Y ) = minf(X; Y ), (Y; X)g. Any agent, given a pattern Y = Y1Y2 and an assembly X = X1 X2 along with values E(X1 ; Y1), E(X2 ; Y2) submitted by children calculates the mereological distance E(X; Y ) = F (E(X1 ; Y1); E(X2 ; Y2)). The only exception to this rule is 9
the following: the agent M assigns E(X; Y ) = 1 to any pair X; Y of objects if X and Y satisfy one of the following constraints (i) xM;3 ^ xM;5 ^ xM;6 (ii) xM;3 ^ xM;5 ^ xM;6 (expressing the identity of symmetrical objects). The root agent calculates values of E(X; Y ) and selects at random the pattern Y satisfying 1 ? E(X; Y ) = minf1 ? E(X; Y ) : any pattern Y g. Then it assigns the quality of Y as the quality of X. Any leaf agent a selects a part X such that 0:85(X; Y ) where Y is a pattern part satisfying the constraint. For example [0(YL2 ; YLi)] = [0:8; 1; 0:4]. For the constraint (1; R) the root agent applies his best t strategy to (X, [0:4, 1, 0:33, 0:4, 0:33, 0:33, 0:25, 0:25, 0:25, 0:4, 0:4]); and selects Y = YB1 YL2 YL3YA2 YA1 YH 4 . Accordingly, Q(X) = LJS. This work was partially supported by the grant 8-S503-019-06 from State Committee for Scienti c Research (Komitet Badan Naukowych).
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