A first-order logic for string diagrams Aleks Kissinger David Quick - - PowerPoint PPT Presentation

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A first-order logic for string diagrams Aleks Kissinger David Quick - - PowerPoint PPT Presentation

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Introduction !-Logic Interpretation Example Summary A first-order logic for string diagrams Aleks Kissinger David Quick Oxford Quantum Group CALCO June 2015 Q UANTUM G ROUP Introduction !-Logic Interpretation Example Summary

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SLIDE 1

Introduction !-Logic Interpretation Example Summary

A first-order logic for string diagrams

Aleks Kissinger David Quick

Oxford Quantum Group

CALCO June 2015 QUANTUM GROUP

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SLIDE 2

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

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SLIDE 3

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

  • ,

Diagram(Σ) :=

  • ,

, , . . .

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SLIDE 4

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

  • ,

Diagram(Σ) :=

  • ,

, , . . .

  • =

,

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SLIDE 5

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

  • ,

Diagram(Σ) :=

  • ,

, , . . .

  • =

,

= =

,

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SLIDE 6

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

  • ,

Diagram(Σ) :=

  • ,

, , . . .

  • =

,

= =

,

=

,

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SLIDE 7

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

  • ,

Diagram(Σ) :=

  • ,

, , . . .

  • =

,

= =

,

=

,

=

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SLIDE 8

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

  • ,

Diagram(Σ) :=

  • ,

, , . . .

  • =

,

= =

,

=

,

=

Reason using substitution:

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SLIDE 9

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

  • ,

Diagram(Σ) :=

  • ,

, , . . .

  • =

,

= =

,

=

,

=

Reason using substitution:

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SLIDE 10

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

  • ,

Diagram(Σ) :=

  • ,

, , . . .

  • =

,

= =

,

=

,

=

Reason using substitution:

=

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SLIDE 11

Introduction !-Logic Interpretation Example Summary

Diagrammatic Reasoning

Given Σ =

  • ,

,

  • ,

Diagram(Σ) :=

  • ,

, , . . .

  • =

,

= =

,

=

,

=

Reason using substitution:

= =

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SLIDE 12

Introduction !-Logic Interpretation Example Summary

Families of Diagrams

Extend theory to arbitrary arity node

. . . . . . =

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SLIDE 13

Introduction !-Logic Interpretation Example Summary

Families of Diagrams

Extend theory to arbitrary arity node

. . . . . . =

and hope

. . . = . . .

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SLIDE 14

Introduction !-Logic Interpretation Example Summary

!-Boxes

. . . replaces

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SLIDE 15

Introduction !-Logic Interpretation Example Summary

!-Boxes

. . . replaces

and represents the concrete diagrams obtained from two !-box operations:

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SLIDE 16

Introduction !-Logic Interpretation Example Summary

!-Boxes

. . . replaces

and represents the concrete diagrams obtained from two !-box operations:

Kill

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SLIDE 17

Introduction !-Logic Interpretation Example Summary

!-Boxes

. . . replaces

and represents the concrete diagrams obtained from two !-box operations:

Kill Exp

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SLIDE 18

Introduction !-Logic Interpretation Example Summary

!-Boxes

. . . replaces

and represents the concrete diagrams obtained from two !-box operations:

Kill Exp Kill

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SLIDE 19

Introduction !-Logic Interpretation Example Summary

!-Boxes

. . . replaces

and represents the concrete diagrams obtained from two !-box operations:

Kill Exp Kill Exp

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SLIDE 20

Introduction !-Logic Interpretation Example Summary

!-Boxes

. . . replaces

and represents the concrete diagrams obtained from two !-box operations:

Kill Exp Kill Exp Kill

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SLIDE 21

Introduction !-Logic Interpretation Example Summary

!-Box Equations

A A

=

replaces

. . . = . . .

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SLIDE 22

Introduction !-Logic Interpretation Example Summary

!-Box Equations

A A

=

replaces

. . . = . . .

and represents

  • A

A

=

  • =
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SLIDE 23

Introduction !-Logic Interpretation Example Summary

!-Box Equations

A A

=

replaces

. . . = . . .

and represents

  • A

A

=

  • =

, =

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SLIDE 24

Introduction !-Logic Interpretation Example Summary

!-Box Equations

A A

=

replaces

. . . = . . .

and represents

  • A

A

=

  • =

, , =

  • =
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SLIDE 25

Introduction !-Logic Interpretation Example Summary

!-Box Equations

A A

=

replaces

. . . = . . .

and represents

  • A

A

=

  • =

, , , =

  • =

=

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SLIDE 26

Introduction !-Logic Interpretation Example Summary

!-Box Equations

A A

=

replaces

. . . = . . .

and represents

  • A

A

=

  • =

, , , . . . =

  • =

=

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SLIDE 27

Introduction !-Logic Interpretation Example Summary

Induction Attempt

=

A A

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SLIDE 28

Introduction !-Logic Interpretation Example Summary

Induction Attempt

=

A A

= (Induct)

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SLIDE 29

Introduction !-Logic Interpretation Example Summary

Induction Attempt

=

A A

=

A A

= → =

A A

(Induct)

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SLIDE 30

Introduction !-Logic Interpretation Example Summary

Induction Attempt

A A

= → =

A A

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SLIDE 31

Introduction !-Logic Interpretation Example Summary

Induction Attempt

A A

= → =

A A

represents:

, , , . . . =

  • =

=

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SLIDE 32

Introduction !-Logic Interpretation Example Summary

Induction Attempt

A A

= → =

A A

represents:

, , , . . . =

  • =

=

→

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SLIDE 33

Introduction !-Logic Interpretation Example Summary

Induction Attempt

A A

= → =

A A

represents:

, , , . . . =

  • =

=

→

, , . . .

  • =

= , =

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SLIDE 34

Introduction !-Logic Interpretation Example Summary

Induction Problem

→ P(0) ∀n P(n) ∀n P(n + 1) ∀n P(n) (Induct)

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SLIDE 35

Introduction !-Logic Interpretation Example Summary

Induction Problem

→ P(0) ∀n P(n) ∀n P(n + 1) ∀n P(n) (Induct)

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SLIDE 36

Introduction !-Logic Interpretation Example Summary

Induction Problem

→ P(0) ∀n P(n) ∀n P(n + 1) ∀n P(n) (Induct) → (Induct) P(0) P(n) P(n + 1) ∀n P(n) ∀n ( )

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SLIDE 37

Introduction !-Logic Interpretation Example Summary

Corrected Induction

=

A A

=

A A

= → =

A A

(Induct)

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SLIDE 38

Introduction !-Logic Interpretation Example Summary

Corrected Induction

=

A A

=

A A

= → =

A A

(Induct)

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SLIDE 39

Introduction !-Logic Interpretation Example Summary

Corrected Induction

=

A A

=

A A

= → =

A A

(Induct) =

A A

=

A A

= → =

A A

(Induct)

  • ∀A.

∀A.

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SLIDE 40

Introduction !-Logic Interpretation Example Summary

Nesting

!-Boxes can be nested

A B

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SLIDE 41

Introduction !-Logic Interpretation Example Summary

Nesting

!-Boxes can be nested

ExpA

B′ A B A B

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SLIDE 42

Introduction !-Logic Interpretation Example Summary

Nesting

!-Boxes can be nested

ExpA ExpB

B′ A B A B A B

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SLIDE 43

Introduction !-Logic Interpretation Example Summary

Nesting

!-Boxes can be nested

ExpA ExpB

B′ A B A B A B

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SLIDE 44

Introduction !-Logic Interpretation Example Summary

Components

B A

=

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SLIDE 45

Introduction !-Logic Interpretation Example Summary

Components

B A

= A B

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SLIDE 46

Introduction !-Logic Interpretation Example Summary

Components

B A

=

  • ∀A.
  • A

B

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SLIDE 47

Introduction !-Logic Interpretation Example Summary

Components

B B′ A

= A B B′

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SLIDE 48

Introduction !-Logic Interpretation Example Summary

Components

B B′ A

=

  • ∀A.
  • A

B B′

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SLIDE 49

Introduction !-Logic Interpretation Example Summary

Components

B B′ A

=

  • ∀B′.
  • A

B B′

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SLIDE 50

Introduction !-Logic Interpretation Example Summary

Components

B B′ A

=

  • ∀B′.
  • ∀A.

A B B′

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SLIDE 51

Introduction !-Logic Interpretation Example Summary

Components

B B′ B A A

= = A B B A B′

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SLIDE 52

Introduction !-Logic Interpretation Example Summary

Components

− →

B B′ B A A

= = A B B′

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SLIDE 53

Introduction !-Logic Interpretation Example Summary

Components

− →

B B′ B A A

∀A.

  • =

= A B B′

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SLIDE 54

Introduction !-Logic Interpretation Example Summary

Components

− →

B B′ B A A

∀B′.

  • =

= A B B′

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SLIDE 55

Introduction !-Logic Interpretation Example Summary

Components

− →

B B′ B A A

∀B′.

  • =

= ∀A. A B B′

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SLIDE 56

Introduction !-Logic Interpretation Example Summary

!-Formulae

Definition

The set of !-formulas, FΣ, for a signature Σ are any of:

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SLIDE 57

Introduction !-Logic Interpretation Example Summary

!-Formulae

Definition

The set of !-formulas, FΣ, for a signature Σ are any of:

  • G = H

G, H compatible !-diagrams, same inputs/outputs

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SLIDE 58

Introduction !-Logic Interpretation Example Summary

!-Formulae

Definition

The set of !-formulas, FΣ, for a signature Σ are any of:

  • G = H

G, H compatible !-diagrams, same inputs/outputs

  • X ∧ Y

compatible X, Y ∈ FΣ

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SLIDE 59

Introduction !-Logic Interpretation Example Summary

!-Formulae

Definition

The set of !-formulas, FΣ, for a signature Σ are any of:

  • G = H

G, H compatible !-diagrams, same inputs/outputs

  • X ∧ Y

compatible X, Y ∈ FΣ

  • X → Y

compatible X, Y ∈ FΣ

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SLIDE 60

Introduction !-Logic Interpretation Example Summary

!-Formulae

Definition

The set of !-formulas, FΣ, for a signature Σ are any of:

  • G = H

G, H compatible !-diagrams, same inputs/outputs

  • X ∧ Y

compatible X, Y ∈ FΣ

  • X → Y

compatible X, Y ∈ FΣ

  • ∀A. X

X ∈ FΣ, A top-level in X

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SLIDE 61

Introduction !-Logic Interpretation Example Summary

Natural Deduction System

Build logic from formulae using sequents: X1, X2, . . . , Xn ⊢ Y .

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SLIDE 62

Introduction !-Logic Interpretation Example Summary

Natural Deduction System

Build logic from formulae using sequents: X1, X2, . . . , Xn ⊢ Y . Based on positive intuitionistic logic. e.g.

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SLIDE 63

Introduction !-Logic Interpretation Example Summary

Natural Deduction System

Build logic from formulae using sequents: X1, X2, . . . , Xn ⊢ Y . Based on positive intuitionistic logic. e.g.

(Ident)

X ⊢ X

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SLIDE 64

Introduction !-Logic Interpretation Example Summary

Natural Deduction System

Build logic from formulae using sequents: X1, X2, . . . , Xn ⊢ Y . Based on positive intuitionistic logic. e.g.

(Ident)

X ⊢ X Γ ⊢ X ∆ ⊢ Y

(∧I)

Γ, ∆ ⊢ X ∧ Y

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SLIDE 65

Introduction !-Logic Interpretation Example Summary

Natural Deduction System

Build logic from formulae using sequents: X1, X2, . . . , Xn ⊢ Y . Based on positive intuitionistic logic. e.g.

(Ident)

X ⊢ X Γ ⊢ X ∆ ⊢ Y

(∧I)

Γ, ∆ ⊢ X ∧ Y Γ, X ⊢ Y

(→ I)

Γ ⊢ X → Y

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SLIDE 66

Introduction !-Logic Interpretation Example Summary

Natural Deduction System

Build logic from formulae using sequents: X1, X2, . . . , Xn ⊢ Y . Based on positive intuitionistic logic. e.g.

(Ident)

X ⊢ X Γ ⊢ X ∆ ⊢ Y

(∧I)

Γ, ∆ ⊢ X ∧ Y Γ, X ⊢ Y

(→ I)

Γ ⊢ X → Y Γ ⊢ X ∆, X ⊢ Y

(Cut)

Γ, ∆ ⊢ Y

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SLIDE 67

Introduction !-Logic Interpretation Example Summary

!-Box Rules

  • Add quantifier intro/elim:

Γ ⊢ ∀A. X

(∀E)

Γ ⊢ X ′ Γ ⊢ X ′

(∀I)

Γ ⊢ ∀A. X Where X ′ is X with the component containing A renamed (to names not present in Γ):

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SLIDE 68

Introduction !-Logic Interpretation Example Summary

!-Box Rules

  • Add quantifier intro/elim:

Γ ⊢ ∀A. X

(∀E)

Γ ⊢ X ′ Γ ⊢ X ′

(∀I)

Γ ⊢ ∀A. X Where X ′ is X with the component containing A renamed (to names not present in Γ):

  • and !-box operation rules:

Γ ⊢ ∀A. X

(KillB)

Γ ⊢ KillB(X) Γ ⊢ ∀A. X

(ExpB)

Γ ⊢ ExpB(X) where B is equal to or nested in A.

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SLIDE 69

Introduction !-Logic Interpretation Example Summary

Induction in !L

Γ ⊢ KillA(X) ∆, X ⊢ ∀B1. . . . ∀Bn. ExpA(X)

(Induct)

Γ, ∆ ⊢ X where B1 to Bn are the fresh names of children of A

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SLIDE 70

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

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SLIDE 71

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

P(nm)

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SLIDE 72

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2

P(nm)

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SLIDE 73

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2

P(nm) P(4)

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SLIDE 74

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2 n → 5 m → 3

P(nm) P(4) P(15)

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SLIDE 75

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2 n → 5 m → 3 n → 2 m → 4

P(nm) P(4) P(15) P(8)

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SLIDE 76

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2 n → 5 m → 3 n → 2 m → 4

P(nm) P(4) P(15) P(8)

P(nm) =

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SLIDE 77

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2 n → 5 m → 3 n → 2 m → 4

P(nm) P(4) P(15) P(8)

P(nm) ={ n → 1

m → 1 , n → 1 m → 2 , n → 2 m → 1 , n → 1 m → 3 , n → 2 m → 2 , . . .}

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SLIDE 78

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2 n → 5 m → 3 n → 2 m → 4

P(nm) P(4) P(15) P(8)

P(nm) ={ n → 1

m → 1 , n → 1 m → 2 , n → 2 m → 1 , n → 1 m → 3 , n → 2 m → 2 , . . .}

∀m P(nm) =

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SLIDE 79

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2 n → 5 m → 3 n → 2 m → 4

P(nm) P(4) P(15) P(8)

P(nm) ={ n → 1

m → 1 , n → 1 m → 2 , n → 2 m → 1 , n → 1 m → 3 , n → 2 m → 2 , . . .}

∀m P(nm) ={ n → 1 , n → 2 , n → 3 , . . .}

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SLIDE 80

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2 n → 5 m → 3 n → 2 m → 4

P(nm) P(4) P(15) P(8)

P(nm) ={ n → 1

m → 1 , n → 1 m → 2 , n → 2 m → 1 , n → 1 m → 3 , n → 2 m → 2 , . . .}

∀m P(nm) ={ n → 1 , n → 2 , n → 3 , . . .} ∀n (∀m P(nm)) = ∅

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SLIDE 81

Introduction !-Logic Interpretation Example Summary

Semantics for predicate logic

Let P(n) be the proposition that n is even.

n → 2 m → 2 n → 5 m → 3 n → 2 m → 4

P(nm) P(4) P(15) P(8)

P(nm) ={ n → 1

m → 1 , n → 1 m → 2 , n → 2 m → 1 , n → 1 m → 3 , n → 2 m → 2 , . . .}

∀m P(nm) ={ n → 1 , n → 2 , n → 3 , . . .} ∀n (∀m P(nm)) = ∅ =: F

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SLIDE 82

Introduction !-Logic Interpretation Example Summary

Valuation

Σ C −

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SLIDE 83

Introduction !-Logic Interpretation Example Summary

Valuation

Σ C Diagram(Σ) −

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SLIDE 84

Introduction !-Logic Interpretation Example Summary

Valuation

Σ C Diagram(Σ) − −

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SLIDE 85

Introduction !-Logic Interpretation Example Summary

Valuation

Σ C Diagram(Σ) − −

For G = H a concrete equation: G = H :=

  • T

if G = H F

  • therwise

(1)

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SLIDE 86

Introduction !-Logic Interpretation Example Summary

XOR Example

Given: Σ =

  • 0 ,

1 , ×

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SLIDE 87

Introduction !-Logic Interpretation Example Summary

XOR Example

Given: Σ =

  • 0 ,

1 , ×

  • := 0
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SLIDE 88

Introduction !-Logic Interpretation Example Summary

XOR Example

Given: Σ =

  • 0 ,

1 , ×

  • := 0
  • 1
  • := 1
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SLIDE 89

Introduction !-Logic Interpretation Example Summary

XOR Example

Given: Σ =

  • 0 ,

1 , ×

  • := 0
  • 1
  • := 1
  • ×
  • :=

(0, 0) → 0

(0, 1) → 1 (1, 0) → 1 (1, 1) → 0

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SLIDE 90

Introduction !-Logic Interpretation Example Summary

XOR Example

Given: Σ =

  • 0 ,

1 , ×

  • := 0
  • 1
  • := 1
  • ×
  • :=

(0, 0) → 0

(0, 1) → 1 (1, 0) → 1 (1, 1) → 0

So:

  • ×

1

=

1

  • = T
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SLIDE 91

Introduction !-Logic Interpretation Example Summary

XOR Example

Given: Σ =

  • 0 ,

1 , ×

  • := 0
  • 1
  • := 1
  • ×
  • :=

(0, 0) → 0

(0, 1) → 1 (1, 0) → 1 (1, 1) → 0

So:

  • ×

1

=

1

  • = T
  • ×

=

1

  • = F
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SLIDE 92

Introduction !-Logic Interpretation Example Summary

XOR Example

Given: Σ =

  • 0 ,

1 , ×

  • := 0
  • 1
  • := 1
  • ×
  • :=

(0, 0) → 0

(0, 1) → 1 (1, 0) → 1 (1, 1) → 0

So:

  • ×

1

=

1

  • = T
  • ×

=

1

  • = F

. . .

× ×

:= . . .

×

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SLIDE 93

Introduction !-Logic Interpretation Example Summary

Semantics for XOR

A 1

=

×

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SLIDE 94

Introduction !-Logic Interpretation Example Summary

Semantics for XOR

A 1

=

×

ExpA KillA

1 ×

=

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SLIDE 95

Introduction !-Logic Interpretation Example Summary

Semantics for XOR

A 1

=

×

ExpA KillA

ExpA ExpA KillA 1 ×

=

1 ×

=

1

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SLIDE 96

Introduction !-Logic Interpretation Example Summary

Semantics for XOR

A 1

=

×

ExpA KillA

ExpA ExpA KillA

ExpA ExpA ExpA ExpA KillA

1 ×

=

1 ×

=

1 ×

=

1 1 1 1

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SLIDE 97

Introduction !-Logic Interpretation Example Summary

Semantics for XOR

A 1

=

×

ExpA KillA

ExpA ExpA KillA

ExpA ExpA ExpA ExpA KillA

1 ×

=

1 ×

=

1 ×

=

1 1 1 1

  • A

1

=

×

  • =
  • KillA,

ExpA KillA ,

ExpA ExpA KillA

,

ExpA ExpA ExpA KillA

,

ExpA ExpA ExpA ExpA KillA

, . . .

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SLIDE 98

Introduction !-Logic Interpretation Example Summary

Semantics for XOR

A 1

=

×

ExpA KillA

ExpA ExpA KillA

ExpA ExpA ExpA ExpA KillA

1 ×

=

1 ×

=

1 ×

=

1 1 1 1

  • A

1

=

×

  • =
  • KillA,

ExpA KillA ,

ExpA ExpA KillA

,

ExpA ExpA ExpA KillA

,

ExpA ExpA ExpA ExpA KillA

, . . .

  • ∀A.

 

A 1

=

×

 

  • = ∅
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SLIDE 99

Introduction !-Logic Interpretation Example Summary

Semantics for XOR

A 1

=

×

ExpA KillA

ExpA ExpA KillA

ExpA ExpA ExpA ExpA KillA

1 ×

=

1 ×

=

1 ×

=

1 1 1 1

  • A

1

=

×

  • =
  • KillA,

ExpA KillA ,

ExpA ExpA KillA

,

ExpA ExpA ExpA KillA

,

ExpA ExpA ExpA ExpA KillA

, . . .

  • ∀A.

 

A 1

=

×

 

  • = ∅ = F
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SLIDE 100

Introduction !-Logic Interpretation Example Summary

Semantics for Copy

A

=

A

slide-101
SLIDE 101

Introduction !-Logic Interpretation Example Summary

Semantics for Copy

A

=

A

= KillA

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SLIDE 102

Introduction !-Logic Interpretation Example Summary

Semantics for Copy

A

=

A

= = KillA

ExpA ExpA KillA

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SLIDE 103

Introduction !-Logic Interpretation Example Summary

Semantics for Copy

A

=

A

= = = KillA

ExpA ExpA KillA

ExpA ExpA ExpA KillA

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SLIDE 104

Introduction !-Logic Interpretation Example Summary

Semantics for Copy

A

=

A

= = = KillA

ExpA ExpA KillA

ExpA ExpA ExpA KillA

  • A

A

=

  • =
  • KillA,

ExpA KillA ,

ExpA ExpA KillA

,

ExpA ExpA ExpA KillA

,

ExpA ExpA ExpA ExpA KillA

, . . .

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SLIDE 105

Introduction !-Logic Interpretation Example Summary

Semantics for Copy

A

=

A

= = = KillA

ExpA ExpA KillA

ExpA ExpA ExpA KillA

  • A

A

=

  • =
  • KillA,

ExpA KillA ,

ExpA ExpA KillA

,

ExpA ExpA ExpA KillA

,

ExpA ExpA ExpA ExpA KillA

, . . .

  • ∀A.
  • A

A

=

= {id}

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SLIDE 106

Introduction !-Logic Interpretation Example Summary

Semantics for Copy

A

=

A

= = = KillA

ExpA ExpA KillA

ExpA ExpA ExpA KillA

  • A

A

=

  • =
  • KillA,

ExpA KillA ,

ExpA ExpA KillA

,

ExpA ExpA ExpA KillA

,

ExpA ExpA ExpA ExpA KillA

, . . .

  • ∀A.
  • A

A

=

= {id} =: T

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SLIDE 107

Introduction !-Logic Interpretation Example Summary

Equations

Definition

The interpretation − of a !-logic formula is defined recursively as:

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SLIDE 108

Introduction !-Logic Interpretation Example Summary

Equations

Definition

The interpretation − of a !-logic formula is defined recursively as: G = H :=

  • i
  • i(G) = i(H)
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SLIDE 109

Introduction !-Logic Interpretation Example Summary

Equations

Definition

The interpretation − of a !-logic formula is defined recursively as: G = H :=

  • i
  • i(G) = i(H)
  • X ∧ Y :=
  • i
  • i|X ∈ X ∧ i|Y ∈ Y
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SLIDE 110

Introduction !-Logic Interpretation Example Summary

Equations

Definition

The interpretation − of a !-logic formula is defined recursively as: G = H :=

  • i
  • i(G) = i(H)
  • X ∧ Y :=
  • i
  • i|X ∈ X ∧ i|Y ∈ Y
  • X → Y :=
  • i
  • i|X ∈ X → i|Y ∈ Y
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SLIDE 111

Introduction !-Logic Interpretation Example Summary

Equations

Definition

The interpretation − of a !-logic formula is defined recursively as: G = H :=

  • i
  • i(G) = i(H)
  • X ∧ Y :=
  • i
  • i|X ∈ X ∧ i|Y ∈ Y
  • X → Y :=
  • i
  • i|X ∈ X → i|Y ∈ Y
  • ∀A. X :=
  • i
  • ∀j ∈ Inst(A) . i ◦ j ∈ X
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SLIDE 112

Introduction !-Logic Interpretation Example Summary

Sequents

We interpret sequents as truth values: X1, . . . , Xn ⊢ Y

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SLIDE 113

Introduction !-Logic Interpretation Example Summary

Sequents

We interpret sequents as truth values: X1, . . . , Xn ⊢ Y := X1 ∧ . . . ∧ Xn ⊢ Y

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SLIDE 114

Introduction !-Logic Interpretation Example Summary

Sequents

We interpret sequents as truth values: X1, . . . , Xn ⊢ Y := X1 ∧ . . . ∧ Xn ⊢ Y := ⊢ (X1 ∧ . . . ∧ Xn) → Y

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SLIDE 115

Introduction !-Logic Interpretation Example Summary

Sequents

We interpret sequents as truth values: X1, . . . , Xn ⊢ Y := X1 ∧ . . . ∧ Xn ⊢ Y := ⊢ (X1 ∧ . . . ∧ Xn) → Y := ∀ . . . .((X1 ∧ . . . ∧ Xn) → Y )

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SLIDE 116

Introduction !-Logic Interpretation Example Summary

Soundness and Completeness

Theorem (Soundness)

If Γ ⊢ X is derivable in !L, then Γ ⊢ X is true for any category C.

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SLIDE 117

Introduction !-Logic Interpretation Example Summary

Soundness and Completeness

Theorem (Soundness)

If Γ ⊢ X is derivable in !L, then Γ ⊢ X is true for any category C. Future work to seek complete !-Logic such that: If Γ ⊢ X is true for any category C then Γ ⊢ X is derivable in !L.

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SLIDE 118

Introduction !-Logic Interpretation Example Summary

N-Fold Copy

Given Σ =

  • ,

,

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SLIDE 119

Introduction !-Logic Interpretation Example Summary

N-Fold Copy

Given Σ =

  • ,

,

  • satisfying

Γ =   

= =

,

slide-120
SLIDE 120

Introduction !-Logic Interpretation Example Summary

N-Fold Copy

Given Σ =

  • ,

,

  • satisfying

Γ =   

= =

,

=

,

slide-121
SLIDE 121

Introduction !-Logic Interpretation Example Summary

N-Fold Copy

Given Σ =

  • ,

,

  • satisfying

Γ =   

= =

,

=

,

=

,

slide-122
SLIDE 122

Introduction !-Logic Interpretation Example Summary

N-Fold Copy

Given Σ =

  • ,

,

  • satisfying

Γ =   

= =

,

=

,

=

,

=

  

slide-123
SLIDE 123

Introduction !-Logic Interpretation Example Summary

N-Fold Copy

Given Σ =

  • ,

,

  • satisfying

Γ =   

= =

,

=

,

=

,

=

  

Theorem

= (Copy) Γ ⊢

slide-124
SLIDE 124

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= Γ ⊢

slide-125
SLIDE 125

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= (Ind) Γ ⊢

slide-126
SLIDE 126

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= = (Ind) Γ ⊢ Γ ⊢

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SLIDE 127

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= = (Ind) Γ ⊢ Γ ⊢ (Axiom)

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SLIDE 128

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= = = ⊢ = (Ind) (Axiom) Γ, Γ ⊢ Γ ⊢

slide-129
SLIDE 129

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= = = ⊢ = (Ind) (Axiom) (≡) (*) Γ, Γ ⊢ Γ ⊢

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SLIDE 130

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= = = ⊢ = (Ind) (Axiom) (≡) (*) Γ, Γ ⊢ Γ ⊢

where we prove (∗) by:

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SLIDE 131

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= = = ⊢ = (Ind) (Axiom) (≡) (*) Γ, Γ ⊢ Γ ⊢

where we prove (∗) by:

slide-132
SLIDE 132

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= = = ⊢ = (Ind) (Axiom) (≡) (*) Γ, Γ ⊢ Γ ⊢

where we prove (∗) by:

=

slide-133
SLIDE 133

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= = = ⊢ = (Ind) (Axiom) (≡) (*) Γ, Γ ⊢ Γ ⊢

where we prove (∗) by:

= =

slide-134
SLIDE 134

Introduction !-Logic Interpretation Example Summary

Proof

Proof.

= = = ⊢ = (Ind) (Axiom) (≡) (*) Γ, Γ ⊢ Γ ⊢

where we prove (∗) by:

= =

i.h.

=

slide-135
SLIDE 135

Introduction !-Logic Interpretation Example Summary

Summary

slide-136
SLIDE 136

Introduction !-Logic Interpretation Example Summary

Summary

  • !-Formulae:

B B

= → =

B B

∀B.

slide-137
SLIDE 137

Introduction !-Logic Interpretation Example Summary

Summary

  • !-Formulae:

B B

= → =

B B

∀B.

  • !-Logic:

Γ ⊢ ∀A. X

(KillB)

Γ ⊢ KillB(X) Γ ⊢ ∀A. X

(ExpB)

Γ ⊢ ExpB(X)

slide-138
SLIDE 138

Introduction !-Logic Interpretation Example Summary

Summary

  • !-Formulae:

B B

= → =

B B

∀B.

  • !-Logic:

Γ ⊢ ∀A. X

(KillB)

Γ ⊢ KillB(X) Γ ⊢ ∀A. X

(ExpB)

Γ ⊢ ExpB(X)

  • Induction:

Γ ⊢ KillA(X) ∆, X ⊢ ∀B1. . . . ∀Bn. ExpA(X)

(Induct)

Γ, ∆ ⊢ X

slide-139
SLIDE 139

Introduction !-Logic Interpretation Example Summary

Summary

  • !-Formulae:

B B

= → =

B B

∀B.

  • !-Logic:

Γ ⊢ ∀A. X

(KillB)

Γ ⊢ KillB(X) Γ ⊢ ∀A. X

(ExpB)

Γ ⊢ ExpB(X)

  • Induction:

Γ ⊢ KillA(X) ∆, X ⊢ ∀B1. . . . ∀Bn. ExpA(X)

(Induct)

Γ, ∆ ⊢ X

  • Soundness:

If Γ ⊢ X in !L, then Γ ⊢ X is true.