Foundational and Mathematical Aspects of Rough Set Theory

– p.1/70UNIVERSITÀ DEGLI STUDI DI MILANO–BICOCCADIP. DI INFORMATICA, SISTEMISTICA E COMUNICAZIONE (DISCO)Foundational and Mathematical Aspectsof Rough Set TheoryGianpiero Cattaneo — cattang@disco.unimib.itMay 2009, Milan (Italy)

Talk OutlineMay 2009, Milan (Italy) – p.2/70 Rough Set Theory by Equivalence Relations Rough Set Theory by Similarity Relations Some Metatheoretical (Metaphysical?)Reflections The Abstract Interior–Closure Approach

Part IMay 2009, Milan (Italy) – p.3/70Rough Set Theoryby Equivalence Relations

Some Terminological/Foundational Remark (1)May 2009, Milan (Italy) – p.4/70UI ⊆ U × U Isomorphism They are categorical isomorphic sinceI −→ π I −→ ( )I πI =Iπ ⊆ P(U)So,⎫⎧(U, I) ⎪⎬⎪⎨ (U, π)EquivalenceSpaces⎪⎭ ←− Isomorphic −→ Partition⎪ ⎩Spaces

Some Terminological/Foundational Remark (2)May 2009, Milan (Italy) – p.5/70I know nothing about ontology (and this is a fault of mine!),but A Complete Information system is a mapping (table)F : U × Att → V alwith associated of Indistinguishability relation betweenobjects I ⊆ U × U:(x, y) ∈ I iff ∀a ∈ Att F(x, a) = F(y, a)I is an Equivalence (reflexive, symmetric, and transitive)binary relation on the universe U

Some Terminological/Foundational Remark (3)May 2009, Milan (Italy) – p.6/70On the other hand, An Incomplete Information system is a mapping (table)F : (U × Att) p → V alpartially defined on a subset (U × Att) p ⊆ U × Att Introduced the definition domain of the object x ∈ U asAtt(x) := {a ∈ Att : (x, a) ∈ (U × Att) p } the associated indistinguishability relation is(x, y) ∈ S iff ∀a ∈ Att(x) ∩ Att(y) F(x, a) = F(y, a)

Incomplete IS: in order to be clearerMay 2009, Milan (Italy) – p.7/70Example of an Incomplete Information SystemFlat Price Rooms Down-Town Furnituref 1 high 2 yes *f 2 high * yes nof 3 * 2 yes nof 4 low * no nof 5 low 1 * nof 6 * 1 yes *∗ stands for unknown, missing valueThe pair (f 4 , Rooms) /∈ (U × Att) pThe pair (f 1 , Price) ∈ (U × Att) pFlats f 2 and f 6 are similar: (f 2 , f 6 ) ∈ S(and also (f 6 , f 5 ) ∈ S, but (f 2 , f 5 ) /∈ S !!!)

Some Terminological/Foundational Remark (4)May 2009, Milan (Italy) – p.8/70Differently from the Complete case, in which equivalencespaces are isomorphic to partition spaces,I −→ π I −→ ( I πI)=I

Some Terminological/Foundational Remark (4)Differently from the Complete case, in which equivalencespaces are isomorphic to partition spaces,I −→ π I −→ ( I πI)=IIn the Incomplete Information Systems, The binary relation S is a Similarity relation (|greenreflexive, symmetric, but not transitive) The collection γ S of all similarity granulesS(x) = {y : (y, x) ∈ S} is a covering of U, which isnot a partition The categorical isomorphism does not hold sinceS −→ γ S −→ ( S γS)⊆ SMay 2009, Milan (Italy) – p.8/70

The Classical Pawlak ApproachMay 2009, Milan (Italy) – p.9/70The classical Pawlak Approach to Rough Set Theory isessentially based on an Equivalence SpaceE = 〈X, I〉For any single object x ∈ X it is possible to construct the Equivalence Class or Granule of Knowledgegenerated by xG(x) := {y ∈ X : (y, x) ∈ I}This is the collection of all objects y which cannot be distinguishedfrom x with respect to the knowledge supported by the knowledgebase E.

The “Local” Definition of Lower–Upper ApproximationsMay 2009, Milan (Italy) – p.10/70For any Approximable subset H ∈ P(X) of the universeX, we can “locally” define The Lower Approximation of Hl(H) := {x ∈ X : G(x) ⊆ H} The Upper Approximation of Hu(H) := {x ∈ X : G(x) ∩ H ≠ ∅}Local in the sense that these definitions refer to points xfrom the universe X.

The “Global” Definition of Lower–Upper ApproximationsMay 2009, Milan (Italy) – p.11/70Making reference to the partition space isomorphic to theequivalence spaceP = 〈X, π I (X)〉we can give an alternative, but equivalent, Globaldefinition of Lower and Upper Approximations of Hl(H) = ∪{G ∈ π I (X) : G ⊆ H}u(H) = ∪{G ∈ π I (X) : G ∩ H = ∅}Global in the sense that these definitions refer to subsets(granules) G from the universe (and not to points !!!).

The Definition of Lower–Upper ApproximationsMay 2009, Milan (Italy) – p.12/70Trivially, the Local and the Global definitions of theLower–Upper Approximation Pair coincidel(H) : = {x ∈ X : G(x) ⊆ H}Local= ∪{G ∈ π I (X) : G ⊆ H} Globalu(H) : = {x ∈ X : G(x) ∩ H ≠ ∅}Local= ∪{G ∈ π I (X) : G ∩ H = ∅} Global

Partition Space: Granules and Crisp SetsMay 2009, Milan (Italy) – p.13/70Given a generic Partition Space 〈X, π(X)〉, we canconstruct The Crisp or Precise SetsE := ∪{G i ∈ π(X) : i ∈ I} The collection of all Crisp setsE(X) := {E ∈ P(X) : crisp sets}which is a (complete, atomic) Boolean latticewith respect to the usual set theoretic operations ofunion ∪, intersection ∩, complementation c ,whose atoms are the granules from π(X).

The “Topological” Definitions of Lower–Upper ApproximationsMay 2009, Milan (Italy) – p.14/70On the basis of the Boolean complete lattice E(X) of allcrisp sets the Lower–Upper approximations of H can alsobe equivalently defined asl(H) = ∪{E ∈ E(X) : E ⊆ H}u(H) = ∩{E ∈ E(X) : H ⊆ E}According to Pawlak [Pa92], and following Frege [Fr1903],approximable sets H from P(X) describe Vague Concepts, whereascrisp sets E from E(X) describe Precise Concepts, since theirboundary b(A) = u(A) \ l(A) = ∅, i.e.,“it is a situation in which it is possible to decide if a generic objectbelongs or not to the concept”

The Equivalent Definitions of Lower–Upper ApproximationsMay 2009, Milan (Italy) – p.15/70Summarizing, we collect the equivalent definitions of theLower–Upper Approximations of an approximable subsetH of Xl(H) : = {x ∈ X : G(x) ⊆ H}Local= ∪{G ∈ π I (X) : G ⊆ H} Global= ∪{E ∈ E(X) : E ⊆ H} Topologicalu(H) : = {x ∈ X : G(x) ∩ H ≠ ∅}Local= ∪{G ∈ π I (X) : G ∩ H = ∅} Global= ∩{E ∈ E(X) : H ⊆ E} Topological

The Approximation Space induced by PartitionMay 2009, Milan (Italy) – p.16/70On the basis of a Partition Space 〈X, π(X)〉 we caninduce the following Approximation Space:where〈P(X), l, u〉 〈P(X), ∩, ∪, c , ∅, X〉 is the complete Boolean Lattice ofApproximable Subsets of the universe X. l : P(X) → P(X) is the Lower Approximation Map:H → l(H) = ∪{E ∈ E : E ⊆ H} u : P(X) → P(X) is the Upper Approximation Map:H → u(H) = ∩{E ∈ E(X) : H ⊆ E}

The Rough Approximation MappingMay 2009, Milan (Italy) – p.17/70From the structure of Approximation Space for RoughTheory〈P(X), l, u〉It is possible to deduce the Rough Approximationr : P(X) → E(X) × E(X) defined for every H as theOrdered Crisp–Pairr(H) := ( l(H), u(H) ) ,with l(H) ⊆ H ⊆ u(H)or, equivalently, introduced the Exterior e(H) = u(H) c , asthe Ortho Crisp–Pairr ⊥ (H) := ( l(H), e(H) ) ,with l(H) ⊆ e(H) c [ = u(H) ]

The Kuratowski Topological Approximation Space (2)May 2009, Milan (Italy) – p.18/70The structure of Approximation Space for Rough Theory〈P(X), l, u〉is such that l : P(X) → P(X) and u : P(X) → P(X)satisfy the properties of being an Interior and a ClosureOperator according to the Kuratowski approach totopology(KI1) l(X) = X (KC1) u(∅) = ∅(KI2) l(A) ⊆ A (KC2) A ⊆ u(A)(KI3) l(A ∩ B) = l(A) ∩ l(B) (KC3) u(A) ∪ u(B) = u(A ∪ B)(KI4) l ( l(A) ) = l(A) (KC4) u ( u(A) ) = u(A)

The Kuratowski Topological Approximation Space (3)May 2009, Milan (Italy) – p.19/70Note that in general the interior and closure Kuratowskioperators satisfy(KI2) l(A) ⊆ A (KC2) A ⊆ u(A)As usual it is possible to introduce the two collectionsO(X) : = {O ∈ P(X) : l(O) = O}C(X) : = {C ∈ P(X) : u(C) = C}(Open Sets)(Closed Sets)which in the present case of topology induced by partition,are characterized by the Clopen PropertyO(X) = C(X)(Clopen Sets)

The Partition Approximation SpaceMay 2009, Milan (Italy) – p.20/70On the basis of a Partition Space 〈X, π(X)〉 we haveconstructed the Partition Approximation Space as the pairwhere〈P(X), E(X)〉 〈P(X), ∩, ∪, c , ∅, X〉 is the complete, atomic BooleanLattice of Approximable Subsets of the universe X,whose atoms are the singletons {x} ∈ P(X) 〈E(X), ∩, ∪, c , ∅, X〉 is the complete, atomic Booleanlattice of Crisp Subsets of the universe X,whose atoms are the granules G ∈ π(X)which satisfies the following Three Conditions (leading tothe first Metatheoretical – Metaphysical consideration) ...

The Partition Approximation Space (2) for any Approximable Set H ∈ P(X) there exist twosubsets l(H) and u(H) s.t.l(H) ⊆ H ⊆ u(H)i.e., they are Lower and Upper Approximations of H Both l(H) ∈ E(X) and u(H) ∈ E(X) are Crisp Sets Moreover, if E ∈ E(X) is a crisp lower approximation E ⊆ H, thenE ⊆ l(H)bottom by crisp sets)(l(H) is the best approximation of H from the if F ∈ E(X) is a crisp upper approximation l(H) ⊆ F , thenH ⊆ Fby crisp sets)(u(H) is the best approximation of H from the topMay 2009, Milan (Italy) – p.21/70

First Metatheoretical Considerations... centered in the following three points The Principle of Roughness Coherence: the Lower Approximationof H is less or equal than the approximable set H and the UpperApproximation is greater or equal than Hl(H) ⊆ H ⊆ u(H)The comparison is made by the partial order relation of setinclusion ⊆, generating the lattice of Approximable sets P(X) The Principle of Crispness: it is possible to single out a class ofcrisp sets E(X) such that either the lower and the upperapproximations are Crisp, i.e, they describe Precise situations The lower and upper approximation are not only crisp, but alsothey give the best approximation of any approximable set by crispsetsMay 2009, Milan (Italy) – p.22/70

Category EquivalenceMay 2009, Milan (Italy) – p.23/70Given a Partition Space P = 〈X,π(X)〉 the Kuratowski interior–closure space 〈P(X),l,u〉with Kuratowski interior l : P(X) → P(X) and closureu : P(X) → P(X) operators, and induced collections of OpenSets O(X) and Closed sets C(X)is isomorphic to the Pawlak approximation space 〈P(X), E(X)〉with collection E(X) of Crisp Sets, satisfying the above threeconditionsUnder the conditionO(X) = C(X) = E(X)

Part IIMay 2009, Milan (Italy) – p.24/70From Approximation by EquivalencesTo Approximations by Similarities

A Summary of Equivalence Space RoughnessMay 2009, Milan (Italy) – p.25/70The Approximation Space induced from an Equivalence(Partition) Space is characterized by the following The Approximation Operators are equivalently defined asl(H) : = {x ∈ X : G(x) ⊆ H}Local= ∪{G ∈ π I (X) : G ⊆ H} Global= ∪{E ∈ E(X) : E ⊆ H} Topologicalu(H) : = {x ∈ X : G(x) ∩ H ≠ ∅}Local= ∪{G ∈ π I (X) : G ∩ H = ∅} Global= ∩{E ∈ E(X) : H ⊆ E} Topological The Topological Open, Closed, and Crisp Sets are identical:O(X) = C(X) = E(X)

The Similarity Approach to RoughnessMay 2009, Milan (Italy) – p.26/70The Pawlak Approach describes roughness by Equivalence Relationsinduced from Complete Information SystemsIn the real situations one has to do with Incomplete InformationSystems in which some information is missing or unknown.In this case the induced Similarity Space is a pairwhereS = 〈X, S〉 X is a (nonempty) Universe of objects S ⊆ X × X is a binary Relation of Indistinguishabilitybetween objects(Sm1) x S x (reflexive)(Sm2) x S y implies y S x (symmetric)

The Similarity Approach to Roughness (2)May 2009, Milan (Italy) – p.27/70From the formal point of view, such indistinguishabilityrelations satisfy the Reflexive and Symmetric conditions,but in general the Transitivity does NOT hold.Relations of this kind are called Similarity relation (after Poincaré La science et l’hypothèse,Flammarion, Paris, 1903) and in the context of Kripke semanticsof modal logic (B. F. Chellas, Modal logic, an introduction,Cambridge Univ. Press, 1988) Tolerance relation (after Zeeman, The topology of the brain andvisual perception, Prentice-Hall, 1962) in the context ofincomplete information systems (see for instance L. Polkowski,A. Skowron, and J. Zytkow, Tolerance based rough sets, 1994)

The Similarity Approach to Roughness (3)May 2009, Milan (Italy) – p.28/70In the context of Similarity SpacesS = (X, S)the three definitions of Approximation Operator (the Local,the Global, and the Topological ones) which in the Equivalent Spaces context are equalamong them turn out to be different definitions leading to threedifferent approaches to Similarity ApproximationSpaces for Roughness

Part IIIMay 2009, Milan (Italy) – p.29/70Approximation Spaces from SimilarityThe Local Approach

The Similarity Local ApproachMay 2009, Milan (Italy) – p.30/70Analogously to the Equivalence case, given a SimilaritySpace S = 〈X, S〉 for any single object x ∈ X it is possible to constructthe Similarity Class or Similarity GranuleS(x) := {y ∈ X : (y, x) ∈ S} the Local Lower Approximation of a subset H isl (l) (H) := {x ∈ X : S(x) ⊆ H} the Local Upper Approximation of the same H isu (l) (H) := {x ∈ X : S(x) ∩ H ≠ ∅}

The Similarity Local Approach (2)May 2009, Milan (Italy) – p.31/70The Local Lower and Local Upper approximation mapssatisfy the following (PRATOPOLOGICAL) conditions only(Il1) l (l) (X) = X(Il2) l (l) (A) ⊆ A(Ul1) u (l) (∅) = ∅(Ul2) A ⊆ u (l) (A)(Il3) l (l) (A ∩ B) = l (l) (A) ∩ l (l) (B) (Ul3) l (l) (A) ∪ l (l) (B) = l (l) (A ∪ B)In general the Idempotent Conditions do NOT holdl (l)( l (l) (A) ) ≠l (l) (A) and u (l)( u (l) (A) ) ≠u (l) (A)and this is a “pathological” behavior of the Local approachof similarity spaces with respect to the properties of theequivalence space case

The Similarity Local Approach (3)This means that, with respect to the equivalence case, wecan only state that the Local Rough Approximation of anysubset H, as usual defined asr (l) (H) = ( l (l) (H), u (l) (H) )satisfies the COHERENCE Condition about anApproximation:l (l) (H) ⊆ H ⊆ u (l) (H)i.e., the lower approximation is really lower and the upperapproximation is really upper, with respect to the settheoretic Inclusion criterium !!!But ...May 2009, Milan (Italy) – p.32/70

The Similarity Local Approach (4)May 2009, Milan (Italy) – p.33/70But,in general we cannot state that the local lower approximation l(H) is an Open Setl(H) /∈ O (l) (X) := { O ∈ P(X) : l(O) ( l(O) ) = l(O) } the local upper approximation u(H) is a Closed Setu(H) /∈ C (l) (X) := { C ∈ P(X) : u(C) ( u(C) ) = u(C) }and so it is meaningless to talk about l (l) (H) or u (l) (H) as“the best approximation of H by crisp sets from the bottom or the top”In this local similarity context it is the notion of CRISP Set which ismeaningless !!!

May 2009, Milan (Italy) – p.34/70Some Foundational Considerations(Metatheoretical, Metaphysics???)

The Similarity Local Approach: Foundational SummaryMay 2009, Milan (Italy) – p.35/70With respect to the equivalence space rough theory, the similarity local approach preserves the important notions oflower and upper approximations of a subset H,as really “lower” and “upper” with respect to the partial ordering ofset inclusion, as the right criterium to state correctly what is lowerand what is upper the notion of “crisp set” is definitively lost.In some sense, if according to Pawlak (1992) any subset Hrepresent a vague concept, it is problematic to state if the “lower”and “upper” approximations are performed relatively to someprivileged class of precise concepts (“lower” and “upper” withrespect to what classes of precise, crisp, concepts?)

A Foundational DiscussionMay 2009, Milan (Italy) – p.36/70All these considerations lead to a Foundational (Metaphysical?)discussion, which should be better the argument of some round table,anyway exemplified in some points, the first of which is what I called: the Coherence Requirement, which asserts that any lower andupper approximation of a set A, whatever be their formaldefinitions, must be such thatl(A) ⊆ A ⊆ u(A)where the partial order relation of set inclusion ⊆ furnishes thecriterium to state that l(A) is lower than A, and that u(A) is upper than A.

A Foundational Discussion (2)May 2009, Milan (Italy) – p.37/70The Coherence is a minimal requirement, which at any rate shouldprevent to consider as “lower” and ”upper” approximations insidesome roughness theory definitions which do not satisfy this criteriumFor instance the “local” definitions based on a serial binary relation inwhich one is able to state that for any subset Al(A) ⊆ u(A)but with no link of these two notions with respect to the original setwhich should be approximated by themOf course, nothing against the formal study of these concepts,for instance in the context of the Kripke Model of the logical modalsystem (KD)But ...

A Foundational Discussion (3)May 2009, Milan (Italy) – p.38/70But,in my opinion, it is very hard to claim that l(A) is a lowerand u(A) an upper approximations of a set A if oneobtains that l(A) ⊆ u(A) ⊆ A(as it happens in some concrete examples), or that A ⊆ l(A) ⊆ u(A)(and also this happens in concrete cases), or that A is incomparable either with l(A) or with u(A)or with both(there are concrete examples with this behavior)

A Foundational Discussion (4)If the previous Coherence Principle is, more or less (sometimeshiddenly), acceptedthe second principle is a little bit problematical The Crispness Principle: in order to state the roughness of anapproximable set it is necessary that the lower and upperapproximations are Crisp Sets from some suitable classes of sets Crisp Sets represent, according to the classical Pawlak[Paw1992] approach, precise concepts as different fromApproximable Sets representing vague concepts.This principle allows one to state not only that l(A) (resp., u(A) is thelower (reps., upper) approximation of the vague concept described byA, but also that they are precise approximationsI am well aware that this principle is sometimes (or by someone) hardto be accepted !!!May 2009, Milan (Italy) – p.39/70

A Foundational Discussion (5)May 2009, Milan (Italy) – p.40/70The third is a principle of Comparison of Approximations If in some situation it is possible to give two different roughapproximations of the generic approximable set Ar 1 (A) = (l 1 (A),u 1 (A)) and r 2 (A) = (l 2 (A),u 2 (A)) Then we can say that the approximation 1 is better than theapproximation 2 iff for any set Al 2 (A) ⊆ l 1 (A) ⊆ A ⊆ u 1 (A) ⊆ u 2 (A)and in this case we can write that for any AA ⊑ r 1 (A) ⊑ r 2 (A)

A Foundational Discussion (6)May 2009, Milan (Italy) – p.41/70This in analogy with the very usual situation in which, for instance, with respect to the irrational numberπ ∈ R the rational rough approximationr 1 (π) = (3.14, 3.15) ∈ Q × Qis better than the rational rough approximationr 2 (π) = (1.72, 7.84) ∈ Q × Qfor the simple fact that1.72 ≤ 3.14 ≤ π ≤ 3.15 ≤ 7.84Written asπ ⊑ r 1 (π) = (3.14, 3.15) ⊑ r 2 (π) = (1.72, 7.84)

Part IIIbMay 2009, Milan (Italy) – p.42/70Approximation Spaces from SimilarityThe Global Approach

The Similarity Global ApproachMay 2009, Milan (Italy) – p.43/70The second approach to roughness in similarity spaces(X, S) is the Global oneIn analogy with the partition case one can introduce thefollowing Global Approximations of the approximable setHl (g) (H) : = ∪{S ∈ γ : S ⊆ H}u (g) (H) : = ∪{S ∈ γ : S ∩ H ≠ ∅}Lower ApproxUpper Approxwhich trivially satisfy the coherence principlel (g) (H) ⊆ H ⊆ u (g) (H)

The Similarity Global Approach (2)May 2009, Milan (Italy) – p.44/70The global approximation operators now introducedsatisfy the following conditions:(Ig1) l (g) (X) = X (Ug1) u (g) (∅) = ∅(Ig2) l (g) (A) ⊆ A (Ug2) A ⊆ u (g) (A)(Ig3) NO (Ug3) A ⊆ B =⇒ u (g) (A) ⊆ u (g) (B)(Ig4) l (g)( l (g) (A) ) = l (g) (A) (Ug4) NO The global lower approximation lacks of monotonicity, a verynatural property The global upper approximation is not idempotent, i.e., lacks ofcrispness, in my opinion also important (but not universallyaccepted) condition of roughness

The Similarity Global Approach (3)May 2009, Milan (Italy) – p.45/70Of course, maintaining the global upper approximationoperator u (g) (A), one can introduce its duall (g d) (H) := [ u (g)( H c)] cwith respect to which one has the Pratopological behavior(Ig1) l (g d) (X) = X(Ig2) l (g d) (A) ⊆ A(Ig3) A ⊆ B ⇒ l (g d) (A) ⊆ l (g d) (B)(Ug1) u (g) (∅) = ∅(Ug2) A ⊆ u (g) (A)(Ug3) A ⊆ B ⇒ u (g) (A) ⊆ u (g) (B)The same properties of the roughness local approximations!!!,Moreover, the following set inclusions holdl (g d) (H) ⊆ l (l) (H) ⊆ H ⊆ u (l) (H) ⊆ u (g) (H)

The Similarity Global Approach (4)May 2009, Milan (Italy) – p.46/70FROMl (g d) (H) ⊆ l (l) (H) ⊆ H ⊆ u (l) (H) ⊆ u (g) (H)IF one accepts the Metatheoretical Principle ofApproximation Comparison one has thatH ⊑ ( l (l) (H), u (l) (H) ) ⊑ ( l (g d) (H)u (g) (H) )i.e., the similarity local approximation is better than thesimilarity (modified) global oneBooth lack the Crispness Requirement !• ( l (l) (H), u (l) (H) ) is a First Type of covering approximation• ( l (g d) (H)u (g) (H) ) is a Second Type of covering approximation

Part IIIcMay 2009, Milan (Italy) – p.47/70Approximation Spaces from Similarity(The Tarski Topological Approach)

Covering Spaces Induced by SimilarityMay 2009, Milan (Italy) – p.48/70Any Similarity Space (X, S) induces a Covering Space,as a pair (X, γ) consisting of a nonempty finite set X, the universe of the discourse,whose elements are the objects a covering γ of the universe X, i.e., a family ofnonempty subsets of X s.t.(Co) ∪ {A : A ∈ γ} = X (Covering)

Covering Spaces Induced by SimilarityMay 2009, Milan (Italy) – p.48/70Any Similarity Space (X, S) induces a Covering Space,as a pair (X, γ) consisting of a nonempty finite set X, the universe of the discourse,whose elements are the objects a covering γ of the universe X, i.e., a family ofnonempty subsets of X s.t.(Co) ∪ {A : A ∈ γ} = X (Covering)Note that a covering is an Open Base for a Topology on X if besidesthe condition (Co) also the further holds(To)If B i ∩ B j ≠ ∅, then B i ∩ B j = ∪ k C kwith C k ∈ B(Open Basis)

The Covering Pseudo–TopologyMay 2009, Milan (Italy) – p.49/70Given a Covering γ for the universe XA set O ∈ P(X) is Pseudo–Open (Lower Crisp) iff∃{B j ∈ γ : j ∈ J} s.t.O = ∪{B j : j ∈ J}A set C ∈ P(X) is Pseudo–Closed (Upper Crisp) iff∃{B d ∈ γ : d ∈ D} s.t.C = ∩{(B d ) c : d ∈ D}Properties: The empty set ∅ and the universe X are pseudo–clopen The family of pseudo–open sets O γ (X) is closed with respect toarbitrary set union The family of pseudo–closed C γ (X) is closed with respect toarbitrary set intersection

Covering ApproximationsMay 2009, Milan (Italy) – p.50/70The T–approximations by the covering γ of a subsetA ⊆ X is defined as (and compared with the globalapproach):l (T) (A) = ∪{O ∈ O γ (X) : O ⊆ A}= ∪{B ∈ γ : B ⊆ A}= l (g) (A)u (T) (A) = ∩{C ∈ C γ (X) : A ⊆ C} (Pseudo–Closed !!!)u (g) (A) = ∪{B ∈ γ : B ∩ A ≠ ∅} (Pseudo–Open !!!)For any subset A ⊆ X it isl (g) (A) = l (T) (A) ⊆ A ⊆ u (T) (A) ⊆u (g) (A)

The Three Types Covering ApproximationsFurther, for any subset A ⊆ X the inclusions hold:l (g d) (A) ⊆ l (l) (A) ⊆l (T) (A) ⊆ A ⊆ u (T) (A)⊆ u (l) (A) ⊆ u (g) (A)Generating a Three Types of covering approximations of a generic A: the First Type Pseudo-Topological rough approximationr γ(T) (A) = (l γ(T) (A),u γ(T) (A)),which is the BEST approximation of A the Second Type Local rough approximationr γ (l) (A) = (l γ (l) (A),u γ (l) (A)),which is an “Intermediate” approximation of A the Third Type Global rough approximationr γ (g) (A) = (l (g d)γ (A),u γ (g) (A)),which is the WORST approximation of AMay 2009, Milan (Italy) – p.51/70

Tarski Approximation SpacesMay 2009, Milan (Italy) – p.52/70Without entering in technical details, we mention here thatthe Tarski Closure–Interior (pseudo–topological) approach toroughness theory on the Power Set Boolean lattice structure〈P(X), ∪, ∩, c 〉 of a universe Xis a Concrete Model of the Abstract Theory of Tarski Closure–Interiorbased on Lattice structure 〈Σ, ∧, ∨, ′ 〉, according to the interpretationABSTRACTCONCRETE UNIVERSEa ∈ Σ =⇒ A ∈ P(X)a ≤ b =⇒ A ⊆ Ba ∧ b =⇒ A ∩ Ba ∨ b =⇒ A ∪ Ba ′ =⇒ A c = X \ A

Part IVMay 2009, Milan (Italy) – p.53/70Abstract Approximation SpacesThe Tarski Interior–Closure Lattices

Tarski Interior Operation on LatticesMay 2009, Milan (Italy) – p.54/70Let 〈Σ, ∧, ∨, 0, 1〉 be a bounded lattice.Then a Tarski Interior Operation on Σ is a mappingo : Σ ↦→ Σ satisfying the following conditions:(I1) 1 o = 1 (normalized)(I2) a o ≤ a (decreasing)(I3) a o = a oo (idempotent)(I4) (a ∧ b) o ≤ a o ∧ b o (sub–multiplicative)

Tarski Interior Operation on LatticesMay 2009, Milan (Italy) – p.54/70Let 〈Σ, ∧, ∨, 0, 1〉 be a bounded lattice.Then a Tarski Interior Operation on Σ is a mappingo : Σ ↦→ Σ satisfying the following conditions:(I1) 1 o = 1 (normalized)(I2) a o ≤ a (decreasing)(I3) a o = a oo (idempotent)(I4) (a ∧ b) o ≤ a o ∧ b o (sub–multiplicative)The sub–multiplicative property (I4) is equivalent to the monotonicitycondition: a ≤ b implies a o ≤ b o .

Tarski Interior Operation on LatticesMay 2009, Milan (Italy) – p.54/70Let 〈Σ, ∧, ∨, 0, 1〉 be a bounded lattice.Then a Tarski Interior Operation on Σ is a mappingo : Σ ↦→ Σ satisfying the following conditions:(I1) 1 o = 1 (normalized)(I2) a o ≤ a (decreasing)(I3) a o = a oo (idempotent)(I4) (a ∧ b) o ≤ a o ∧ b o (sub–multiplicative)The sub–multiplicative property (I4) is equivalent to the monotonicitycondition: a ≤ b implies a o ≤ b o .The subset of open elements is the collection of all elements whichare equal to their interior:O(Σ) = {a ∈ Σ : a = a o } ∋ 0, 1

Tarski Closure Operation on LatticesMay 2009, Milan (Italy) – p.55/70Let 〈Σ, ∧, ∨, 0, 1〉 be a bounded lattice.Then a Tarski Closure Operation on Σ is a mapping∗ : Σ ↦→ Σ satisfying the following conditions:(C1) 0 ∗ = 0 (normalized)(C2) a ≤ a ∗ (increasing)(C3) a ∗ = a ∗∗ (idempotent)(C4) a ∗ ∨ b ∗ ≤ (a ∨ b) ∗ (sub–additive)

Tarski Closure Operation on LatticesLet 〈Σ, ∧, ∨, 0, 1〉 be a bounded lattice.Then a Tarski Closure Operation on Σ is a mapping∗ : Σ ↦→ Σ satisfying the following conditions:(C1) 0 ∗ = 0 (normalized)(C2) a ≤ a ∗ (increasing)(C3) a ∗ = a ∗∗ (idempotent)(C4) a ∗ ∨ b ∗ ≤ (a ∨ b) ∗ (sub–additive)The sub–additive property (C4) is equivalent to the monotonicitycondition: a ≤ b implies a ∗ ≤ b ∗ May 2009, Milan (Italy) – p.55/70

Tarski Closure Operation on LatticesMay 2009, Milan (Italy) – p.55/70Let 〈Σ, ∧, ∨, 0, 1〉 be a bounded lattice.Then a Tarski Closure Operation on Σ is a mapping∗ : Σ ↦→ Σ satisfying the following conditions:(C1) 0 ∗ = 0 (normalized)(C2) a ≤ a ∗ (increasing)(C3) a ∗ = a ∗∗ (idempotent)(C4) a ∗ ∨ b ∗ ≤ (a ∨ b) ∗ (sub–additive)The sub–additive property (C4) is equivalent to the monotonicitycondition:a ≤ b implies a ∗ ≤ b ∗The subset of closed elements is the collection of all elements whichare equal to their closure:C(Σ) = {a ∈ Σ : a = a ∗ } ∋ 0, 1

A Categorical Equivalence (Isomorphism)May 2009, Milan (Italy) – p.56/70The Tarski Interior–Closure lattice structure 〈Σ, o , ∗ 〉based on the (bounded) lattice ( Σ, ∧, ∨, 0, 1 ) , equipped with an interior o and a closure ∗ operation, satisfying Tarski axioms (I1)–(I4) and (C1)–(C4).is Categorically Equivalent toThe Approximation Space structure 〈Σ, L(Σ), U(Σ)〉 based on the same (bounded) lattice ( Σ, ∧, ∨, 0, 1 ) , two sub–lattices of lower crisp L(Σ) and upper crispU(Σ) elements.Where ...

Abstract Approximation SpaceMay 2009, Milan (Italy) – p.57/70Where:R := 〈Σ, L(Σ), U(Σ)〉 〈Σ, ∧, ∨, 0, 1〉 is a complete (bounded) lattice L(Σ) and U(Σ) are sub–lattices of Σ Elements from Σ are interpreted as impreciseconcepts, data, etc., and are said to be theapproximable elements Elements from L(Σ) and U(Σ) represent lower andupper crisp, precise knowledge, respectivelyUnder the following (Ax1) and (Ax2) Assumptions ...

Rough Approximation Space: AxiomsMay 2009, Milan (Italy) – p.58/70(Ax1) For any approximable element x ∈ Σ, there exists one elementl(x) such that:(In1) l(x) ≤ x (Lower Coherence)(In2) l(x) ∈ L(Σ) (Lower Crispness)(In3) ∀α ∈ L(Σ), (α ≤ x implies α ≤ l(x))

Rough Approximation Space: AxiomsMay 2009, Milan (Italy) – p.58/70(Ax1) For any approximable element x ∈ Σ, there exists one elementl(x) such that:(In1) l(x) ≤ x (Lower Coherence)(In2) l(x) ∈ L(Σ) (Lower Crispness)(In3) ∀α ∈ L(Σ), (α ≤ x implies α ≤ l(x))(Ax2) For any approximable element x ∈ Σ, there exists one elementu(x) such that:(Up1) x ≤ u(x) (Upper Coherence)(Up2) u(x) ∈ U(Σ) (Upper Crispness)(Up3) ∀γ ∈ U(Σ), (x ≤ γ implies u(x) ≤ γ)

Rough Approximation Space: AxiomsMay 2009, Milan (Italy) – p.58/70(Ax1) For any approximable element x ∈ Σ, there exists one elementl(x) such that:(In1) l(x) ≤ x (Lower Coherence)(In2) l(x) ∈ L(Σ) (Lower Crispness)(In3) ∀α ∈ L(Σ), (α ≤ x implies α ≤ l(x))(Ax2) For any approximable element x ∈ Σ, there exists one elementu(x) such that:(Up1) x ≤ u(x) (Upper Coherence)(Up2) u(x) ∈ U(Σ) (Upper Crispness)(Up3) ∀γ ∈ U(Σ), (x ≤ γ implies u(x) ≤ γ)Therefore, l(x) [resp., u(x)] is the best approximation of the “vague”,“imprecise”, “uncertain” approximable element x from the bottom [resp., top]by lower [resp., upper] “precise”, “crisp” elements.

A Hierarchy of Closure LatticesMay 2009, Milan (Italy) – p.59/70A hierarchy of Closure Lattices under the basic conditions(C 1 ) 0 ∗ = 0 and (C 2 ) a ≤ a ∗ (Coherence) The weakest Tarski Closure Lattice:(C3) a ∗ = a ∗∗ (idempotent)(C4T) a ∗ ∨ b ∗ ≤ (a ∨ b) ∗ (sub–additive) The “intermediate” Kuratowski Closure Lattice:(C3) a ∗ = a ∗∗ (idempotent)(C4K) a ∗ ∨ b ∗ = (a ∨ b) ∗ (additive) The strongest Halmos Closure Lattice:(C3H) a o = a o ∗ (interconnection)(C4K) a ∗ ∨ b ∗ = (a ∨ b) ∗ (additive)

Summary of the Concrete Models of Approximation SpacesMay 2009, Milan (Italy) – p.60/70Concrete P(X)Covering Spaces⇑Topological Spaces⇑Partition SpacesModel of−−−−−→Model of−−−−−→Model of−−−−−→Model of−−−−−→Abstract Σ{Tarski Closurea ∗ ∨ b ∗ ≤ (a ∨ b) ∗{Kuratowski Closurea ∗ ∨ b ∗ = (a ∨ b) ∗{Halmos closurea o = a o ∗

Two Others Similarity Approximation TypesMay 2009, Milan (Italy) – p.61/70For the lake of time, I omitted to mention two otherapproaches: W. Zhu, Topological approaches to covering roughsets, Information Sciences 177 (2007) 1499–1508 Y. Y. Yao, Relational interpretations of neighborhoodoperators ..., Information Sciences 111 (1998)239–259arriving to Five Types of covering approximationsThe Zhu approach is incomparable with the Tarski one proposed bymyself, but it is better than the local one, summarized by⎧ ⎫ ⎧ ⎫⎨l (gd) (A) ⊆ l (l) l (T) (A) ⎬ ⎨(A) ⊆⎩l (Z) (A) ⎭ ⊆ A ⊆ u (T) (A) ⎬⎩u (Z) (A) ⎭ ⊆ u(l) (A) ⊆ u (g) (A)

Two Others Similarity Approximation Types (2)May 2009, Milan (Italy) – p.62/70The Yao one is “sufficiently” ⊑ bad. Example:The Universe X = {1, 2, 3, 4, 5}The Covering γ = {{1, 2}, {2, 3, 4}, {4, 5}}• The Pomikala (1987) neighborhood of x:N(x) : = ∪{S ∈ γ : x ∈ S}• The Yao (1998) “local” approximations of A:l (y)γ (A) = {x ∈ X : N(x) ⊆ A}u (y)γ (A) = {x ∈ X : N(x) ∩ A ≠ ∅}The “approximations” chain of the set A = {3, 4, 5}:l (y)γ (A) ⊂ l (g d)γ(A) ⊂ l (T)γ(A) ⊆ A ⊆ u γ(T) (A) ⊂ u γ (g) (A) = u (y)γ (A)

Canonical Generation of Partition by a CoveringMay 2009, Milan (Italy) – p.63/70Canonical procedure to generate a Partition from a Covering of Xγ(X) = {K 1 ,K 2 ,...,K N } Let us construct the anti–covering by the set complement ofgranulesγ ′ (X) = {K1,K c 2,...,K c N}c Let us form the super-coveringγ s (X) = {K 1 ,K 2 ,...,K N ; K c 1,K c 2,...,K c N} For any point x ∈ X let us construct the Super–GranuleObtaining in this way a Partition !!![x] s = ∩{S ∈ γ s (X) : x ∈ S}

Canonical Generation of Partition by a Covering (2)May 2009, Milan (Italy) – p.64/70CBCovering {A,B,C}AXCAXBc cCovering {A,B,C,A , B ,Cc}PartitionThe lower l (p) (A) and upper u (p) (A) (6th Type) approximationsgenerated in this way are the absolutely best ones:⎧ ⎫⎧ ⎫⎨l (T) (A) ⎬⎨l (l) (A) ⊆⎩l (Z) (A) ⎭ ⊆ u (T) (A) ⎬l(p) (A) ⊆ A ⊆ u (p) (A) ⊆⎩u (Z) (A) ⎭ ⊆ u(l) (A)

Part VMay 2009, Milan (Italy) – p.65/70Local and Tarski Dynamical Evolutions

Local and Tarski Dynamical Evolutions (1)May 2009, Milan (Italy) – p.66/70Let us consider a dynamical evolution (increase ofknowledge) of an Incomplete Information System in which the set of objects X and the set of attributes Att(X)stay invariant in passing from the situation t 1 to thesituation t 2 (t 1 → t 2 ) but in which there is an increase of knowledge in thesense that some missing information ∗ at time t 1transforms in a known value at time t 2 , and all theknown (object–attribute) values remain invariantl (l)t 1(H) ⊆l (l)Best Approximation{ }} {l (T)t 1(H) ⊆ H ⊆ u (T)t 1(H) ⊆ u (l)t 1(H)t 2(H) ⊆ l (T)t 2(H) ⊆ H ⊆ u (T)t 2(H)} {{ }Best Approximation⊆ u (l)t 2(H)

Local and Tarski Dynamical Evolutions: (Cattaneo& Ciucci, LNCS 2004)If during the time evolution t 1 → t 2 there is an increasing ofknowledge (i.e., some ∗ transforms in a known value), then At any time t the Tarski Topological approximation is always betterthan the local onel (l)t (H) ⊆ l (T)t (H) ⊆ H ⊆ u (T)t (H)} {{ }Best Approximation⊆ u (l)t (H) The “local” lower approximation increases and the “local” upperapproximation decreases:l (l)t 1(H) ⊆ l (l)t 2(H) ⊆ H ⊆ u (l)t 2(H) ⊇ u (l)t 1(H) The variation in time of the “Tarski” lower and upperapproximations is unpredictable (there are example of increasingan example of decreasing, without any evident regularity)May 2009, Milan (Italy) – p.67/70

Local and Tarski Dynamical Evolutions: SummaryMay 2009, Milan (Italy) – p.68/70 increase of local lower approximation decrease of local upper approximation unpredictability of Tarski topological approximations but in some sense, the local situation gives an upper and a lowercontrol of the Topological dynamics, which is always the best oneat any time instant r (T)t (A) ⊑ r (l)t (A)U_R(H)U_#(H)HL_#(H)HtL_R(H)

BookMay 2009, Milan (Italy) – p.69/70G. CATTANEO, Rough Bookhttp://www.fislab.disco.unimib.it/Books/RGH2 Book-RGH2-Index Book-RGH2-Part1 Book-RGH2-Part?X? (in preparation) Book-RGH2-Bib

THE ENDMay 2009, Milan (Italy) – p.70/70