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

Jul 30, 2023 131 likes •347 views

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

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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
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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

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Relations and Multirelations

  • state space A = {1, 2, 3}
  • relation ⊆ A × A

multirelation ⊆ A × 2A

1 2 3 1 2 3 1 2 3 ∅ 3 2 23 1 13 12 123

  • Boolean algebra with ∪, ∩,
  • composition
  • converse ·c, dual ·d

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 3

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Multirelational Constants

O =

1 2 3 ∅ 3 2 23 1 13 12 123

T =

1 2 3 ∅ 3 2 23 1 13 12 123

E =

1 2 3 ∅ 3 2 23 1 13 12 123

U =

1 2 3 ∅ 3 2 23 1 13 12 123

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 4

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Relational Composition

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

(QR)x,z ⇔ ∃y ∈ A : Qx,y ∧ Ry,z

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 5

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Multirelational Composition

1 2 3 ∅ 3 2 23 1 13 12 123 1 2 3 ∅ 3 2 23 1 13 12 123 1 2 3 ∅ 3 2 23 1 13 12 123

(Q ;R)x,Z ⇔ ∃Y ∈ 2A : Qx,Y ∧ ∀y ∈ Y : Ry,Z

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 6

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Up-closed Multirelations

not up-closed

1 2 3 ∅ 3 2 23 1 13 12 123

up-closed

1 2 3 ∅ 3 2 23 1 13 12 123

∀x ∈ A : ∀Y , Z ∈ 2A : Rx,Y ∧ Y ⊆ Z ⇒ Rx,Z

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 7

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Relational Operations for Multirelations

right residual Q \ R = QcR symmetric quotient Q÷R = (Q \ R) ∩ (R \ Q)c subset relation : 2A ↔ 2A S = E \ E multirelational composition Q ;R = Q(E \ R) R up-closed if R = RS

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 8

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Unit and Zero of Multirelations

left unit E;R = E(E \ R) = R right unit R ;E = R(E \ E) = RS = R if R up-closed left zero O;R = O T;R = T

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 9

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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

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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

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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 (µf ) ≤ µf f (x) ≤ x ⇒ µf ≤ x if µf , µg, µh exist then µf ≤ µg = µh

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 12

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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

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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 · yd (x + y)d = xd yd xdd = x 1d = 1

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 14

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Relationships between Properties

idempotent dense transitive total co-total reflexive co-reflexive contact up-closed kernel ∪-distributive ∩-distributive ∪-distributive contact ∩-distributive kernel test co-test vector up-closed

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 15

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Closure Properties

O E T ∪ ∩ ;

d

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

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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 → 2T

  • contact multirelation R : A ↔ 2A

Rx,Y ⇔ t(x) ⊆

  • y∈Y

t(y)

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 17

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Axioms of Contact Relations

(K0) ¬∃x ∈ A : Rx,∅ (K1) ∀x ∈ A : Rx,{x} (K2) ∀x ∈ A : ∀Y , Z ∈ 2A : Rx,Y ∧ Y ⊆ Z ⇒ Rx,Z (K3) ∀x ∈ A : ∀Y , Z ∈ 2A : Rx,Y ∧ (∀y ∈ Y : Ry,Z) ⇒ Rx,Z (K4) ∀x ∈ A : ∀Y , Z ∈ 2A : Rx,Y ∪Z ⇔ Rx,Y ∨ Rx,Z (K1)–(K3) contact relation (K0)–(K4) topological contact relation

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 18

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Examples of Topological Contact

  • ∈

A ↔ 2A

  • Rx,Y ⇔ ∃y ∈ Y : f (x) = f (y) where f : A → B
  • Rx,Y ⇔ ∃y ∈ Y : x ≤ y

N ↔ 2N

  • Rx,Y ⇔ ∃y1, y2 ∈ Y : y1 ≤ x ≤ y2
  • Rx,Y ⇔ ∃yi ∈ Y : ∃ri ∈ Q : x = riyi

Rn ↔ 2Rn

  • Rx,Y ⇔ ∃yi ∈ Y : ∃ri ∈ Q+

0 : ri = 1 ∧ x = riyi

  • Rx,Y ⇔ ∀ε > 0 : ∃y ∈ Y : d(x, y) < ε

satisfy (K0)–(K3), some also (K4)

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 19

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Axioms using Multirelational Operations

(K0) R ;O = O co-total (K1) E ⊆ R (if R up-closed) reflexive (K2) R ;E = R up-closed (K3) R ;R ⊆ R transitive (K4) R ;(P ∪ Q) = R ;P ∪ R ;Q (if R up-closed) ∪-distributive

Rudolf Berghammer, Walter Guttmann · RAMiCS · 2015-09-30 20

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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