Closure, Properties and Closure Properties of Multirelations Rudolf - PowerPoint PPT Presentation

Closure, Properties and Closure Properties of Multirelations Rudolf Berghammer Walter Guttmann Christian-Albrechts-Universit¨ at zu Kiel University of Canterbury 1. Multirelations 2. Reflexive - Transitive Closure 3. Properties and their Closure 4. Topological Contact

Context and Method • multirelations in program semantics, games, topological contact • systematically investigate their properties • express multirelational operations using relations • study properties of operations • abstract properties to weak algebras • derive theory in these algebras Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 2

Relations and Multirelations • state space A = { 1 , 2 , 3 } • relation ⊆ A × A multirelation ⊆ A × 2 A 123 23 13 12 1 2 3 ∅ 3 2 1 1 1 2 2 3 3 • Boolean algebra with ∪ , ∩ , • composition • converse · c , dual · d Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 3

Multirelational Constants 123 123 23 13 12 23 13 12 ∅ 3 2 1 ∅ 3 2 1 O = T = 1 1 2 2 3 3 123 123 23 13 12 23 13 12 ∅ 3 2 1 ∅ 3 2 1 E = U = 1 1 2 2 3 3 Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 4

Relational Composition 1 2 3 1 2 3 1 2 3 1 2 3 1 1 2 2 3 3 ( QR ) x , z ⇔ ∃ y ∈ A : Q x , y ∧ R y , z Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 5

Multirelational Composition 123 23 13 12 ∅ 3 2 1 1 2 3 123 123 23 13 12 23 13 12 ∅ 3 2 1 ∅ 3 2 1 1 1 2 2 3 3 ( Q ; R ) x , Z ⇔ ∃ Y ∈ 2 A : Q x , Y ∧ ∀ y ∈ Y : R y , Z Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 6

Up-closed Multirelations 123 23 13 12 ∅ 3 2 1 not up-closed 1 2 3 123 23 13 12 ∅ 3 2 1 up-closed 1 2 3 ∀ x ∈ A : ∀ Y , Z ∈ 2 A : R x , Y ∧ Y ⊆ Z ⇒ R x , Z Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 7

Relational Operations for Multirelations Q c R right residual Q \ R = ( Q \ R ) ∩ ( R \ Q ) c symmetric quotient Q ÷ R = subset relation : 2 A ↔ 2 A S = E \ E multirelational composition Q ; R = Q (E \ R ) R up-closed if = R S R Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 8

Unit and Zero of Multirelations left unit E; R = E(E \ R ) = R right unit R ;E = R (E \ E) = R S = R if R up-closed left zero O; R = O T; R = T Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 9

Laws of Multirelations all multirelations up-closed multirelations O; R = O E; R = R T; R = T R ;E ⊇ R R ;E = R Q ⊆ R ⇒ P ; Q ⊆ P ; R ( P ∪ Q ); R = P ; R ∪ Q ; R ( P ∩ Q ); R ⊆ P ; R ∩ Q ; R ( P ∩ Q ); R = P ; R ∩ Q ; R ( P ; Q ); R ⊆ P ;( Q ; R ) ( P ; Q ); R = P ;( Q ; R ) Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 10

Algebraic Structures bounded join-semilattice x + ( y + z ) = ( x + y ) + z x + x = x x + y = y + x 0 + x = x pre-left semiring ( x · y ) + ( x · z ) ≤ x · ( y + z ) ( x · y ) · z ≤ x · ( y · z ) ( x · z ) + ( y · z ) = ( x + y ) · z x ≤ x · 1 0 = 0 · x x = 1 · x left residual x · y ≤ z ⇔ x ≤ z / y Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 11

Reflexive-Transitive Closure recursion modelled by f ( x ) = 1 + x · y g ( x ) = 1 + y · x h ( x ) = 1 + y + x · x least prefixpoint f ( µ f ) ≤ µ f f ( x ) ≤ x ⇒ µ f ≤ x if µ f , µ g , µ h exist then µ f ≤ µ g = µ h Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 12

Properties of Multirelations up-closed R ;E = R total R ;T = T co-total R ;O = O ∪ -distributive R ;( P ∪ Q ) = R ; P ∪ R ; Q ∩ -distributive R ;( P ∩ Q ) = R ; P ∩ R ; Q reflexive E ⊆ R co-reflexive R ⊆ E transitive R ; R ⊆ R dense R ⊆ R ; R idempotent R ; R = R contact R ; R ∪ E = R kernel R ; R ∩ E = R ;E test R ;T ∩ E = R co-test R ;O ∪ E = R vector R ;T = R Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 13

Algebraic Structures ( S , + , � , 0 , ⊤ ) bounded distributive lattice, ( S , + , · , 0 , 1) pre-left semiring and ⊤ = ⊤ · x x · ( y · z ) = ( x · ( y · 1)) · z ( x · z ) � ( y · z ) = (( x · 1) � ( y · 1)) · z dual ( x · y ) d = ( x · 1) d · y d ( x + y ) d = x d � y d x dd = x 1 d = 1 Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 14

Relationships between Properties co-total transitive dense total idempotent co-reflexive reflexive up-closed up-closed ∩ -distributive ∪ -distributive contact kernel ∩ -distributive kernel vector ∪ -distributive contact test co-test Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 15

Closure Properties d O E T ∪ ∩ ; total − � � � � � ▽ co-total − � � � � � � transitive − − � � � � � dense − − � � � � △ reflexive − � � � � � � − co-reflexive � � � � � � − − − idempotent � � � � up-closed � � � � � � � ∪ -distributive − � � � � � ▽ ∩ -distributive − � � � � � △ − − − a contact � � � � − − − a kernel � � � � a ∪ -distributive contact − − − − � � � a ∩ -distributive kernel − − − − � � � − a test � � � � � � − a co-test � � � � � � − a vector � � � � � � Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 16

Topological Contact • according to G. Aumann (1970) • set of persons A • set of topics T • t ( x ) = topics person x is interested in t : A → 2 T • contact multirelation R : A ↔ 2 A � R x , Y ⇔ t ( x ) ⊆ t ( y ) y ∈ Y Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 17

Axioms of Contact Relations ( K 0 ) ¬∃ x ∈ A : R x , ∅ ( K 1 ) ∀ x ∈ A : R x , { x } ∀ x ∈ A : ∀ Y , Z ∈ 2 A : R x , Y ∧ Y ⊆ Z ⇒ R x , Z ( K 2 ) ∀ x ∈ A : ∀ Y , Z ∈ 2 A : R x , Y ∧ ( ∀ y ∈ Y : R y , Z ) ⇒ R x , Z ( K 3 ) ∀ x ∈ A : ∀ Y , Z ∈ 2 A : R x , Y ∪ Z ⇔ R x , Y ∨ R x , Z ( K 4 ) ( K 1 )–( K 3 ) contact relation ( K 0 )–( K 4 ) topological contact relation Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 18

Examples of Topological Contact A ↔ 2 A • ∈ • R x , Y ⇔ ∃ y ∈ Y : f ( x ) = f ( y ) where f : A → B N ↔ 2 N • R x , Y ⇔ ∃ y ∈ Y : x ≤ y • R x , Y ⇔ ∃ y 1 , y 2 ∈ Y : y 1 ≤ x ≤ y 2 • R x , Y ⇔ ∃ y i ∈ Y : ∃ r i ∈ Q : x = � r i y i R n ↔ 2 R n 0 : � r i = 1 ∧ x = � r i y i • R x , Y ⇔ ∃ y i ∈ Y : ∃ r i ∈ Q + • R x , Y ⇔ ∀ ε > 0 : ∃ y ∈ Y : d ( x , y ) < ε satisfy ( K 0 )–( K 3 ), some also ( K 4 ) Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 19

Axioms using Multirelational Operations ( K 0 ) R ;O = O co-total ( K 1 ) E ⊆ R (if R up-closed) reflexive ( K 2 ) R ;E = R up-closed ( K 3 ) R ; R ⊆ R transitive ( K 4 ) R ;( P ∪ Q ) = R ; P ∪ R ; Q (if R up-closed) ∪ -distributive Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 20

Conclusion • multirelations describe topological contact • also consider not up-closed multirelations • many results hold in weak algebras • study connections to topology and closure systems • generate further counterexamples • give complete axioms Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 21