We investigate a model for rough mereology based reasoning in which things in the universe of mereology are endowed with positive masses. We define mass based rough inclusions and establish its properties. This model does encompass inter alia set theoretical universes of finite sets with masses as cardinalities, probability universes with masses as probabilities of possible events, sets of satisfiable formulas with values of satisfiability. We give an abstract version of the Bayes theorem which does extend the classical Bayes theorem as well as the Łukasiewicz logical version of the Bayes formula. We define the mass-based rough mereological logic (m-RM logic) and we show that m-RM logic contains as a particular case the Łukasiewicz many-valued logic and that m-RM logic becomes the Łukasiewicz logic of probability in the case when masses are satisfiability values of indefinite formulas. We also establish an abstract form of the betweenness relation which has proved itself important in problems of data analysis and behavioral robotics. We point finally to applications in clustering and in the evidence theory.
4. Mass assignment on a mereological universe. The mass based rough mereological logic (m-RM logic) 5. The Bayes formula 6. Rough inclusions induced by mass assignments 7. The Łukasiewicz logic of probability as a particular case of m-RM logic. Relations of m-RM logic to many-valued logics and a generalization of the Łukasiewicz ‘fuzzy’ logic.
In our investigations into problems of data analysis and behavioral robotics as a model for intelligent
agents, we have come at the theory of rough mereology as a useful environment for these investigations.
Rough mereology (or, as Achille Varzi has called it, the fuzzified mereology) rests on the notion of a
rough inclusion (x; y; r), a relation of being a part to a degree. In our studies of rough mereology,
we have applied some forms of rough inclusions, e.g., derived from Archimedean t-norms via their
Hilbertian representations or as residua of continuous t-norms, see, e.g. [
We define the mass based rough mereological logics and show that it translates into the Łukasiewicz
logic of probability calculus in [
We regard this work as a preliminary study into a new area of rough mereology which offers an enriching of the repertoire of rough inclusions with applications in Data Analysis.
We accept here the standard version of mereology as proposed in Les´niewski [
M1 Irreflexive: For each thing x, it is not true that (x; x). M2 Transitive: For each triple x; y; z of things, if (x; y) and (y; z), then (x; z).
which is clearly a partial order on things in U . The basic relation involving the notion of an improper part is the relation of overlapping, Ov(x; y) in symbols, defined as follows.
(x; y) _ x = y;
Ov(x; y) , 9z: (z; x) ^
(z; y):
The notion of overlapping in turn is instrumental in definition of the class operator in the sense of
Les´niewski [
(IR) for each pair x; y of things, if for each thing z such that (z; x) there exists a thing w such that (w; y) and Ov(z; w), then (x; y).
Proposition 2.1. For each thing x, it is true that x = Clsfz : (z; x)g.
We are now in a position to recall here two fusion operators due to Tarski [
(z; y)g x y = Clsfz : (z; x) ^ (z; y)g (1) (2) (3) (4) (5) (6) (7)
The things x; y are disjoint, dis(x; y) in symbols, whenever there is no thing z such that (z; x) and (z; y) (a fortiori, the product of x and y is not defined). Hence, dis(x; y) holds true if and only if
x y = Clsfz 2 U : (z; x) ^ : (z; y)g:
V = ClsU: x = V x: which we call the universal thing.
Helped by V , for each thing x 2 U , we define the complement to x denoted x as follows
We recall that those operations induce in the mereological universe the structure of a complete
Finally, we introduce the operation denoted x ,! y which we call the mereological implication and define as x ,! y = x + y: (8)
(T) x ,! y holds true if and only if x + y = V .
Rough (fuzzified, see Varzi [
Rough inclusions satisfy the following postulates, relative to a given part relation and the induced by relation of an improper part on U :
RINC1 (x; y; 1) if and only if (x; y):
RINC2 if (x; y; 1) then for each thing z and r 2 [0; 1] [if (z; x; r) then (z; y; r)]: This postulate does express a feature of partial containment that a ‘bigger’ thing contains a given thing ‘more’ than a ‘smaller’ thing. It can be called the monotonicity condition for rough inclusions.
RINC3 (x; y; r) and s < r then (x; y; s).
We mention here the fact that rough inclusions have been extensively studied from the point of view
of granular computing and granular algorithms for classification in data in Polkowski and Artiemjew
[
Mass assignment on a mereological universe. The mass based rough
mereological logic (m-RM logic)
We introduce a new scheme for inducement of rough inclusions on the mereological unverse (U; ) by
means of the mass assignment m : (0; 1] ! U which assigns the value of mass m(x) to each thing
x in U . The notion of a mass is known from the evidence theory of Dempster and Schaffer, see, e.g.,
Dempster [
We assume that masses on things in U are positive. We introduce the symbol 0 to denote non-existing
things and we extend the mass assignment by letting
m(0) = 0:
(9)
m-RM-logic We define the logic which settles the properties of mass assignments. This logic is a
generalization, inter alia, of the logic of probability calculus in Łukasiewicz [
(A1) (x = V ) , (m(x) = 1). (A2) (x = 0) , (m(x) = 0). (A3) (x ,! y) ) [m(y) = m(x) + m( x y)].
Theses (T1) m(V ) = 1. By substitution x=V into A1. (T2) m(0) = 0. By substitution x=0 into A2.
(T3) (x = y) ) [m(x) = m(y)]: Proof: x = y is equivalent to (x; y) ^ (y; x), hence, to x(,! y) ^ (y ,! x) which implies by A3 and positiveness of m that m(y) m(x) and m(x) m(y) and finally m(y) = m(x).
(T4) m(x) + m( x) = 1. Proof: substitute x=V in A3. Get m(V ) = m(x) + m( x V ), i.e., 1 = m(x) + m( x) by T1.
(T5) m(x + y) = m(x) + m( x y). Proof: Substitute y=x + y into A3 and get m(x + y) = m(x) + m( x x + x y) = m(x) + m( x y).
(T6) m(y) = m(x y) + m( x y). Proof: Substitute x=x y into A3 and obtain m(y) = m(x y) + m( (x y) y)=m(x y) + m(( x + y) y)= m(x y) + m( x y).
(T7) m(x + y) = m(x) + m(y) m(x y). Proof: hint in Łukasiewicz [
(T8) dis(x; y) ) m(x + y) = m(x) + m(y). Proof: dis(x; y) is equivalent to x y = 0, hence, m(x y) = 0 by T2 and the thesis follows by T6.
(T9) m(x + y) = m(x) + m(y) ) x y = 0. Proof: by T7, m(x y) = 0 so by T2 x y = 0. (T10) m(x) + m( x y) = m(y) ) (x ,! y). Proof: by T8, m(y) = m(x) + m( x y) = m(x + x y) = m(x + y) so x + y = V , i.e. x ,! y holds true.
Def:1: my(x) =
Def 2: I(x; y) , my(x) = my( x): (T14) I(x; y) , mm(x(xy)) = mm(( xxy)) . Proof: by Def 2 and Def. 1.
(T15) I(x; y) , m(x y) = m(x) m(y). Proof: Łukasiewicz (loc.cit.) shows the most elegant proof so we apply his idea here. From mm(x(xy)) = m( x y) , on assumption that I(x; y), it follows that m( x) It follows from above that mass m is additive: m(x + y) = m(x) + m(y) when x and y are disjoint. (10) (11)
The simple Bayes formula has been proved as T13 in the preceding section. Now, we prove its general form (T 16) +i6=j yi yj = 0 ^ +iyi = V ) mz(x) = m(z)mx(z) Pk j=1 m(yj )myj (x)
: ; yk of things in U (the reference set) such that for Proposition 5.1. There exists a set Y = y1; y2; each thing x it is true that m(x) =
k X m(yj )myj (x): j=1 Proof: Let Y be a maximal set of pairwise disjoint things. Then, +Y = V . Given an arbitrarily chosen thing x, there exists the set Y (x) Y such that if Ov(x; yj ) then yj 2 Y for j = 1; 2; ; k. We need to establish the truth of some claims.
Claim 1. x = Clsfx yj : yj 2 Y (x)g: Proof: We recall that x = Clsfz : (z; x)g and we use the inference rule (IR) to establish that things on both sides of Claim 1 are improper elements of each other, hence, they are equal.
Let (z; x); there exists yj in Y (x) such that z overlaps with yj and (z yj ; x yj ) holds true, hence, by (IR), (x; Clsfx yj : yj 2 Y (x)g) is true.
For the converse, let (z; Clsfx yj : yj 2 Y (x)g); there exist things u; v such that (u; z), (u; v), w = x yp for some yp 2 Y (x) , hence (w; x) and, by (IR), (Clsfx yj : yj 2 Y (x)g; x).
It follows finally that x = Clsfx yj : yj 2 Y (x)g. tu Claim 2. Clsfx yj : yj 2 Y (x)g = +yj2Y (x)(x yj ).
Proof:
Claim 3. x = +jk=1(x yj ). Proof:
m(x) =
k X m(x yj ) = j=1 k X m(yj ) myj (x): j=1 mz(x) = m(z) mx(z) Pk j=1 m(yj ) myj (x) : tu tu (12) (13)
For things in a universe U and a mass assignment m, we define the relation m(x; y; r), where x; y 2 U and r 2 [0; 1], as follows
Hence, m(x; y; r) coincides on the left-hand side with already discussed relative mass my(x).We denote with the symbol m(x; y) the greatest value of r possible, i.e., m(x; y; r) ,
For things x; y 2 U , we define the notion of an improper part (x; y) as follows Definition 6.1.
(x; y) , (x y = x): We establish the following properties of the relation and of function . (T17) (x; y) ) m(x; y) = 1. Proof: by definition6.1 and equation (12). (T18) (x; y) ^ (y; x) ) (x = y): Proof: by definition 6.1, x y = x and x y = y, hence, x = y. (T19) (x; x). Proof: x x = x.
(T20) (x; y) ^ (y; z) ) (x; z). Proof: from x y = x and y z = y, we obtain x z = (x y) z=x (y z) = x y = x.
(T18), T(19) and (T20) show that is a genuine notion of an improper part and (x; y) , (x; y) ^ x 6= y is the corresponding notion of a part.
(T21) ( m(x; y) = 1) ) (x; y). Proof: as x y ,! x, we have by A3: m(x) = m(x y) + m(x x y), hence, m(x x y) = 0 so x x y = 0, hence, x = x y, i.e., (x; y).
(T22) (x; y) ) x ,! y. Proof: from x y = x, we obtain x+y = (x y)+y = x+( y)+y = x + V = V .
(T23) (x ,! y) ) (x; y). Proof: from x + y = V it follows that x x + y x = x, i.e., y x = x. (T24) (x; y) , x ,! y , m(x; y) = 1. Proof: by T17, T21, T22, T23.
(T25) (x; y) ) m(x) m(y). Proof: by T22 and A3, we have m(y) = m(x) + m(y ( x)), hence, m(y) m(x).
(T26) [( m(x; y) = 1) ^( m(z; x) = r)] ) ( m(z; y) r): Proof: as (x; y), hence, (z x; z y),
T24, T26 imply that defined by us is a genuine rough inclusion satisfying (RINC1), (RINC2) and in an obvious way (RINC3).
Let us recall the simple form of the Bayes theorem T13 rendered with the help of the rough inclusion : (T13)* m(x; y) = m(y)m(mx)(y;x) .
We infer from this theorem some new facts on the rough inclusion . (T27) (x; y) ) m(y; x) = mm((xy)) . Proof: let m(x; y) = 1 in T13*. (T28) mm((yx;;xy)) = mm((xy)) . Proof. Direct from T13*. (T29) [Transitivity formula] m(x;y) m(;x) mm((zy;;yz)) = mm((zx;;xz)) : Proof: from T28. (T30) m(x; y) = 1 m(x; y). Proof: as x y ,! x, we have by A3 that m(x) = m(x y) + m( (x y) x) = m(x y) + m(( y) x) and thus m(( y) x) = m(x) m(x y). Hence, m(x; y) = m(m((yx)) x) = m(x)m(mx)(x y) = mm((xx)) mm(x(xy)) =1 m(x; y).
and,
M (x; y) = minfx; yg;
P (x; y) = x y; L(x; y) = maxf0; x + y 1g:
Each of those t-norms does induce the implication in the form of its residuum x !T y, where T stands for the corresponding t-norm. When x y, each residuum takes on the value 1, so only the case x > y does require definitions:
The Łukasiewicz logic of probability as a particular case of m-RM
logic. Relations of m-RM logic to many-valued logics and a
generalization of the Łukasiewicz ‘fuzzy’ logic
In Łukasiewicz [
We define the function tr from symbols of arguments and operations of m RM logic onto symbols and operations of the logic of probability. For a thesis of m RM logic, the formula such that tr ) is a thesis of the logic of probability.
For symbols: x tr a; y tr b; m tr w; 1 tr 1; 0 tr 0; V tr 1.
For operations: x tr a; y tr b; x + y tr a + b; xy tr ab; (x; y) tr wa(b): Hence, x ,! y tr a > b. 7.1.
Many-valued ‘fuzzy’ logics
The name for those many-valued logics comes from their importance for Artificial Intelligence and
Machine Learning, see Hajek [
They are induced by t-norms, respectively, the minimum t-norm M (x; y), the product t-norm P (x; y) and the Łukasiewicz t-norm L(x; y) defined as follows on the unit square [0; 1]2 (14) (15) (16) (17) (18) (19) (20) x !M y = y;
y x !P y = x
; x !L y = minf1; 1 x + yg:
Along with negations defined as x = x ! 0, the residua !M ; !P ; !L define logics of, respectively, Goedel, Goguen, Łukasiewicz.
Each of these residua defines the corresponding rough inclusion Goedel, Goguen, Lukasiewicz by
the formula (x; y; r) if and only if x ! y r, see [
m-RM logic against many-valued logics We observe the following relationships showing that m-RM logic coincides to a substantial extent with logics of Goedel, Goguen and Łukasiewicz and it offers a generalization of the last one.
(T31) If (x; y) then Goedel, Goguen, Łukasiewicz rough inclusions, and, the mass based rough inclusion all give the value 1.
(T32) If x; y are independent, i.e., m(x y) = m(x) m(y), then m(x; y) = m(y), i.e., m(x; y) = Goedel(x; y) : on collections of pairwise independent things, the degree of inclusion is given by the Goedel logic; e.g. on collections of pairwise independent events in a finite probability space (or, more generally, on collections of pairwise independent sets).
(T33) If (x; y) then m(y) m(x), hence, m(y; x) = mm((xy)) , i.e., m(y; x) = Goguen(y; x), so when x is an improper part in y with respect to the rough inclusion m, then the reciprocal rough inclusion m(y; x) of y into x is defined also by the Goguen logic.
Finally, we consider the rough mereological implication x ,! y. We have (T34) (i) If (x; y) then m(x ,! y) = 1; (ii) If (y; x) them m(x ,! y) = 1 m(x) + m(y); (iii) If neither (x; y) nor (y; x) then m(x ,! y) = 1 m(x y).
Proofs. For (i): T24. For (ii): m(x ,! y) = m( x + y)=m(V (x y)) = 1 m(x y) = 1 [m(x) m(y)]=1 m(x)+m(y). For (iii): m(x ,! y) = m( x+y)=m(V (x y)) = 1 m(x y).
Letting for x 2 [0; 1]: m(x) = x, we obtain the logic of Łukasiewicz: x = 1 x; x !L y =
minf1; 1 x + yg. The case (iii) was not considered by Łukasiewicz as in his case the set of values was
the linearly ordered unit interval [
The notion of betweenness relation due to Tarski [
We introduce first the notion of distance (x; y) between two things x; y in the universe U . We let (x; y) = m(x y) + m(y x):
We say that the thing z is between things x; y, in symbols Btw(z; x; y), when the following condition is satisfied (B) For each thing w not identical to z,and for each thing u such that (u; w) there exists a thing t such that (t; z) and either (t; x) < (u; x) or (t; y) < (u; y).
Proof: Proof: assume that w 6= x + y so w does not satisfy the class definition for x + y, hence, there exists a thing u such that (u; w) and u does overlap neither with x nor with y. Letting v as x or y, we have: (u; x) = m(u) + m(x) > 0 while (x; x) = 0, similarly in case of y. tu The condition (B) as well as the notion of betweenness can be extended for finite sets of things to the notion GBtw(z; T ), where T is a finite set of things, of the generalized betweenness relation which holds when the codition (GB) is satisfied
(GB) For each thing w 6= z and each thing u with (u; w) there exist a thing v such that (v; z) and a thing t 2 T such that (u; t) > (v; t).
Remark 8.1. In particular cases, important for applications, the mereological sum acquires specific renditions in the context of betweenness.
In behavioral robotics, when mobile robots, or more generally intelligent agents, are modeled as
planar rectangles,the mereological sum of two things a; b is the extent ext(a; b), i.e., the smallest rectangle
containing a and b cf. Polkowski and Os´miałowski [
In the case of information/decision systems, where things are represented by means of their
information sets, i.e., for a system with attributes in the set A and with the values of attributes in the set V , the
information set for a thing u is the set Inf (u) = fa(u) : a 2 Ag, the mereological sum of things u and v
relative to a partition P = fA1; A2g is a thing w such that Inf (w) = fa(u) : a 2 A1g[fa(v) : a 2 A2g
cf. Polkowski [
Granular computing in mass-based rough mereology For a thing x and a real number r 2 [0; 1] called the granularity radius, we define the granule about x of radius r denoted gr(x) as the class gr(x) = Clsfy : m(y; x) rg: (22) From the class definition condition (C2), we obtain the following characterization of the improper part for granules.
Proposition 8.2.
(u; gr(x)) iff there exist things w and v with the properties that (i) m(u; w) = mm((wu)) (ii) m(v; w) = mm((wv)) (iii) m(v; x) r. Proof:
tu (23) (24) The algorithm for checking whether (u; gr(x) 1. Find the ordered set W (u) = fw : m(u; w) = mm((wu)) = fw : m(u w) = m(w)g. 2. For each w 2 W (u) according to the order find the set VW (w) = fv : m(v w) = m(w) ^ m(v; x) rg going in the order of the set fv : (v; x) rg.
Let us consider a possible mechanism for clustering based on mass rough inclusions. To this end, we introduce another distance function (x; y) given by the formula (x; y) = j (x; y)
1 (y; x)j = m(x y) j m(x)
1 m(y) j:
Example 9.1. Consider things in a collection U being landscapes or photographs of a countryland on which we have trees, figures of people, houses. For a particular thing x we assign the mass m(x) as the sum m1(x) + m2(x) + m3(x), where m1(x) = if and only if there are at least 3 trees on x, m2(x) = 1 if and only if there are at least 2 people on x, and m3(x) = 1 if and only if there is at least one house on x. We assume that m(x) is at least 1 for each x in U . Figure 1 shows possible outcomes for pairs of things and values of " for which these things may fall into one cluster. We have three possible types of things: Type I with m=3, Type II with m=2, and, Type III with m=1. We include into x y a unit if and only if both x; y satisfy conditions for this unit, for instance if both x; y have m1 = 1 then m1(x y) = 1. We do not consider in this example the sum operation.
It follows that for " < 31 , clustering makes into clusters things of the same type: cluster 1 with things of Type I, cluster 2 with things of Type II, and, cluster 3 with things of Type III. 10.
In evidence theory (cf. Dempster, loc.cit.), mass assignments are also called basic probability assignments (b.p.a.’s) and they are assigned to all subsets of a set of possible outcomes called the frame of discernment. We illustrate our approach with an example. These values are therefore : M (red; yellow) = 0:54, M (red; green) = 1:0, M (yellow; green) = 0:85. It follows that the maximally independent, having the smallest intersection/dependence on other sets is red; yellow. One decides that the light on the main road at the moment of crossing the crossroads was either red or yellow. Now in this set, the proportion of evidence for red to evidence for yellow is like 0.27:0.5 so the decision is on yellow light. We have introduced the notion of a mass into rough mereology which has allowed us to express the reciprocal relations of partial containment in the form characteristic to the Bayes formula in probability theory. We have expressed the betweenness relation in an abstract mass-based framework. We proposed an application to clustering that allows for inducing various sets of clusters dependent on the threshold distance ".We demonstrated the mass based rough mereological logic which does encompass the Łukasiewicz logic of probability and we extended the ‘fuzzy’, i.e., many-valued logic of Łukasiewicz to the logic based on properties of mass assignments. We hope that this abstract formulation will prove a convenient vehicle for some forms of approximate reasoning to be developed in future. At the end, we proposed a decision procedure involving mass based rough inclusions derived from belief values in evidence theory.
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Available: [8] Pawlak, Z.: Rough Sets: Theoetical Aspects of Reasoning about Data. Kluwer, Dordrecht, 1991. [9] Pawlak, Z.: A rough set view on Bayes’s theorem. Intern. J. Intell. Systems 18(5), pp 478-498, 2003.
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