Approximation and Pretopologies

Transcription

1 Piero Pagliani Approximation and Pretopologies

2 Approximation Spaces given a set U given a relation R which structures U (U), lr, ur R is an equivalence relation lr is an interior operator of a 0-dimensional topological space ur is a closure operator of a 0-dimensional topological space

3 Approximation and Pretopologies First part: Observations and Pre-topological Approximation Operators Second part: Dynamic Property Systems and Pre-topological Approximation Operators Third part: Formal Topological Systems and Pre-topological Approximation Operators

4 The basic ingredients to represent objects through properties: a map from objects to properties a map from properties to objects A set of properties M f g G A set of objects Gegenstand result of the factorisation through f and g G Same set of objects Objekt

5 An ideal situation: f is an injective function Objects a a G M b b Observable properties b a b a M f RETRACTION G 1 G G

6 The best model of a horse is a horse? (it must be injective) Horse-ness f M property RETRACTION G 1 G G Modelling means abstraction Abstraction means loosing something

7 An almost ideal situation: f is a surjective function Objects a G M b Observable properties a Sorts (fibres, stalks) a b a G SECTION f (it must be surjective) M 1 M M

8 The basic picture Objects and properties are linked by means of an arbitrary fulfillment relation G M G a a M b b b a b a We call G, M, a Property System or a Basic Pair

9 Observations, Property Systems and Attribute Systems R p 1 p 2 p 3 p 4 a b c d e a Property system R a 1 a 2 a 3 a 4 a 5 a 3 b 1/ b 3 b 1/ c 1 a d 1 c 1/ e 3 b an Attribute system Observation systems

10 Observation and substantiation G obs (observations) M Substance neighbourhoods a a b b Observation neighbourhoods b obs(a ) sub(b ) a b a sub (substance) m obs(g) if and only if g m if and only if g sub(m)

11 Approximations and best approximations. 1 A subset of objects O fulfils a subset of properties P (M) A subset of properties P describes a subset of objects O obs sub (G) (G) Usually, sub(obs(o)) O However, is sub(obs(o)) O an approximation of O by means of the set of properties obs(o)? No, because obs(sub(obs(o))) obs(o). That is, O is defined by means of properties which do not have any essential relationship with the set obs(o).

12 Approximations and best approximations. 2 How to define two operators φ and ψ by means of obs and sub (i. e. ) such that the following holds: (M) φ ψ (G) ( ) la ( ua ) (G)

13 Towards a solution. 1 Operators must fulfil maximality (minimality), i. e. limit, properties best (largest) lower approximation la(x) boundary region X { best (least) upper approximation ua(x)

14 Towards a solution. 2 Maximality (minimality), implies isotonicity If the approximations are not defined by means of isotonic operators, then the results could be incomparable: we could hardly speak of a best approximation

15 Continuity and approximations: 1 Let ψ be isotonic and Y G. If ψ 1 ( Y) belongs to ψ 1 ( Y), then it is the least X s. t. ψ(x) Y i. e. the best approximation from above of Y via ψ. But if this is the case, then ψ 1 ( Y)= X, for X= ψ 1 ( Y) that is, the pre-image of a principal filter is a principal filter Otherwise stated: ψ is lower-residuated Dually, if ψ is upper-residuated, then the pre-image of a principal ideal is a principal ideal and ψ 1 ( Y)) is the largest X such that ψ (X) Y i. e. the best approximation from below of Y via ψ.

16 Continuity and approximations: 2 If ψ is isotonic and lower-residuated, then set ψ =min(ψ 1 ( Y)) ψ is called the lower residual of ψ If ψ is isotonic and upper-residuated, then set ψ + =max(ψ 1 ( Y)) ψ + is called the upper residual of ψ

17 Continuity and approximations: 3 φ (G) (M) ( ) la ( ua ) (G) If ψ is isotonic and lower residuated, set φ=ψ. Then ψφ(y) Y and for any other X such that ψ(x) Y, φ(y) X ψ If ψ is isotonic and upper-residuated, set φ = ψ +. Then ψφ(y) Y and for any other X such that ψ(x) Y, φ(y) X

18 Approximations and Galois Adjunctions. 1 Let O and O be two ordered sets. Let : O O and : O O, such that p O, p O, (p ) p if and only if p (p) Then is called the lower adjoint of and is called the upper adjoint of O O The pair, is called a Galois adjunction or an axiality and the following holds: is a closure operator is an interior operator If O O op, then, is called a Galois connection or a polarity. In this case, both and are closure operators.

19 Approximations and Galois Adjunctions. 2 φ (G) (M) ( ) la ( ua ) ψ (G) If φ = ψ, then ψ = φ + and we have (G) φ ψ (M) Problem: How one can define a Galois adjunction on an Observation System, in a natural way?

20 Basic perception constructors. 1 Let us extend the constructors obs and sub from tokens to types e : (M) (G) ; e (Y)={g G : m(m Y g sub(m))} existential extension of sub Extensional operators [[e]]: (M) (G) ; [[e]](y)={g G : m(m Y g sub(m))} universal extension of sub [e]: (M) (G) ; [e](y)={g G : m(g sub(m) m Y)} co-universal extension of sub i : (G) (M) ; i (X)={m M : g(g X m obs(g))} existential extension of obs Intensional operators [[i]]: (G) (M) ; [[i]](x)={m M : g(g X m obs(g))} universal extension of obs [i]: (G) (M) ; [i](x)={m M : g(m obs(g) g X)} co-universal extension of obs

21 Basic perception constructors. 2 [[e]]({b, b }) [e]({b, b, b }) e ({b, b, b }) G a a a a M b b b b [i]({a, a }) [[i]]({a, a }) i ({a, a }) i : (G) (M) ; i (X)={m M : g(g X m obs(g))} existential extension of obs [[i]]: (G) (M) ; [[i]](x)={m M : g(g X m obs(g))} universal extension of obs [i]: (G) (M) ; [i](x)={m M : g(m obs(g) g X)} co-universal extension of obs

22 Intuitive and modal reading of the basic perception constructors it is possible for elements of X to fulfil b to fulfil b it is necessary to be in X to fulfil b it is sufficient to be in X <i >(X) b [i](x) b [[i]](x) b X X X at least one there is an example of elements of X which fulfils b at most all there are not examples of elements outside X which fulfill b at least all there are not examples of elements in X which do not fulfill b

23 Adjunction properties of the basic perception constructors i (X) Y if and only if X [e](y ) e (Y) X if and only if Y [i](x ) [[e]] (Y) X if and only if [[i]](x ) Y for all X G, for all Y M adjunction lower adjoint additive isotonic i [e] e [i] upper adjoint multiplicative isotonic Galois [ ] is an interior operator [ ] is an closure operator connection anti-multiplicative anti-isotonic [[e]] op [[i]] anti-multiplicative anti-isotonic

24 Combining adjoint constructors: perception operators modal-style operators Qualitative Data Analysis? intent-extent operators Formal Concept Analysis int : (G) (G) ; int (X)= e [i](x) cl : (G) (G) ; cl (X)= [e] i (X) C : (M) (M) ; C (Y)= i [e] (Y) A : (M) (M) ; A (Y)= [i] e (Y) est : (G) (G) ; est (X)= [[e]][[i]] (X) ITS : (M) (M) ; ITS (Y)= [[i]][[e]] (Y) int({a,a }) cl({a, a }) G a a M b b b A ({b, b }) ITS({b, b }) est({a, a }) a b C ({b, b }) a

25 But what do int, cl, C and A represent, actually? Preliminary question: what does M represent? Each element m of M represents the set e (m) of elements of G which fulfil property m. Hence m collects elements of G by means of the nearness relation to fulfil m. m is a formal neighbourhood a proxy for a subset of G [i](x) is the set of formal neighbourhoods m included in X, i. e. such that e (m) X e [i](x) is the extension of such neighbourhoods. Therefore: e [i](x) denotes in a formal way the interior of X. i (X) is the set of formal neighbourhoods m with non void intersection with X. i. e. such that X e (m) [e] i (X) is the extension of all and only such neighbourhoods. Therefore: [e] i (X) denotes in a formal way the closure of X.

26 dual (Pre) topological properties of the perception operators CONCRETE open : <e>[i] int : <e> symmetric FORMAL closed <i>[e] C : <i> : Straightforwardly from the adjunction properties between the basic constructors dual We obtain G. Sambin s results on Formal Topology (point-free topology) : [e] symmetric [i] : : [e]<i> cl closed CONCRETE [i]<e> A : open FORMAL

27 Algebraic properties of the perception operators Given the following set of fixpoints of the operators one obtains the following lattices

28 Example P b b 1 b 2 b 3 a a a a

29 Approximation properties of perception operators e [i](x) X [e] i (X ) i. e. int (X) X cl (X) with maximality, respectively, minimality property with respect to inclusion for all X G (immediate from the adjunction properties) int is a pre-topological lower approximation operator cl is a pre-topological upper approximation operator

30 Choosing the initial perception act. Information quanta J. L. Bell s approach to Quantum Logic: any object g is perceived together with all the elements which are indiscernible from g in the field of perception. Hence any object is a location and it is perceived embedded in a quantum at that location: minimum perceptibilium at that location. Orthologic, forcing and the manifestation of attributes. We take this phenomenological approach and claim that any object g is perceived together with all the elements which manifest at least all the same properties as g: information quantum at g. g Q g if and only if g fulfils at least all the properties fulfilled by g: i ({g}) i ({g })

31 Information quantum relation and Information quantum systems Given a Property System S g, g R S if and only if g Q g Q(S) = G, G, R S information quantum relation reflexive transitive Information quantum relation system it is a property system!! If g, g R S then g Q g, thus i (g) i (g ). Hence g is less defined than g in S. Indeed, R S is a (anti)specialisation preorder in the usual topological sense.

32 Adjuncion and algebraic properties of i-quantum operators In Q(S) the sets of fixpoints of adjoint i-quantum operators coincide Let cl, int, C and A be induced by Q(S). Then cl = i int = [i] C = [e] A = e cl A C int Since cl is lower adjoint, cl is additive, hence topological Since int is upper adjoint, int is multiplicative, hence topological Similarly for A and C

33 Example P b b 1 b 2 b 3 a a a a R P a a 1 a 2 a 3 a a a a Sat cl (Q(P))

34 Topological approximation operators Let cl, int, C and A in Q(S). Then cl = i : inverse upper approximation cl(x) = set of elements approximated by some member of X int = [i]: inverse lower approximation int(x) = set of elements approximated just by members of X C = [e]: direct lower approximation C (X) = set of elements specialised just by members of X A = e : direct upper approximation A(X) = set of elements specialised by some member of X

35 A particular case If S is an Attribute System, its nominalisation is a Perception System N(S) such that: R N(S) is an equivalence relation and in Q(N(S)) cl(x) = upper approximation of X int(x) = lower approximation of X in the sense of Pawlak s classical Rough Set Theory Indeed, in an Attribute System to fulfill at least the same properties is tantamount to saying to have the same attribute value, which induces an equivalence relation The same is true of dichotomic Property Systems

36 Second part: Dynamic Property Systems and Pre-topological Approximation Operators

37 Z. Pawlak Approximation Spaces <U, E> for E an equivalence relation T. Y. Lin & Y. Y. Yao Neighbourhood Spaces <U, R> for R any binary relation A. Skowron & J. Stepaniuk P. Pagliani Approximation of relations <U, {E i } i I > for {E i } i I a family of equivalence relations P. Pagliani Dynamic Approximation Spaces <U, {R i } i I > for {R i } i I a family of binary relations

38 Sources of dynamics 1) Evolution of the observation process over time 2) Change of context 3) 1+2 R time 1 time 2 R 2 time m R m context 1 R 1 1 context 2 R 12 context n R 1n

39 Time A layered observation system Dynamic Rough Sets Object 1 Object 2 Object 3 Object n

40 Why more than one neighbourhood? More than 1 context More than 1 neighbourhood Context U : applicable criteria (metrics) of type T Context N x f(y) f(w) f(x) f(z) U: object level y w w x z q z z y w w x Neighbourhood of x z q z z

41 What if more than one neighbourhood? More than 1 neighbourhood More neighbourhood families Neighbourhood families Neighbourhood systems Neighbourhood of x Neighbourhood of y

42 Core maps and vicinity maps Let us denote n(x) with N x N x N y N z A x, y, z G(A)

43 Sectioning topological systems For the sake of semplicity, let U =U and let f be the identity function. One can distinguish the following properties, which define a topological space, for any x U, A U: 1. U N x 0. N x Id. if x G(A) then G(A) N x N1. x N, for all N N x N2. if N N x and N N then N N x N3. if N, N N x then N N N x N4. there is an N such that N x = N

44 Conditions and properties of neighbourhood systems Conditions 0, 1, Id, N1, N2 and N3 characterise topological spaces

45 Neighbourhood systems and systems of relations.

46 Expansion and contraction processes A x y z c a b g(a) x y z x y z a c b f(a)

47 Expansion and contraction are not necessarily isotonic A y x z w b a y x z w B b a c c f(a) y x z b y x z w a f(b) c c

48 Pre-topological spaces and neighbourhood systems A pre-topological space is a triple U, ε, κ such that: (i) U is a set, (ii) ε: (U) (U) is an expansion map such that ε( )=, (iii) κ : (U) (U) is a contraction map dual of ε. The core map (vicinity map) induced by a neighbourhood system of type (at least) N 1 is a contraction map (an expansion map)

49 Dynamic Approximation Spaces U, {R i } 1 i n, m, m for {R i } 1 i n a family of binary relations, 1 m n n use case involving the expansion process, and n use cases involving the contraction process

50 Pre-topological spaces and Dynamic Spaces

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52 Associating systems of relations with pre-topological spaces

53 Piero Pagliani Approximation and pre-topologies Third part Formal Topological Systems and Pre-topological Approximation Operators

54 Concrete and formal pre-topology Let P= G, M, R, be any property system, for R G M any binary relation. We remind that int (G)={X G :int(x)=x} is the family of concrete open subsets of P e [i] A (M)={Y M: A(X)=X} is the family of formal open subsets of P Sat int (G)= int (G),,,, G where i I X i =int( i I X i ) is a complete lattice Sat A (M)= A (M),,,, M where i I X i =A( i I X i ) is a complete lattice

55 A formal semi-covering relation between neighbourhoods Let G, M, R be a basic pair. Then, for any b M and Y, Y M, a formal semi-cover relation is defined as follows: (basis) b Y iff b A(Y), (step) Y Y iff y Y, y Y From this definition, it follows: b Y b Y b Y Y Y b Y reflexivity transitivity Y Y identity

56 Combination of formal neighbourhoods and semi-covering relation Properties may be combined by means of a formal meet operation that we call fusion and denote with The operation is inherited by subsets of properties in the following way: X Y= x y: x X & y Y (point-meet) Formal opens inherit the operation in the following way: X Y=A(X Y)

57 Properties which can be added to a semi-cover For all Y, Y A (M), and b, b M b Y b b Y b Y b Y b Y Y b Y b Y b b Y Y left right stability

58 Property systems and formal systems Let G, M, R be a basic pair (i. e. property system), let M,, 1 be commutative monoid, set = m M : R(m)=. Then: M,, 1,, is called a semi-topological formal system induced by the basic pair G, M, R A semi-topological formal system in which stability holds, is called a pre-topological formal system, and is called a precover A semi-topological formal system in which left and right hold, is called a topological formal system, and is called a cover. If M,, 1,, is a topological formal system, then A (M),,, M, A( ), for any subset of M, is a complete lattice with complete distributivity and ordering. In any semi-topological formal system in which is idempotent and stability holds, right is derivable. Any semi-topological formal system in which is idempotent and stability and left hold, is a topological formal system.

59 A bridge between concrete neighborhood systems and formal systems A, (A), R is called a basic neighborhood pair and, therefore, (A),, A,, will be called a formal neighborhood system Since is idempotent, any formal neighborhood system in which stability and left hold, is a topological formal system.

60 Towards the main connection between concrete pre-topological systems and formal systems Let (A),, A,, be induced by A, (A), R. Then: a) If R(x): x A fulfils N3, then right holds. That is: for all X, X (A), x A and Y, Y (A), (X, X R(x) X X R(x)) (X Y and X Y X Y Y ) b) R(x): x A fulfils N2 if and only if left holds. That is: for all X, X (A), x A and Y (A), (X R(x) & X X X R(x)) (X Y X X Y)

61 Main connection between concrete pre-topolgical systems and formal systems Let (A),, A,, be induced by A, (A), R. Then, for all X (A), G(X)=int(X) if and only if R(x): x A is a neighborhood system of type N 2Id. A formal neighborhood system (A),, A,, such that for all X (A), G(x)=int(X), will be called a pseudo-topological formal system.

62 From concrete pre-topolgical systems to concrete topological systems, through formal systems Let (A),, A,, be a pseudo-topological formal system. Then: A ( (A)), is a meet semilattice with ordering. However, in general does not distribute over in Sat A ( (A)). To obtain distributivity of over, we need stability: In any pseudo-topological formal system, if stability holds, then A ( (A)),, is a complete distributive lattice, hence a Heyting algebra. But this is not a surprise, because: In any pseudo-topological formal system N2 holds, by definition. Hence left holds, too. Moreover, the monoidal operation is, which is idempotent. It follows that if we add stability we obtain right. But we know that a pre-topological formal system in which right and left hold, is a topological formal system.

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66 A logical interpretation Pretopological formal systems provide models for limited resource logics Logic formula Pretopological interpretation a a b A A (M) A B additive conjunction a b A B a & b A B additive conjunction a b A B multiplicative conjunction (left) A C A B C Γ, a c Γ, a b c left & (right) A B A C A B C Γ a Γ a b Γ b right &

67 Neighborhood systems [Lin, Yao, Pagliani] Linear Logic Intermediate (non-standard) constructive logics Quantum logic Ortholattices [Bell] Duality theory [Stone-Birkfoff] Formal (pre) topology [Sambin] Heyting co-heyting ALGEBRAS [Rauszer, Lawvere, Pagliani, Iturrioz] Lukasiewicz Nelson Post Stone ALGEBRAS [Pomikała, Düntsch, Pagliani] Topological quasi Boolean algebras [Chakraborty, Banerje] Rough Sets Logics with strong negation Topological Bilattices [Fitting] Semi-Post Algebras [Rasiowa, Cat Ho] Intuitionistic logic Quanta of information Modal-style operators [Yao, Düntsch, Gegida, Pagliani] Galois adjunctions [Ore, Grothendieck, Lowvere, Scott] Kripke models Relational modal logic [Orlowska, Düntsch, Pagliani] Modal Logic Chu Spaces Information Flow [Pratt, Barwise, Seligman] Formal Concept Analysis [Wille, Kent] Program semantics [Parikh, He, Hoare] Categorical modal logic [Meloni, Ghilardi]