Piero Pagliani Approximation and Pretopologies

Piero Pagliani Università di Milano Bicocca maggio 2009

Piero Pagliani

Approximation and Pretopologies


Approximation Spaces

(U), lR, uR

•given a set U

•given a relation R which structures U

•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


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

Approximation and Pretopologies


The basic ingredients to represent objects through properties:

a map from objects to properties

a map from properties to objects

result of the factorisation

through f and g

A set of properties

A set of objects

G G

M

f g

Same set of objects

Gegenstand Objekt


a

a’

a’’

b

b’

b’’

b’’’

An “ideal” situation: f is an injective function

G M

a’’’

Objects

Observable

properties

G G

M

1G

RETRACTIONf


G G

M

1G

RETRACTION

The best model of a horse is a horse?

Modelling means abstraction

Abstraction means loosing something

(it must be injective)

Horse-ness property

f


a

a’

a’’

b

b’’’

G M

a’’’

ObjectsObservable

properties

M M

G

1M

SECTION

Sorts

(fibres, stalks)

An almost ideal situation: f is a surjective function

(it must be surjective)f


a

a’

a’’’

b

b’

b’’

b’’’

G M

a’’’

Objects and properties are linked

by means of an arbitrary fulfillment relation

⊩ G M

We call

G, M, ⊩

a Property System or a Basic Pair

The basic picture


Observations, Property Systems and Attribute Systems

R p1 p2 p3 p4

a 1 0 1 1

b 0 0 1 1

c 0 0 1 1

d 1 0 0 1

e 0 1 1 0

a Property system

R a1 a2 a3 a4 a5

a 3 b 1/3 6.5 0

b 3 b 1/3 6.5 0

c 1 a 2 7 1

d 1 c 1/2 3.3 1

e 3 b 1 6 0

an Attribute system

Observation systems


Observation and substantiation

a

a’

a’’

b

b’

b’’

b’’’

sub (substance)

G M

a’’’

sub(b’’’)

obs(a’)

obs

(observations)

Substance

neighbourhoods

Observation

neighbourhoods

m obs(g) if and only if g⊩ m if and only if g sub(m)


(G)

(M)

(G)

A subset of objects O

fulfils a

subset of properties P

A subset of properties P’

describes a

subset of objects O’

Usually, sub(obs(O)) O

Approximations and best approximations. 1

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).

obs sub


(G) (G)

(M)

( )

la ( ua )

φ ψ

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:


Towards a solution. 1

Operators must fulfil maximality (minimality),

i. e. limit, properties

X

best (least)

upper

approximation

ua(X)

best (largest)

lower

approximation

la(X)

{boundary

region


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


Continuity and approximations: 1

Let ψ be isotonic and YG.

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.



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



(G) (G)

(M)

( )

la ( ua )

φ ψ

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

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


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’ ┫ Oop,

then , is called a Galois connection or a polarity.

In this case, both and are closure operators.


Approximations and Galois Adjunctions. 2

If φ = ψ– , then ψ = φ+ and we have

(G) φ ┫ψ (M)

(G) (G)

(M)

( )

la ( ua )

φ ψ

Problem:

How one can define a Galois adjunction on an

Observation System, in a natural way?


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

[[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

Extensional

operators

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

Intensional

operators

Basic perception constructors. 1


Basic perception constructors. 2

aa’

a’’

b

b’

b’’

b’’’

G M

i ({a, a’})

e ({b, b’, b’’’})

[e]({b, b’, b’’’})

[[e]]({b, b’})

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


<i >(X)

X

b

[i](X)

X

b

[[i]](X) b

X

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 Xto fulfil b it is sufficient

to be in 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


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

Galois

connection

i ┫ [e]

e ┫ [i]upper adjointlower adjoint

multiplicative

isotonic

additive

isotonic

[[e]] ┫op [[i]] anti-multiplicative

anti-isotonic

anti-multiplicative

anti-isotonic

[ ] is an interior operator

[ ] is an closure operator


a

a’

a’’

b

b’

b’’

b’’’

G M

ITS({b’, b’’})

est({a, a’})

cl({a, a’})

int({a,a’})

a’’’

C ({b’’, b’’’})

A ({b’, b’’})

Combining adjoint constructors: perception operators

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)

modal-style operators

Qualitative Data Analysis

?

intent-extent operators

Formal Concept Analysis


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.


FORMAL

(Pre) topological properties of the perception operators

We obtain G. Sambin’s

results

on Formal Topology

(point-free topology)

Straightforwardly

from the

adjunction properties

between the basic

constructors

closed

CONCRETE

: <e>[i] int <i>[e] C :

: [e]<i> cl [i]<e> A :

CONCRETE

open closed

symmetric

symmetric

: <e> <i> :

open

[i] : : [e]

du

al d

ua

l

FORMAL


Algebraic properties of the perception operators

Given the following set of fixpoints of the operators

one obtains the following lattices


Example

P b b1 b2 b3

a 1 1 0 0

a1 0 1 0 1

a2 0 1 1 1

a3 0 0 0 1


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


Choosing the initial perception act.

Information quanta

g’ Qg if and only if g’ fulfils at least all the properties fulfilled by g:

i ({g}) i ({g’})

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, g’ RS if and only if g’ Qg

Information quantum relation

and

Information quantum systems

information quantum relation

• reflexive

• transitive

Given a Property System S

Q(S) = G, G, RS

Information quantum relation system

• it is a property system!!

If g’, g RS then g’ Qg , thus i (g) i (g’).

Hence g is less defined than g’ in S.

Indeed, RS is a (anti)specialisation preorder in the usual topological sense.


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

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

cl C

A int


Example

Satcl(Q(P))

P b b1 b2 b3

a 1 1 0 0

a1 0 1 0 1

a2 0 1 1 1

a3 0 0 0 1

RP a a1 a2 a3

a 1 0 0 0

a1 0 1 1 0

a2 0 0 1 0

a3 0 1 1 1


Let cl, int, C and A in Q(S). Then

cl = i : inverse upper approximationcl(X) = set of elements approximated by some member of X

int = [i]: inverse lower approximationint(X) = set of elements approximated just by members of X

C = [e]: direct lower approximationC (X) = set of elements specialised just by members of X

A = e : direct upper approximationA(X) = set of elements specialised by some member of X

Topological approximation operators


If S is an Attribute System, its nominalisation

is a Perception System N(S) such that:

RN(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

A particular case


Second part:

Dynamic Property Systems

and

Pre-topological Approximation Operators


•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, {Ei}iI >

for {Ei}iI a family of equivalence relations

•P. Pagliani

Dynamic Approximation Spaces

<U, {Ri}iI >

for {Ri}iI a family of binary relations


R

context 1R1

time mRmtime 2R2time 1

1 context 2R12context nR1n

Sources of dynamics

1) Evolution of the observation process over time

2) Change of context

3) 1+2


Object 1

Object 2

Object 3

Object n

Tim

e

Dynamic Rough

Sets

A layered observation system


f(x)f(y)

f(w)

f(z)

U ’: applicable criteria (metrics) of type TContext Context

xy

w z

U: object level

w’

Nx

q

z’z’’

More than 1 context More than 1 neighbourhood

xy

w

w’ z q

z’z’’

Neighbourhood

of x

Why more than one neighbourhood?


More than 1 neighbourhood Neighbourhood families

More neighbourhood families Neighbourhood systems

Neighbourhood

of x

Neighbourhood

of y

What if more than one neighbourhood?


A

NzNy

x, y, z G(A)

Nx

Let us denote n(x) with Nx

Core maps and vicinity maps


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 xU, AU:

1. UNx

0. Nx

Id. if xG(A) then G(A) Nx

N1. xN, for all NNx

N2. if NNx and N N’ then N’Nx

N3. if N, N’Nx then NN’Nx

N4. there is an N such that Nx =N

Sectioning topological systems


Conditions and properties of neighbourhood systems

Conditions 0, 1, Id, N1, N2 and N3 characterise topological spaces


Neighbourhood systems and systems of relations.


x

yz

c

b

a

x

y zc

b

af(A)

x

y

zg(A)

Expansion and contraction processes

A


c

A

c

b

y

c

w

a

y

x

z

w

b

f(A)y

x

z

a

f(B)

x

z

a

b

B

c

w

y

x

z

Expansion and contraction are

not necessarily isotonic


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) N1 is a contraction map (an expansion map)


Dynamic Approximation Spaces

U, {Ri}1 i n, m, m

for {Ri}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


Pre-topological spaces and Dynamic Spaces


Associating systems of relations with pre-topological spaces


Piero Pagliani

Approximation and pre-topologies

Third part

Formal Topological Systems

and

Pre-topological Approximation Operators


Concrete and formal pre-topology

Satint(G)=int (G), , , , G where iI Xi=int(iI Xi)

is a complete lattice

SatA(M)=A (M), , , , M where iI Xi=A(iI Xi)

is a complete lattice

Let P=G, M, R, be any property system, for R GM any binary relation.

We remind that

int(G)={X G :int(X)=X} is the family of concrete open subsets of P

A(M)={YM: A(X)=X} is the family of formal open subsets of P

e [i]


A formal semi-covering relation

between neighbourhoods

bY

bY

bY YY’

b Y’

YY

reflexivity

transitivity

identity

Let G, M, R be a basic pair. Then, for any bM and Y, Y’M,

a formal semi-cover relation is defined as follows:

(basis) bY iff bA(Y), (step) YY’ iff y Y, yY’

From this definition, it follows:


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:

XY=xy: xX & yY (point-meet)

Formal opens inherit the operation “” in the following way:

X•Y=A(XY)


For all Y, Y’A(M), and b, b’ M

Properties which can

be added to a semi-cover

b Y

b b’Y

bY bY’

b Y Y’

left

right

stabilitybY b’Y’

b b’ Y Y’


Let G, M, R be a basic pair (i. e. property system),

let M, , 1 be commutative monoid, set =mM : 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.

Property 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.

A bridge between

concrete neighborhood systems and formal systems


Let (A), , A, , be induced by A, (A), R. Then:

a) If R(x): xA fulfils N3, then right holds. That is:

for all X, X’ (A), x A and Y, Y’(A),

(X, X’ R(x) XX’ R(x)) (X Y and X Y’ X Y·Y’)

b) R(x): xA 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)

Towards the main connection between

concrete pre-topological 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): xA is a neighborhood system of type N2Id.

A formal neighborhood system (A), , A, , such that for all X(A),

G(x)=int(X), will be called a pseudo-topological formal system.

Main connection between

concrete pre-topolgical systems and 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 SatA ((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.

From concrete pre-topolgical systems to

concrete topological systems, through formal systems


Pretopological formal systems provide models for limited resource logics

Logic formula Pretopological interpretation

a

a b ABa b AB

A logical interpretation

multiplicative conjunction

a & b AB additive conjunction

AB

additive conjunction

a ⊢ b

left &

right &

(left)

(right)

AC

A ·BCΓ, a ⊢ c

Γ, ab ⊢ c

AB AC

A B ·C

Γ⊢ a Γ⊢ b

Γ⊢ ab

A A (M)


Categorical

modal logic

[Meloni, Ghilardi]

Ortholattices

[Bell]

Quantum logic

Lukasiewicz

Nelson

Post

Stone

ALGEBRAS

[Pomikała,

Düntsch,

Pagliani]

Topological Bilattices

[Fitting]

Semi-Post Algebras

[Rasiowa, Cat Ho]

Formal Concept

Analysis

[Wille, Kent]

Linear Logic

Program semantics

[Parikh, He, Hoare]

Formal

(pre) topology

[Sambin]

Quanta of

information

Modal Logic

Rough

Sets

Intermediate

(non-standard)

constructive

logics

Topological quasi

Boolean algebras

[Chakraborty,

Banerje]

Chu Spaces

Information Flow

[Pratt, Barwise,

Seligman]

Galois adjunctions

[Ore, Grothendieck,

Lowvere, Scott]

Logics with

strong negation

Duality

theory

[Stone-Birkfoff]

Kripke

models

Relational

modal logic

[Orlowska, Düntsch,

Pagliani]

Neighborhood systems

[Lin, Yao, Pagliani]

Heyting

co-Heyting

ALGEBRAS

[Rauszer, Lawvere,

Pagliani, Iturrioz]

Intuitionistic logic

Modal-style

operators

[Yao, Düntsch,

Gegida, Pagliani]

Piero Pagliani Approximation and Pretopologies

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