Rough Sets. V.W. Marek. General introduction and one theorem. Department of Computer Science University of Kentucky. October 2013.

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1 General introduction and one theorem V.W. Marek Department of Computer Science University of Kentucky October 2013

2 What it is about? is a popular formalism for talking about approximations Esp. studied in Poland, Canada, US, China, India Actually, invented by late Zdzisław Pawlak Ties several areas of science: Statistics, Logic, Universal Algebra, Topology, Combinatorics even Functional Analysis Motivated by situations when the language is inadequate to describe collections of objects (I wrote few papers on RS and was a coauthor of the first paper on RS)

3 What it is about, cont d Also I read a paper by Yanfang Liu and William Zhu, of Zhangzhou Normal University Parameterized matroid of rough set and will present one theorem (and its proof) The reason is that, originally, I doubted the result is true, and the proof did not make sense This specific result ties with an important area of Combinatorial Optimization and explains why some algorithms for work This is (unfortunately) the only actual proof that I will present in this series of lectures

4 Plan One-table bags of records, and associated equivalence relation Rough Liu theorem

5 Database background A table is a collection of records, possibly with repetition In other words a bag, not set, of records Let us assume now that we assign to each of these records a unique identifier Then there is an equivalence relation on the set of the identifiers, namely: i 1 i 2 if i 1, i 2 are identifiers of the same record

6 Example Here is a table patients: Id lname fname temp 1 marek victor morek vector marek victor marek victor morek vector (But remember that the id s are NOT the part of data) Here the relation has three equivalence classes: {1, 3}, {2, 5}, and {4}

7 Not every subset is describable If we implement this table in SQL (what is SQL?) the set consisting of records with identifiers 1, 2, and 3, can not be described The point is that from the point of view of SQL, records where we set id 2 and 5 can not be distinguished Only that are unions of equivalence classes of can be described The set {1, 3, 4} can be described by: SELECT FROM patients WHERE lname = marek

8 Two kinds of linguistic inadequacy Say, we have a language for description (think medicine, the original motivation of Pawlak) There may be of objects we can not describe (given that language) It is also possible that there is a description, but it is just too big This happens when we have plenty of attributes and need to perform attribute reduction to get a human-readable description When you do this records may become indistinguishable (Ever heard about Johnson-Lindenstrauss Theorem?)

9 So, we have to approximate X A a1 a2 a3 a4 a5 B b1 b2 b3 b4 b5 There is a largest definable set included in a given set X, often called interior of X, X There is a smallest definable set containing a given set, often called closure of X, X We see them in our figure There is a large number of obvious identities for interior and closure S

10 Topological angle (What about Alexandrov Topology in our context?) It is just that we are not topologists, and think about our objects as database objects This, of course, has consequences; we implement

11 Few important facts An equivalence class of x, [x] is {y : y x} The interior of the set X consists of the union of equivalence classes included in X Closure of the set X consists of the union of equivalence classes that have a nonempty intersection with X There are various characterizations of interior and closure, in various terms One such characterization, by Mirek Truszczynski and myself is that the pair X, X is the best approximation of X in the Kleene ordering of pairs of definable (Kleene ordering is the order of approximations where the lower class goes up and the upper class goes down )

12 Rough, formally Given an equivalence relation in a set U, a rough set determined by a set X such that X U is the pair X, X Then a rough subset of U is a pair determined by any subset X of U Besides of characterization mentioned above, there are other characterizations of rough : in terms of topology, in terms of Boolean Algebras with operators, etc. The person who invented (no longer with us), Professor Zdzisław Pawlak, wrote a often quoted book on the subject There is a journal Transactions on and even a Society There is plenty of conferences on, in all sort of places

13 Matroid Matroid is a combinatorial structure that attempts to capture notions behind concepts such as independent set of vectors in a vector space But also cycle-free subgraphs of an undirected graph Formally, a matroid is a pair A,M where M consists of (some) sub of A and satisfies the following conditions: M If A B and B M then A M If A, B M, A < B then for some x B \ A, A {x} M (This definition of a matroid abstracts out of linearly independent subset of a vector space)

14 , cont d This last property is called Steinitz exchange property and whoever had a class of linear algebra must have heard about it The concept of matroid is one of fundamental combinatorial structures There are many other characterizations of matroids in various terms One important connection of matroids and Computer Science is so-called Rado-Edmonds Theorem that characterizes greedy algorithms in terms of (weighted) matroids (Look up an absolute classic: Witold Lipski, jr., Kombinatoryka dla programistow, ISBN )

15 Why are matroids important? They occur in many places, but the important point is the characterization of Greedy algorithms via matroids Say, we have a set A and a weight function, wt : A R + Weight of set S A is Σ x S wt(x)

16 Rado-Edmonds Theorem We sort the set A according to weights in descending order Rado-Edmonds Theorem tells us that if a family F P(A) is a matroid, and we select greedily (i.e. we initialize X to the empty set and in each step we select fresh maximum weight element x so that X {x} is in F and then set X := X {x}) then we will compute a base of maximum weight (what is a base?) When no fresh x A so that X {x} belongs to F can be found, we return X

17 Rado-Edmonds Theorem, cont d Conversely, if F is not a family of independent of a matroid, then there is a weight function where we will not get a maximum-weight element of F (If you had a serious data-structures course, then certainly these facts were learned - if you were paying attention)

18 associated with rough Let U be a set of objects, and an equivalence relation on U. Let Y U Then Y determines a collection M Y of sub of U namely {A U : A Y} In our simple example, with 1 3, 2 5 and 4 in relation with itself only, The set X = {1, 2, 3} determines the following class M X : empty set. one-element {1}, {2}, {3}, {5} (but not {4}). What about {1, 2}, {1, 3}, {2, 3}? And are there more?

19 and rough, cont d Here is the result of Liu : Let be an equivalence relation in the set U. For every set Y U, the structure M Y is a matroid (We will prove that) Since the structure M Y obviously is closed under sub ( if a set grows(?) smaller, the interior grows smaller ) first two conditions are obvious So now, let us assume that we have two, A, B in M Y, with A < B. We need to find a object x (B \ A) so that A {x} also belongs to M Y

20 Case 1 Some x B \ A has the property that [x] = {x} Specifically, that x is in relation with itself but not with any other object We claim that this specific x has the property that A {x} M Y Indeed, because [x] is a singleton, x / A, A {x} = A {x} Now, A Y (because A M Y ) and also x Y because B M Y, and {x} B Y Thus in this case the matter is easy

21 Case 2 No x B \ A has the property that [x] = {x} Our idea now is to assume that for no x B \ A, A {x} belongs to M Y and work for a contradiction The fact that A {x} / M Y means that A {x} is strictly larger than A But there are only two possibilities: either A {x} is A or it is A [x] The first possibility does not hold - so the other one must hold

22 What does it mean? This means that all y x, y x, are in A! And this happens for all x B \ A Next question we ask if it is possible that for some x, y B \ A, x y, x y If that would be the case then [x] = [y] and [x]\{x} A and also [y]\{y} A But then [x] A, contradicting the fact that x / A

23 What is going on? For each x (B \ A) all the elements y such that x y are in A! Moreover, because we are in Case 2, every x B \ A is in relation with some element of A (actually of A\B) Let us select, for each x (B \ A) one element y such that x y, x y Then, because [x]\{x} A, this function maps B \ A into A (in fact into A\B, because no object in B \ A is to any object in B \ A

24 A bit of combinatorics A\B B\A Figure: The injection of B \ A into A\B Then, because [x]\{x} A, this function maps B \ A into A (in fact into A\B, because no object in B \ A is -related to any different object in B \ A In fact it is an injection of B \ A into A\B! But then B \ A A\B! This contradicts the fact that A < B and completes the argument

25 Few other things Here is a characterization of M Y M Y = {A : A Y} And one more: M Y = {A : (A\Y) = } There are plenty of other similarly easy characterizations of M Y

26 More stuff Discussing with Professor M. Truszczyński of my Department, I learned few things It is quite possible that these things are in many papers of W. Zhu and his coauthors We will present Mirek s suggestions now

27 Selectors Let F be a family of pairwise disjoint nonempty A set Z is a selector for F if for all T F, Z T = 1 Something called Axiom of Choice (what is it?) requires that selectors exist, but if F is a finite family of finite, then no special axioms are needed

28 Family F Y Now, let Y be a subset of U The family F Y is defined as follows: {[x] : [x] Y = } Thus F Y consists of equivalence classes of which are disjoint with Y This family F Y, if nonempty, consists of nonempty only and so, has nonempty selectors

29 Family F Y, cont d All the selectors S for F Y are of the same size, namely F Y Here is what Prof. Truszczyński observed: The maximal in M Y are precisely the of the form U \ S where S is a selector for F Y Therefore the family M Y can be characterizes as follows: M Y = {X : T (T is a selector for F Y and X T = )} (This can be used for an alternative proof of the fact that M Y is a matroid)

30 Conclusions Even though rough are such a fundamental data structure, people still find new and interesting facts In the case of the result we presented there was a new technology (matroids) that we used Maybe it will be useful in further investigations