Orthopairs Davide Ciucci Dipartimento di Informatica, Sistemistica - - PowerPoint PPT Presentation

Introduction Models Mathematical aspects

Orthopairs

Davide Ciucci

Dipartimento di Informatica, Sistemistica e Comunicazione Università di Milano Bicocca

May 2009/Milano

Davide Ciucci (DISCo) Orthopairs RST 1 / 33

Introduction Models Mathematical aspects

Outline

1

Introduction

2

Models

3

Mathematical aspects On three values and orthopairs Operations on orthopairs Operations on rough sets

Davide Ciucci (DISCo) Orthopairs RST 2 / 33

Introduction Models Mathematical aspects

Outline

1

Introduction

2

Models

3

Mathematical aspects On three values and orthopairs Operations on orthopairs Operations on rough sets

Davide Ciucci (DISCo) Orthopairs RST 3 / 33

Introduction Models Mathematical aspects

Orthopairs: definition

Definition Let X a given universe, A, B ⊆ X form an orthopair iff they are disjoint or orthogonal (A, B) with A ∩ B = ∅ Any orthopair (A, B) is equal to a subset pair (A, C) such that A ⊆ C through the mapping C := Bc The collection of all orthopairs on a universe X is denoted as O(X)

Davide Ciucci (DISCo) Orthopairs RST 4 / 33

Introduction Models Mathematical aspects

Orthopairs: definition

Definition Let X a given universe, A, B ⊆ X form an orthopair iff they are disjoint or orthogonal (A, B) with A ∩ B = ∅ Any orthopair (A, B) is equal to a subset pair (A, C) such that A ⊆ C through the mapping C := Bc The collection of all orthopairs on a universe X is denoted as O(X)

Davide Ciucci (DISCo) Orthopairs RST 4 / 33

Introduction Models Mathematical aspects

Orthopairs: definition

Definition Let X a given universe, A, B ⊆ X form an orthopair iff they are disjoint or orthogonal (A, B) with A ∩ B = ∅ Any orthopair (A, B) is equal to a subset pair (A, C) such that A ⊆ C through the mapping C := Bc The collection of all orthopairs on a universe X is denoted as O(X)

Davide Ciucci (DISCo) Orthopairs RST 4 / 33

Introduction Models Mathematical aspects

Historical notes

Orthopairs studied in Narinyani, 1980: Sub-definite sets

elements belonging/not belonging to a set A

Cattaneo,Nistico, 1989: classical preclusivity propositions

an example of BZ–lattices induced by a preclusivity relation

Coker, 1996: Intuitionistic sets

a weakening to the classical case of Atanassov intuitionistic fuzzy sets

Subset pairs studied in Gentilhomme, 1968: Flou sets Klaua, 1968: Partial sets Iwinski, 1987 Yao, Li, 1997: Interval sets

Davide Ciucci (DISCo) Orthopairs RST 5 / 33

Introduction Models Mathematical aspects

Historical notes

Orthopairs studied in Narinyani, 1980: Sub-definite sets

elements belonging/not belonging to a set A

Cattaneo,Nistico, 1989: classical preclusivity propositions

an example of BZ–lattices induced by a preclusivity relation

Coker, 1996: Intuitionistic sets

a weakening to the classical case of Atanassov intuitionistic fuzzy sets

Subset pairs studied in Gentilhomme, 1968: Flou sets Klaua, 1968: Partial sets Iwinski, 1987 Yao, Li, 1997: Interval sets

Davide Ciucci (DISCo) Orthopairs RST 5 / 33

Introduction Models Mathematical aspects

Historical notes

Orthopairs studied in Narinyani, 1980: Sub-definite sets

elements belonging/not belonging to a set A

Cattaneo,Nistico, 1989: classical preclusivity propositions

an example of BZ–lattices induced by a preclusivity relation

Coker, 1996: Intuitionistic sets

a weakening to the classical case of Atanassov intuitionistic fuzzy sets

Subset pairs studied in Gentilhomme, 1968: Flou sets Klaua, 1968: Partial sets Iwinski, 1987 Yao, Li, 1997: Interval sets

Davide Ciucci (DISCo) Orthopairs RST 5 / 33

Introduction Models Mathematical aspects

Historical notes

Orthopairs studied in Narinyani, 1980: Sub-definite sets

elements belonging/not belonging to a set A

Cattaneo,Nistico, 1989: classical preclusivity propositions

an example of BZ–lattices induced by a preclusivity relation

Coker, 1996: Intuitionistic sets

a weakening to the classical case of Atanassov intuitionistic fuzzy sets

Subset pairs studied in Gentilhomme, 1968: Flou sets Klaua, 1968: Partial sets Iwinski, 1987 Yao, Li, 1997: Interval sets

Davide Ciucci (DISCo) Orthopairs RST 5 / 33

Introduction Models Mathematical aspects

Models

Several model which generate orthopairs (subset pairs) Boolean rough sets

Pawlak, VPRS, decision theoretic, ...

Twofold sets Shadowed sets AIM (OF THIS PRESENTATION) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets

Davide Ciucci (DISCo) Orthopairs RST 6 / 33

Introduction Models Mathematical aspects

Models

Several model which generate orthopairs (subset pairs) Boolean rough sets

Pawlak, VPRS, decision theoretic, ...

Twofold sets Shadowed sets AIM (OF THIS PRESENTATION) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets

Davide Ciucci (DISCo) Orthopairs RST 6 / 33

Introduction Models Mathematical aspects

Models

Several model which generate orthopairs (subset pairs) Boolean rough sets

Pawlak, VPRS, decision theoretic, ...

Twofold sets Shadowed sets AIM (OF THIS PRESENTATION) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets

Davide Ciucci (DISCo) Orthopairs RST 6 / 33

Introduction Models Mathematical aspects

Models

Several model which generate orthopairs (subset pairs) Boolean rough sets

Pawlak, VPRS, decision theoretic, ...

Twofold sets Shadowed sets AIM (OF THIS PRESENTATION) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets

Davide Ciucci (DISCo) Orthopairs RST 6 / 33

Introduction Models Mathematical aspects

Models

Several model which generate orthopairs (subset pairs) Boolean rough sets

Pawlak, VPRS, decision theoretic, ...

Twofold sets Shadowed sets AIM (OF THIS PRESENTATION) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets

Davide Ciucci (DISCo) Orthopairs RST 6 / 33

Introduction Models Mathematical aspects

Models

Several model which generate orthopairs (subset pairs) Boolean rough sets

Pawlak, VPRS, decision theoretic, ...

Twofold sets Shadowed sets AIM (OF THIS PRESENTATION) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets

Davide Ciucci (DISCo) Orthopairs RST 6 / 33

Introduction Models Mathematical aspects

Outline

1

Introduction

2

Models

3

Mathematical aspects On three values and orthopairs Operations on orthopairs Operations on rough sets

Davide Ciucci (DISCo) Orthopairs RST 7 / 33

Introduction Models Mathematical aspects

Pawlak Rough Sets

Definition An approximation space is a pair (X, E) with X a set of objects and E an equivalence (reflexive, simmetric, transitive) relation on X. Equivalence classes are denoted as [x]E. Rough membership function ∀H ⊆ X µH : X → [0, 1] µH(y) := |[y]E ∩ H| |[y]E|

Davide Ciucci (DISCo) Orthopairs RST 8 / 33

Introduction Models Mathematical aspects

Pawlak Rough Sets

Definition An approximation space is a pair (X, E) with X a set of objects and E an equivalence (reflexive, simmetric, transitive) relation on X. Equivalence classes are denoted as [x]E. Rough membership function ∀H ⊆ X µH : X → [0, 1] µH(y) := |[y]E ∩ H| |[y]E|

Davide Ciucci (DISCo) Orthopairs RST 8 / 33

Introduction Models Mathematical aspects

Pawlak rough sets

Lower approximation L(H) = {y ∈ X : [y]E ⊆ H} Upper approximation U(H) = {y ∈ X : y ∈ X : [y]E ∩ H = ∅} Exterior region E(H) = Uc(H) (L(H), E(H)) is an orthopair

Davide Ciucci (DISCo) Orthopairs RST 9 / 33

Introduction Models Mathematical aspects

Pawlak rough sets

Lower approximation L(H) = {y ∈ X : [y]E ⊆ H} Upper approximation U(H) = {y ∈ X : y ∈ X : [y]E ∩ H = ∅} Exterior region E(H) = Uc(H) (L(H), E(H)) is an orthopair

Davide Ciucci (DISCo) Orthopairs RST 9 / 33

Introduction Models Mathematical aspects

Pawlak rough sets

Lower approximation L(H) = {y ∈ X : [y]E ⊆ H} Upper approximation U(H) = {y ∈ X : y ∈ X : [y]E ∩ H = ∅} Exterior region E(H) = Uc(H) (L(H), E(H)) is an orthopair

Davide Ciucci (DISCo) Orthopairs RST 9 / 33

Introduction Models Mathematical aspects

Pawlak rough sets

Lower approximation L(H) = {y ∈ X : [y]E ⊆ H} Upper approximation U(H) = {y ∈ X : y ∈ X : [y]E ∩ H = ∅} Exterior region E(H) = Uc(H) (L(H), E(H)) is an orthopair

Davide Ciucci (DISCo) Orthopairs RST 9 / 33

Introduction Models Mathematical aspects

Twofold sets

Given a universe X, a multivalued attribute a with domain D Γa(x) ⊆ D set of possible values for a of x, Γa : X → P(D) Let E ⊆ D a subset of values The ill-known set Γ−1

a (E) ⊆ X is approximated by the pair Γ−1 a∗ (E), Γ−1∗ a

(E) Γ−1

a∗ (E) := {x ∈ X, Γa(x) ⊆ E}

Γ−1∗

a

(E) := {x ∈ X, Γa(x) ∩ E = ∅}

• D. Dubois and H. Prade

Twofold fuzzy sets and rough sets – some issues in knowledge representation Fuzzy Sets and Systems 23 (1987), 3–18

Davide Ciucci (DISCo) Orthopairs RST 10 / 33

Introduction Models Mathematical aspects

Twofold sets

Given a universe X, a multivalued attribute a with domain D Γa(x) ⊆ D set of possible values for a of x, Γa : X → P(D) Let E ⊆ D a subset of values The ill-known set Γ−1

a (E) ⊆ X is approximated by the pair Γ−1 a∗ (E), Γ−1∗ a

(E) Γ−1

a∗ (E) := {x ∈ X, Γa(x) ⊆ E}

Γ−1∗

a

(E) := {x ∈ X, Γa(x) ∩ E = ∅}

• D. Dubois and H. Prade

Twofold fuzzy sets and rough sets – some issues in knowledge representation Fuzzy Sets and Systems 23 (1987), 3–18

Davide Ciucci (DISCo) Orthopairs RST 10 / 33

Introduction Models Mathematical aspects

Twofold sets

Given a universe X, a multivalued attribute a with domain D Γa(x) ⊆ D set of possible values for a of x, Γa : X → P(D) Let E ⊆ D a subset of values The ill-known set Γ−1

a (E) ⊆ X is approximated by the pair Γ−1 a∗ (E), Γ−1∗ a

(E) Γ−1

a∗ (E) := {x ∈ X, Γa(x) ⊆ E}

Γ−1∗

a

(E) := {x ∈ X, Γa(x) ∩ E = ∅}

• D. Dubois and H. Prade

Twofold fuzzy sets and rough sets – some issues in knowledge representation Fuzzy Sets and Systems 23 (1987), 3–18

Davide Ciucci (DISCo) Orthopairs RST 10 / 33

Introduction Models Mathematical aspects

Twofold sets

Given a universe X, a multivalued attribute a with domain D Γa(x) ⊆ D set of possible values for a of x, Γa : X → P(D) Let E ⊆ D a subset of values The ill-known set Γ−1

a (E) ⊆ X is approximated by the pair Γ−1 a∗ (E), Γ−1∗ a

(E) Γ−1

a∗ (E) := {x ∈ X, Γa(x) ⊆ E}

Γ−1∗

a

(E) := {x ∈ X, Γa(x) ∩ E = ∅}

• D. Dubois and H. Prade

Twofold fuzzy sets and rough sets – some issues in knowledge representation Fuzzy Sets and Systems 23 (1987), 3–18

Davide Ciucci (DISCo) Orthopairs RST 10 / 33

Introduction Models Mathematical aspects

Twofold sets

Given a universe X, a multivalued attribute a with domain D Γa(x) ⊆ D set of possible values for a of x, Γa : X → P(D) Let E ⊆ D a subset of values The ill-known set Γ−1

a (E) ⊆ X is approximated by the pair Γ−1 a∗ (E), Γ−1∗ a

(E) Γ−1

a∗ (E) := {x ∈ X, Γa(x) ⊆ E}

Γ−1∗

a

(E) := {x ∈ X, Γa(x) ∩ E = ∅}

• D. Dubois and H. Prade

Twofold fuzzy sets and rough sets – some issues in knowledge representation Fuzzy Sets and Systems 23 (1987), 3–18

Davide Ciucci (DISCo) Orthopairs RST 10 / 33

Introduction Models Mathematical aspects

Rough and twofold sets

Formally From Γ−1

a∗ (E) := {x ∈ X, Γa(x) ⊆ E}

Γ−1∗

a

(E) := {x ∈ X, Γa(x) ∩ E = ∅} we recover rough sets if ΓR(x) = [x]R Semantic Different point of views Two-fold sets: precise set of attributes which lead to ill-known sets Rough sets: well-known set but not precisely characterized by the attributes

Davide Ciucci (DISCo) Orthopairs RST 11 / 33

Introduction Models Mathematical aspects

Rough and twofold sets

Formally From Γ−1

a∗ (E) := {x ∈ X, Γa(x) ⊆ E}

Γ−1∗

a

(E) := {x ∈ X, Γa(x) ∩ E = ∅} we recover rough sets if ΓR(x) = [x]R Semantic Different point of views Two-fold sets: precise set of attributes which lead to ill-known sets Rough sets: well-known set but not precisely characterized by the attributes

Davide Ciucci (DISCo) Orthopairs RST 11 / 33

Introduction Models Mathematical aspects

Rough and twofold sets

Formally From Γ−1

a∗ (E) := {x ∈ X, Γa(x) ⊆ E}

Γ−1∗

a

(E) := {x ∈ X, Γa(x) ∩ E = ∅} we recover rough sets if ΓR(x) = [x]R Semantic Different point of views Two-fold sets: precise set of attributes which lead to ill-known sets Rough sets: well-known set but not precisely characterized by the attributes

Davide Ciucci (DISCo) Orthopairs RST 11 / 33

Introduction Models Mathematical aspects

Rough and twofold sets

Formally From Γ−1

a∗ (E) := {x ∈ X, Γa(x) ⊆ E}

Γ−1∗

a

(E) := {x ∈ X, Γa(x) ∩ E = ∅} we recover rough sets if ΓR(x) = [x]R Semantic Different point of views Two-fold sets: precise set of attributes which lead to ill-known sets Rough sets: well-known set but not precisely characterized by the attributes

Davide Ciucci (DISCo) Orthopairs RST 11 / 33

Introduction Models Mathematical aspects

Upper and lower probabilities

X, S two universes Γ multivalued mapping from X to S Γ : X → P(S) Problem Let µ be a probability measure on X: µ : F → [0, 1] with F ⊆ P(X) If s ∈ Γ(x), what is the probability of s? Solution Let T ⊆ S T∗ := {x ∈ X, Γ(x) ⊆ T, Γ(x) = ∅} T ∗ := {x ∈ X, Γ(x) ∩ T = ∅} Upper probability of T: P∗(T) = µ(T ∗)

µ(S∗)

Lower probability of T: P∗(T) = µ(T∗)

µ(S∗)

A.P . Dempster Upper and lower probabilities induced by a multivalued mapping Annals of Mathematical Statistics 38 (1967), 325–339.

Davide Ciucci (DISCo) Orthopairs RST 12 / 33

Introduction Models Mathematical aspects

Upper and lower probabilities

X, S two universes Γ multivalued mapping from X to S Γ : X → P(S) Problem Let µ be a probability measure on X: µ : F → [0, 1] with F ⊆ P(X) If s ∈ Γ(x), what is the probability of s? Solution Let T ⊆ S T∗ := {x ∈ X, Γ(x) ⊆ T, Γ(x) = ∅} T ∗ := {x ∈ X, Γ(x) ∩ T = ∅} Upper probability of T: P∗(T) = µ(T ∗)

µ(S∗)

Lower probability of T: P∗(T) = µ(T∗)

µ(S∗)

A.P . Dempster Upper and lower probabilities induced by a multivalued mapping Annals of Mathematical Statistics 38 (1967), 325–339.

Davide Ciucci (DISCo) Orthopairs RST 12 / 33

Introduction Models Mathematical aspects

Upper and lower probabilities

X, S two universes Γ multivalued mapping from X to S Γ : X → P(S) Problem Let µ be a probability measure on X: µ : F → [0, 1] with F ⊆ P(X) If s ∈ Γ(x), what is the probability of s? Solution Let T ⊆ S T∗ := {x ∈ X, Γ(x) ⊆ T, Γ(x) = ∅} T ∗ := {x ∈ X, Γ(x) ∩ T = ∅} Upper probability of T: P∗(T) = µ(T ∗)

µ(S∗)

Lower probability of T: P∗(T) = µ(T∗)

µ(S∗)

A.P . Dempster Upper and lower probabilities induced by a multivalued mapping Annals of Mathematical Statistics 38 (1967), 325–339.

Davide Ciucci (DISCo) Orthopairs RST 12 / 33

Introduction Models Mathematical aspects

Upper and lower probabilities

X, S two universes Γ multivalued mapping from X to S Γ : X → P(S) Problem Let µ be a probability measure on X: µ : F → [0, 1] with F ⊆ P(X) If s ∈ Γ(x), what is the probability of s? Solution Let T ⊆ S T∗ := {x ∈ X, Γ(x) ⊆ T, Γ(x) = ∅} T ∗ := {x ∈ X, Γ(x) ∩ T = ∅} Upper probability of T: P∗(T) = µ(T ∗)

µ(S∗)

Lower probability of T: P∗(T) = µ(T∗)

µ(S∗)

A.P . Dempster Upper and lower probabilities induced by a multivalued mapping Annals of Mathematical Statistics 38 (1967), 325–339.

Davide Ciucci (DISCo) Orthopairs RST 12 / 33

Introduction Models Mathematical aspects

Upper and lower probabilities

X, S two universes Γ multivalued mapping from X to S Γ : X → P(S) Problem Let µ be a probability measure on X: µ : F → [0, 1] with F ⊆ P(X) If s ∈ Γ(x), what is the probability of s? Solution Let T ⊆ S T∗ := {x ∈ X, Γ(x) ⊆ T, Γ(x) = ∅} T ∗ := {x ∈ X, Γ(x) ∩ T = ∅} Upper probability of T: P∗(T) = µ(T ∗)

µ(S∗)

Lower probability of T: P∗(T) = µ(T∗)

µ(S∗)

A.P . Dempster Upper and lower probabilities induced by a multivalued mapping Annals of Mathematical Statistics 38 (1967), 325–339.

Davide Ciucci (DISCo) Orthopairs RST 12 / 33

Introduction Models Mathematical aspects

Shadowed sets

• W. Pedrycz

Shadowed sets: Representing and processing fuzzy sets IEEE Transaction on Systems, Man and Cybernetics - PART B: Cybernetics 28, 1998, 103–109. An attempt to ”capture uncertainty in a non-numeric fashion” ”provide a transparent and computationally appealing representation of fuzzy sets” From a fuzzy set f : X → [0, 1] to s : X → {0, (0, 1), 1} α ∈ [0, 1 2) sα(f)(x) :=      f(x) ≤ α exclusion region 1 f(x) ≥ 1 − α core (0, 1)

• therwise

shadow α not fixed a priori but chosen according with a balance of vagueness

Davide Ciucci (DISCo) Orthopairs RST 13 / 33

Introduction Models Mathematical aspects

Shadowed sets

An attempt to ”capture uncertainty in a non-numeric fashion” ”provide a transparent and computationally appealing representation of fuzzy sets” From a fuzzy set f : X → [0, 1] to s : X → {0, (0, 1), 1} α ∈ [0, 1 2) sα(f)(x) :=      f(x) ≤ α exclusion region 1 f(x) ≥ 1 − α core (0, 1)

• therwise

shadow α not fixed a priori but chosen according with a balance of vagueness

Davide Ciucci (DISCo) Orthopairs RST 13 / 33

Introduction Models Mathematical aspects

Shadowed sets

An attempt to ”capture uncertainty in a non-numeric fashion” ”provide a transparent and computationally appealing representation of fuzzy sets” From a fuzzy set f : X → [0, 1] to s : X → {0, (0, 1), 1} α ∈ [0, 1 2) sα(f)(x) :=      f(x) ≤ α exclusion region 1 f(x) ≥ 1 − α core (0, 1)

• therwise

shadow α not fixed a priori but chosen according with a balance of vagueness

Davide Ciucci (DISCo) Orthopairs RST 13 / 33

Introduction Models Mathematical aspects

Shadowed sets

An attempt to ”capture uncertainty in a non-numeric fashion” ”provide a transparent and computationally appealing representation of fuzzy sets” From a fuzzy set f : X → [0, 1] to s : X → {0, (0, 1), 1} α ∈ [0, 1 2) sα(f)(x) :=      f(x) ≤ α exclusion region 1 f(x) ≥ 1 − α core (0, 1)

• therwise

shadow α not fixed a priori but chosen according with a balance of vagueness

Davide Ciucci (DISCo) Orthopairs RST 13 / 33

Introduction Models Mathematical aspects

Shadowed sets

An attempt to ”capture uncertainty in a non-numeric fashion” ”provide a transparent and computationally appealing representation of fuzzy sets” From a fuzzy set f : X → [0, 1] to s : X → {0, (0, 1), 1} α ∈ [0, 1 2) sα(f)(x) :=      f(x) ≤ α exclusion region 1 f(x) ≥ 1 − α core (0, 1)

• therwise

shadow α not fixed a priori but chosen according with a balance of vagueness

Davide Ciucci (DISCo) Orthopairs RST 13 / 33

Introduction Models Mathematical aspects

Shadowed sets

An attempt to ”capture uncertainty in a non-numeric fashion” ”provide a transparent and computationally appealing representation of fuzzy sets” From a fuzzy set f : X → [0, 1] to s : X → {0, (0, 1), 1} α ∈ [0, 1 2) sα(f)(x) :=      f(x) ≤ α exclusion region 1 f(x) ≥ 1 − α core (0, 1)

• therwise

shadow α not fixed a priori but chosen according with a balance of vagueness

Davide Ciucci (DISCo) Orthopairs RST 13 / 33

Introduction Models Mathematical aspects

Shadowed and rough sets

Formal point of view Let µH be the membership function of set H We can induce the α-shadowed set generated by µH as      if µH(y) ≤ α 1 if µH(y) ≥ 1 − α

1 2

• therwise

Semantic point of view Different foundations Rough Sets: the approximation space is defined in advance and the equivalence classes are kept fixed Shadowed sets: the three regions are assigned dynamically

Davide Ciucci (DISCo) Orthopairs RST 14 / 33

Introduction Models Mathematical aspects

Shadowed and rough sets

Formal point of view Let µH be the membership function of set H We can induce the α-shadowed set generated by µH as      if µH(y) ≤ α 1 if µH(y) ≥ 1 − α

1 2

• therwise

Semantic point of view Different foundations Rough Sets: the approximation space is defined in advance and the equivalence classes are kept fixed Shadowed sets: the three regions are assigned dynamically

Davide Ciucci (DISCo) Orthopairs RST 14 / 33

Introduction Models Mathematical aspects

Shadowed and rough sets

Formal point of view Let µH be the membership function of set H We can induce the α-shadowed set generated by µH as      if µH(y) ≤ α 1 if µH(y) ≥ 1 − α

1 2

• therwise

Semantic point of view Different foundations Rough Sets: the approximation space is defined in advance and the equivalence classes are kept fixed Shadowed sets: the three regions are assigned dynamically

Davide Ciucci (DISCo) Orthopairs RST 14 / 33

Introduction Models Mathematical aspects

Shadowed and rough sets

Formal point of view Let µH be the membership function of set H We can induce the α-shadowed set generated by µH as      if µH(y) ≤ α 1 if µH(y) ≥ 1 − α

1 2

• therwise

Semantic point of view Different foundations Rough Sets: the approximation space is defined in advance and the equivalence classes are kept fixed Shadowed sets: the three regions are assigned dynamically

Davide Ciucci (DISCo) Orthopairs RST 14 / 33

Introduction Models Mathematical aspects

Shadowed and rough sets

Formal point of view Let µH be the membership function of set H We can induce the α-shadowed set generated by µH as      if µH(y) ≤ α 1 if µH(y) ≥ 1 − α

1 2

• therwise

Semantic point of view Different foundations Rough Sets: the approximation space is defined in advance and the equivalence classes are kept fixed Shadowed sets: the three regions are assigned dynamically

Davide Ciucci (DISCo) Orthopairs RST 14 / 33

Introduction Models Mathematical aspects

Outline

1

Introduction

2

Models

3

Mathematical aspects On three values and orthopairs Operations on orthopairs Operations on rough sets

Davide Ciucci (DISCo) Orthopairs RST 15 / 33

Introduction Models Mathematical aspects

Outline

1

Introduction

2

Models

3

Mathematical aspects On three values and orthopairs Operations on orthopairs Operations on rough sets

Davide Ciucci (DISCo) Orthopairs RST 16 / 33

Introduction Models Mathematical aspects

Three values

Let f be a three valued set on the universe X, f : X → {0, 1

2, 1}

From f to an orthopair (A1, A0) A1 := {x : f(x) = 1} The certainty domain A0 := {x : f(x) = 0} The impossibility domain Au := {x : f(x) = 1

2}

The uncertainty domain From an orthopair to a three valued set f(x) = 1 if x ∈ A1 f(x) = 0 if x ∈ A0 f(x) = 1

2

• rtherwise

Davide Ciucci (DISCo) Orthopairs RST 17 / 33

Introduction Models Mathematical aspects

Three values

Let f be a three valued set on the universe X, f : X → {0, 1

2, 1}

From f to an orthopair (A1, A0) A1 := {x : f(x) = 1} The certainty domain A0 := {x : f(x) = 0} The impossibility domain Au := {x : f(x) = 1

2}

The uncertainty domain From an orthopair to a three valued set f(x) = 1 if x ∈ A1 f(x) = 0 if x ∈ A0 f(x) = 1

2

• rtherwise

Davide Ciucci (DISCo) Orthopairs RST 17 / 33

Introduction Models Mathematical aspects

Three values

Let f be a three valued set on the universe X, f : X → {0, 1

2, 1}

From f to an orthopair (A1, A0) A1 := {x : f(x) = 1} The certainty domain A0 := {x : f(x) = 0} The impossibility domain Au := {x : f(x) = 1

2}

The uncertainty domain From an orthopair to a three valued set f(x) = 1 if x ∈ A1 f(x) = 0 if x ∈ A0 f(x) = 1

2

• rtherwise

Davide Ciucci (DISCo) Orthopairs RST 17 / 33

Introduction Models Mathematical aspects

Three values (2)

Isomorphism between the collection of fuzzy sets F 1

2 (X) := {f|f : X → {0, 1

2, 1}}

and the collection of orthopairs O(X) := {(A1, A0)|A1, A0 ∈ X; A1 ∩ A0 = ∅} The negation of impossibility, i.e., the possibility domain: Ap := Ac From another perspective A1 is the core of the fuzzy set f and Ap its support.

Davide Ciucci (DISCo) Orthopairs RST 18 / 33

Introduction Models Mathematical aspects

Three values (2)

Isomorphism between the collection of fuzzy sets F 1

2 (X) := {f|f : X → {0, 1

2, 1}}

and the collection of orthopairs O(X) := {(A1, A0)|A1, A0 ∈ X; A1 ∩ A0 = ∅} The negation of impossibility, i.e., the possibility domain: Ap := Ac From another perspective A1 is the core of the fuzzy set f and Ap its support.

Davide Ciucci (DISCo) Orthopairs RST 18 / 33

Introduction Models Mathematical aspects

Three values (2)

Isomorphism between the collection of fuzzy sets F 1

2 (X) := {f|f : X → {0, 1

2, 1}}

and the collection of orthopairs O(X) := {(A1, A0)|A1, A0 ∈ X; A1 ∩ A0 = ∅} The negation of impossibility, i.e., the possibility domain: Ap := Ac From another perspective A1 is the core of the fuzzy set f and Ap its support.

Davide Ciucci (DISCo) Orthopairs RST 18 / 33

Introduction Models Mathematical aspects

Bipolar Information

”Bipolarity refers to the propensity of the human mind to reason and make decisions on the basis of positive and negative affects” (Dubois,Prade 2008) examples/counterexamples, pros/cons, . . . Orthopairs: a way to represent bipolar information Orthopair models are usually of type II: Symmetric bivariate unipolarity A1, A0 are defined on the basis of the same data A1, A0 are defined by two independent judgements on unipolar scales three-valued logics is of type I: Symmetric univariate bipolarity single evaluation on a bipolar scale: false, half-true, true International Journal of Intelligent Systems, vol.23 (8-9-10), Special issue

• n Bipolar Information

Davide Ciucci (DISCo) Orthopairs RST 19 / 33

Introduction Models Mathematical aspects

Bipolar Information

”Bipolarity refers to the propensity of the human mind to reason and make decisions on the basis of positive and negative affects” (Dubois,Prade 2008) examples/counterexamples, pros/cons, . . . Orthopairs: a way to represent bipolar information Orthopair models are usually of type II: Symmetric bivariate unipolarity A1, A0 are defined on the basis of the same data A1, A0 are defined by two independent judgements on unipolar scales three-valued logics is of type I: Symmetric univariate bipolarity single evaluation on a bipolar scale: false, half-true, true International Journal of Intelligent Systems, vol.23 (8-9-10), Special issue

• n Bipolar Information

Davide Ciucci (DISCo) Orthopairs RST 19 / 33

Introduction Models Mathematical aspects

Bipolar Information

”Bipolarity refers to the propensity of the human mind to reason and make decisions on the basis of positive and negative affects” (Dubois,Prade 2008) examples/counterexamples, pros/cons, . . . Orthopairs: a way to represent bipolar information Orthopair models are usually of type II: Symmetric bivariate unipolarity A1, A0 are defined on the basis of the same data A1, A0 are defined by two independent judgements on unipolar scales three-valued logics is of type I: Symmetric univariate bipolarity single evaluation on a bipolar scale: false, half-true, true International Journal of Intelligent Systems, vol.23 (8-9-10), Special issue

• n Bipolar Information

Davide Ciucci (DISCo) Orthopairs RST 19 / 33

Introduction Models Mathematical aspects

Bipolar Information

”Bipolarity refers to the propensity of the human mind to reason and make decisions on the basis of positive and negative affects” (Dubois,Prade 2008) examples/counterexamples, pros/cons, . . . Orthopairs: a way to represent bipolar information Orthopair models are usually of type II: Symmetric bivariate unipolarity A1, A0 are defined on the basis of the same data A1, A0 are defined by two independent judgements on unipolar scales three-valued logics is of type I: Symmetric univariate bipolarity single evaluation on a bipolar scale: false, half-true, true International Journal of Intelligent Systems, vol.23 (8-9-10), Special issue

• n Bipolar Information

Davide Ciucci (DISCo) Orthopairs RST 19 / 33

Introduction Models Mathematical aspects

Bipolar Information

”Bipolarity refers to the propensity of the human mind to reason and make decisions on the basis of positive and negative affects” (Dubois,Prade 2008) examples/counterexamples, pros/cons, . . . Orthopairs: a way to represent bipolar information Orthopair models are usually of type II: Symmetric bivariate unipolarity A1, A0 are defined on the basis of the same data A1, A0 are defined by two independent judgements on unipolar scales three-valued logics is of type I: Symmetric univariate bipolarity single evaluation on a bipolar scale: false, half-true, true International Journal of Intelligent Systems, vol.23 (8-9-10), Special issue

• n Bipolar Information

Davide Ciucci (DISCo) Orthopairs RST 19 / 33

Introduction Models Mathematical aspects

Bipolar Information

”Bipolarity refers to the propensity of the human mind to reason and make decisions on the basis of positive and negative affects” (Dubois,Prade 2008) examples/counterexamples, pros/cons, . . . Orthopairs: a way to represent bipolar information Orthopair models are usually of type II: Symmetric bivariate unipolarity A1, A0 are defined on the basis of the same data A1, A0 are defined by two independent judgements on unipolar scales three-valued logics is of type I: Symmetric univariate bipolarity single evaluation on a bipolar scale: false, half-true, true International Journal of Intelligent Systems, vol.23 (8-9-10), Special issue

• n Bipolar Information

Davide Ciucci (DISCo) Orthopairs RST 19 / 33

Introduction Models Mathematical aspects

Outline

1

Introduction

2

Models

3

Mathematical aspects On three values and orthopairs Operations on orthopairs Operations on rough sets

Davide Ciucci (DISCo) Orthopairs RST 20 / 33

Introduction Models Mathematical aspects

Operations

Define operations on three values F 1

2 (X) and then report on orthopairs

O(X) Problem What happens if we are considering a subset of O(X)? Given an operation f : O(X) → O(X) and a subset S ⊆ O(X), is it true that f : S → S?

Davide Ciucci (DISCo) Orthopairs RST 21 / 33

Introduction Models Mathematical aspects

Operations

Define operations on three values F 1

2 (X) and then report on orthopairs

O(X) Problem What happens if we are considering a subset of O(X)? Given an operation f : O(X) → O(X) and a subset S ⊆ O(X), is it true that f : S → S?

Davide Ciucci (DISCo) Orthopairs RST 21 / 33

Introduction Models Mathematical aspects

Conjuntion: T-norms

On three values it is possible to define only two (discrete) t-norms: Gödel and Łukasiewicz

• T. Bartušek and M. Navara

Conjunctions of many-valued criteria, Proceedings of the International Conference Uncertainty Modelling’2001 (Bratislava, Slovakia) (M. Komorníková and R. Mesiar, eds.), Slovak Technical University, 2001, pp. 67–77. ∧

1 2

1

1 2 1 2 1 2

1

1 2

1 ⊙

1 2

1

1 2 1 2

1

1 2

1

Davide Ciucci (DISCo) Orthopairs RST 22 / 33

Introduction Models Mathematical aspects

Conjuntion: T-norms

On three values it is possible to define only two (discrete) t-norms: Gödel and Łukasiewicz

• T. Bartušek and M. Navara

Conjunctions of many-valued criteria, Proceedings of the International Conference Uncertainty Modelling’2001 (Bratislava, Slovakia) (M. Komorníková and R. Mesiar, eds.), Slovak Technical University, 2001, pp. 67–77. ∧

1 2

1

1 2 1 2 1 2

1

1 2

1 ⊙

1 2

1

1 2 1 2

1

1 2

1

Davide Ciucci (DISCo) Orthopairs RST 22 / 33

Introduction Models Mathematical aspects

Conjuntion: T-norms

On three values it is possible to define only two (discrete) t-norms: Gödel and Łukasiewicz

• T. Bartušek and M. Navara

Conjunctions of many-valued criteria, Proceedings of the International Conference Uncertainty Modelling’2001 (Bratislava, Slovakia) (M. Komorníková and R. Mesiar, eds.), Slovak Technical University, 2001, pp. 67–77. ∧

1 2

1

1 2 1 2 1 2

1

1 2

1 ⊙

1 2

1

1 2 1 2

1

1 2

1

Davide Ciucci (DISCo) Orthopairs RST 22 / 33

Introduction Models Mathematical aspects

T-norms on orthopairs

Gödel t-norm and t-conorm (A1, A0) ⊓ (B1, B0) := (A1 ∩ B1, A0 ∪ B0) (A1, A0) ⊔ (B1, B0) := (A1 ∪ B1, A0 ∩ B0) The structure O(X), ∩, ∪, (∅, X), (X, ∅) is a lattice bounded by the minimum element (∅, X) and the maximum (X, ∅) Łukasiewicz t-norm and t-conorm (A1, A0) ⊠ (B1, B0) := (A1 ∩ B1, (Ac

1 ∩ Bc 1) ∪ A0 ∪ B0))

(A1, A0) ⊞ (B1, B0) := ((Ac

0 ∩ Bc 0) ∪ A1 ∪ B1, (A0 ∩ B0))

Davide Ciucci (DISCo) Orthopairs RST 23 / 33

Introduction Models Mathematical aspects

T-norms on orthopairs

Gödel t-norm and t-conorm (A1, A0) ⊓ (B1, B0) := (A1 ∩ B1, A0 ∪ B0) (A1, A0) ⊔ (B1, B0) := (A1 ∪ B1, A0 ∩ B0) The structure O(X), ∩, ∪, (∅, X), (X, ∅) is a lattice bounded by the minimum element (∅, X) and the maximum (X, ∅) Łukasiewicz t-norm and t-conorm (A1, A0) ⊠ (B1, B0) := (A1 ∩ B1, (Ac

1 ∩ Bc 1) ∪ A0 ∪ B0))

(A1, A0) ⊞ (B1, B0) := ((Ac

0 ∩ Bc 0) ∪ A1 ∪ B1, (A0 ∩ B0))

Davide Ciucci (DISCo) Orthopairs RST 23 / 33

Introduction Models Mathematical aspects

Implications

Definition (Many valued implication) An implication on [0, 1] is a binary mapping →: [0, 1] × [0, 1] → [0, 1] such that (I1) If x ≤ y then x → z ≥ y → z; (I2) If x ≤ y then z → x ≤ z → y; (I3) 0 → 0 = 1 → 1 = 1 and 1 → 0 = 0. On three values there are 14 operations which satisfy the above definition Among all these possibilities, only two implications are the restriction

• f a t-norm residuum: the Łukasiewicz and Gödel implications

Davide Ciucci (DISCo) Orthopairs RST 24 / 33

Introduction Models Mathematical aspects

Implications

Definition (Many valued implication) An implication on [0, 1] is a binary mapping →: [0, 1] × [0, 1] → [0, 1] such that (I1) If x ≤ y then x → z ≥ y → z; (I2) If x ≤ y then z → x ≤ z → y; (I3) 0 → 0 = 1 → 1 = 1 and 1 → 0 = 0. On three values there are 14 operations which satisfy the above definition Among all these possibilities, only two implications are the restriction

• f a t-norm residuum: the Łukasiewicz and Gödel implications

Davide Ciucci (DISCo) Orthopairs RST 24 / 33

Introduction Models Mathematical aspects

Implications

Definition (Many valued implication) An implication on [0, 1] is a binary mapping →: [0, 1] × [0, 1] → [0, 1] such that (I1) If x ≤ y then x → z ≥ y → z; (I2) If x ≤ y then z → x ≤ z → y; (I3) 0 → 0 = 1 → 1 = 1 and 1 → 0 = 0. On three values there are 14 operations which satisfy the above definition Among all these possibilities, only two implications are the restriction

• f a t-norm residuum: the Łukasiewicz and Gödel implications

Davide Ciucci (DISCo) Orthopairs RST 24 / 33

Introduction Models Mathematical aspects

Residual Implications

On three values x →L y := min{1, 1 − x + y} x →G y :=

• 1

x ≤ y y x > y On orthopairs (A1, A0) ⇒L (B1, B0) := ((Ac

1 ∩ Bc 0) ∪ A0 ∪ B1, A1 ∩ B0)

(A1, A0) ⇒G (B1, B0) := ((Ac

1 ∩ Bc 0) ∪ A0 ∪ B1, Ac 0 ∩ B0)

Let us note, that they are the implications introduced on rough sets by (Pagliani 1998) and on abstract rough approximations (Cattaneo, Ciucci 2003).

Davide Ciucci (DISCo) Orthopairs RST 25 / 33

Introduction Models Mathematical aspects

Residual Implications

On three values x →L y := min{1, 1 − x + y} x →G y :=

• 1

x ≤ y y x > y On orthopairs (A1, A0) ⇒L (B1, B0) := ((Ac

1 ∩ Bc 0) ∪ A0 ∪ B1, A1 ∩ B0)

(A1, A0) ⇒G (B1, B0) := ((Ac

1 ∩ Bc 0) ∪ A0 ∪ B1, Ac 0 ∩ B0)

Let us note, that they are the implications introduced on rough sets by (Pagliani 1998) and on abstract rough approximations (Cattaneo, Ciucci 2003).

Davide Ciucci (DISCo) Orthopairs RST 25 / 33

Introduction Models Mathematical aspects

Residual Implications

On three values x →L y := min{1, 1 − x + y} x →G y :=

• 1

x ≤ y y x > y On orthopairs (A1, A0) ⇒L (B1, B0) := ((Ac

1 ∩ Bc 0) ∪ A0 ∪ B1, A1 ∩ B0)

(A1, A0) ⇒G (B1, B0) := ((Ac

1 ∩ Bc 0) ∪ A0 ∪ B1, Ac 0 ∩ B0)

Let us note, that they are the implications introduced on rough sets by (Pagliani 1998) and on abstract rough approximations (Cattaneo, Ciucci 2003).

Davide Ciucci (DISCo) Orthopairs RST 25 / 33

Introduction Models Mathematical aspects

HW algebra

The algebraic structure defined in order to treat at the same time Łukasiewicz and Gödel logic is known as Heyting Wajsberg (HW) algebra (Cattaneo,Ciucci,Giuntini,Konig,2004) HW algebras are equivalent to other well known structures: MV∆ algebras, obtained adding a Baaz’s operator ∆ to MV algebras (Baaz 1996,Hajek 1998) Stonean MV algebra, a particular class of MV algebras introduced by Belluce (Belluce 1995) BZMVdM algebras, a pasting of MV algebras and BZ lattices (Cattaneo,Giuntini,Pilla 1999)

Davide Ciucci (DISCo) Orthopairs RST 26 / 33

Introduction Models Mathematical aspects

HW algebra

The algebraic structure defined in order to treat at the same time Łukasiewicz and Gödel logic is known as Heyting Wajsberg (HW) algebra (Cattaneo,Ciucci,Giuntini,Konig,2004) HW algebras are equivalent to other well known structures: MV∆ algebras, obtained adding a Baaz’s operator ∆ to MV algebras (Baaz 1996,Hajek 1998) Stonean MV algebra, a particular class of MV algebras introduced by Belluce (Belluce 1995) BZMVdM algebras, a pasting of MV algebras and BZ lattices (Cattaneo,Giuntini,Pilla 1999)

Davide Ciucci (DISCo) Orthopairs RST 26 / 33

Introduction Models Mathematical aspects

Outline

1

Introduction

2

Models

3

Mathematical aspects On three values and orthopairs Operations on orthopairs Operations on rough sets

Davide Ciucci (DISCo) Orthopairs RST 27 / 33

Introduction Models Mathematical aspects

The lattice operation case (1)

Let re(A) = l(A), e(A), re(B) = l(B), e(B) two rough sets Since re(A), re(B) are orthopairs we can define re(A) ⊓ re(B) = (l(A) ∩ l(B), e(A) ∪ e(B)) re(A) ⊔ re(B) = (l(A) ∪ l(B), e(A) ∩ e(B)) The elements re(A) ⊓ re(B) and re(A) ⊔ re(B) are orthopairs Are also rough sets? That is we ask if there exists elements C, D such that re(C) = re(A) ⊓ re(B) and re(D) = re(A) ⊔ re(B)

Davide Ciucci (DISCo) Orthopairs RST 28 / 33

Introduction Models Mathematical aspects

The lattice operation case (1)

Let re(A) = l(A), e(A), re(B) = l(B), e(B) two rough sets Since re(A), re(B) are orthopairs we can define re(A) ⊓ re(B) = (l(A) ∩ l(B), e(A) ∪ e(B)) re(A) ⊔ re(B) = (l(A) ∪ l(B), e(A) ∩ e(B)) The elements re(A) ⊓ re(B) and re(A) ⊔ re(B) are orthopairs Are also rough sets? That is we ask if there exists elements C, D such that re(C) = re(A) ⊓ re(B) and re(D) = re(A) ⊔ re(B)

Davide Ciucci (DISCo) Orthopairs RST 28 / 33

Introduction Models Mathematical aspects

The lattice operation case (1)

Let re(A) = l(A), e(A), re(B) = l(B), e(B) two rough sets Since re(A), re(B) are orthopairs we can define re(A) ⊓ re(B) = (l(A) ∩ l(B), e(A) ∪ e(B)) re(A) ⊔ re(B) = (l(A) ∪ l(B), e(A) ∩ e(B)) The elements re(A) ⊓ re(B) and re(A) ⊔ re(B) are orthopairs Are also rough sets? That is we ask if there exists elements C, D such that re(C) = re(A) ⊓ re(B) and re(D) = re(A) ⊔ re(B)

Davide Ciucci (DISCo) Orthopairs RST 28 / 33

Introduction Models Mathematical aspects

The lattice operation case (1)

Let re(A) = l(A), e(A), re(B) = l(B), e(B) two rough sets Since re(A), re(B) are orthopairs we can define re(A) ⊓ re(B) = (l(A) ∩ l(B), e(A) ∪ e(B)) re(A) ⊔ re(B) = (l(A) ∪ l(B), e(A) ∩ e(B)) The elements re(A) ⊓ re(B) and re(A) ⊔ re(B) are orthopairs Are also rough sets? That is we ask if there exists elements C, D such that re(C) = re(A) ⊓ re(B) and re(D) = re(A) ⊔ re(B)

Davide Ciucci (DISCo) Orthopairs RST 28 / 33

Introduction Models Mathematical aspects

The lattice operation case (2)

Theoretical point of view There exists C, D such that r(C) = re(A) ⊓ re(B) and re(D) = re(A) ⊔ re(B) (Bonikowski,1992) Semantic point of view C, D cannot be computed simply by use of A and B C, D are not uniquely defined C, D strongly depends on the partition

Davide Ciucci (DISCo) Orthopairs RST 29 / 33

Introduction Models Mathematical aspects

The lattice operation case (2)

Theoretical point of view There exists C, D such that r(C) = re(A) ⊓ re(B) and re(D) = re(A) ⊔ re(B) (Bonikowski,1992) Semantic point of view C, D cannot be computed simply by use of A and B C, D are not uniquely defined C, D strongly depends on the partition

Davide Ciucci (DISCo) Orthopairs RST 29 / 33

Introduction Models Mathematical aspects

Example

Let X = {a, b, c, d, e, f} and π1, π2 two partitions: π1 = {a, b}, {c, d}, {e, f} π2 = {a, b, c}, {d, e, f} Consider the two sets X1 = {b, c} and X2 = {c, d} and compute the approximations with respect to Partition 1 re({b, c}) = (∅, {a, b, c, d}), re({c, d}) = ({c, d}, {c, d}) re({b, c}) ⊓ re({c, d}) = (∅, {a, b, c, d}) = re({x, y}) with x ∈ {a, b} and y ∈ {c, d} Partition 2 re({b, c}) = (∅, {a, b, c}), re({c, d}) = (∅, X) re({b, c}) ⊓ re({c, d}) = (∅, X) = re(∅) = re({x, y}) with x ∈ {a, b, c} and y ∈ {d, e, f}

Davide Ciucci (DISCo) Orthopairs RST 30 / 33

Introduction Models Mathematical aspects

Example

Let X = {a, b, c, d, e, f} and π1, π2 two partitions: π1 = {a, b}, {c, d}, {e, f} π2 = {a, b, c}, {d, e, f} Consider the two sets X1 = {b, c} and X2 = {c, d} and compute the approximations with respect to Partition 1 re({b, c}) = (∅, {a, b, c, d}), re({c, d}) = ({c, d}, {c, d}) re({b, c}) ⊓ re({c, d}) = (∅, {a, b, c, d}) = re({x, y}) with x ∈ {a, b} and y ∈ {c, d} Partition 2 re({b, c}) = (∅, {a, b, c}), re({c, d}) = (∅, X) re({b, c}) ⊓ re({c, d}) = (∅, X) = re(∅) = re({x, y}) with x ∈ {a, b, c} and y ∈ {d, e, f}

Davide Ciucci (DISCo) Orthopairs RST 30 / 33

Introduction Models Mathematical aspects

Example

Let X = {a, b, c, d, e, f} and π1, π2 two partitions: π1 = {a, b}, {c, d}, {e, f} π2 = {a, b, c}, {d, e, f} Consider the two sets X1 = {b, c} and X2 = {c, d} and compute the approximations with respect to Partition 1 re({b, c}) = (∅, {a, b, c, d}), re({c, d}) = ({c, d}, {c, d}) re({b, c}) ⊓ re({c, d}) = (∅, {a, b, c, d}) = re({x, y}) with x ∈ {a, b} and y ∈ {c, d} Partition 2 re({b, c}) = (∅, {a, b, c}), re({c, d}) = (∅, X) re({b, c}) ⊓ re({c, d}) = (∅, X) = re(∅) = re({x, y}) with x ∈ {a, b, c} and y ∈ {d, e, f}

Davide Ciucci (DISCo) Orthopairs RST 30 / 33

Introduction Models Mathematical aspects

Implications

Same problem all the 14 implications re(A) → re(B) depend on ∪ and ∩ Another solution define the implication directly on sets A, B to obtain A → B then compute its approximation re(A → B) = re(Ac ∪ B) Thus, re(A) ⇒ re(B) := re(Ac ∪ B) depends only on A, B but we cannot compute its explicit form starting from re(A), re(B)

Davide Ciucci (DISCo) Orthopairs RST 31 / 33

Introduction Models Mathematical aspects

Implications

Same problem all the 14 implications re(A) → re(B) depend on ∪ and ∩ Another solution define the implication directly on sets A, B to obtain A → B then compute its approximation re(A → B) = re(Ac ∪ B) Thus, re(A) ⇒ re(B) := re(Ac ∪ B) depends only on A, B but we cannot compute its explicit form starting from re(A), re(B)

Davide Ciucci (DISCo) Orthopairs RST 31 / 33

Introduction Models Mathematical aspects

Implications

Same problem all the 14 implications re(A) → re(B) depend on ∪ and ∩ Another solution define the implication directly on sets A, B to obtain A → B then compute its approximation re(A → B) = re(Ac ∪ B) Thus, re(A) ⇒ re(B) := re(Ac ∪ B) depends only on A, B but we cannot compute its explicit form starting from re(A), re(B)

Davide Ciucci (DISCo) Orthopairs RST 31 / 33

Introduction Models Mathematical aspects

Implications

Same problem all the 14 implications re(A) → re(B) depend on ∪ and ∩ Another solution define the implication directly on sets A, B to obtain A → B then compute its approximation re(A → B) = re(Ac ∪ B) Thus, re(A) ⇒ re(B) := re(Ac ∪ B) depends only on A, B but we cannot compute its explicit form starting from re(A), re(B)

Davide Ciucci (DISCo) Orthopairs RST 31 / 33

Introduction Models Mathematical aspects

Implications (2)

Another solution (Avron,Konikoswka 2008) the non-deterministic behaviour of rough sets is brought directly into a logical calculus The semantics of the implication on three values is given by a non-deterministic NMatrix →

1 2

1 1 1 1

1 2 1 2

{1

2, 1}

1 1

1 2

1 The problem of non-determinism still remains, it is just shifted on a different level

Davide Ciucci (DISCo) Orthopairs RST 32 / 33

Introduction Models Mathematical aspects

Conclusions

Orthopairs are a general mathematical tool to describe several approaches to reasoning with uncertainty and bipolar information From a mathematical standpoint: define operations, structures on three values and then report on orthopairs Pay attention we translating results from orthopairs to its models Rough sets: intrinsic non-determinism which reflects on the

• perations

Davide Ciucci (DISCo) Orthopairs RST 33 / 33

Introduction Models Mathematical aspects

Conclusions

Orthopairs are a general mathematical tool to describe several approaches to reasoning with uncertainty and bipolar information From a mathematical standpoint: define operations, structures on three values and then report on orthopairs Pay attention we translating results from orthopairs to its models Rough sets: intrinsic non-determinism which reflects on the

• perations

Davide Ciucci (DISCo) Orthopairs RST 33 / 33

Introduction Models Mathematical aspects

Conclusions

Orthopairs are a general mathematical tool to describe several approaches to reasoning with uncertainty and bipolar information From a mathematical standpoint: define operations, structures on three values and then report on orthopairs Pay attention we translating results from orthopairs to its models Rough sets: intrinsic non-determinism which reflects on the

• perations

Davide Ciucci (DISCo) Orthopairs RST 33 / 33

Introduction Models Mathematical aspects

Conclusions

Orthopairs are a general mathematical tool to describe several approaches to reasoning with uncertainty and bipolar information From a mathematical standpoint: define operations, structures on three values and then report on orthopairs Pay attention we translating results from orthopairs to its models Rough sets: intrinsic non-determinism which reflects on the

• perations

Davide Ciucci (DISCo) Orthopairs RST 33 / 33