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

Introduction !-Logic Interpretation Example Summary Natural Deduction System Build logic from formulae using sequents: X 1 , X 2 , . . . , X n ⊢ Y .

Introduction !-Logic Interpretation Example Summary Natural Deduction System Build logic from formulae using sequents: X 1 , X 2 , . . . , X n ⊢ Y . Based on positive intuitionistic logic. e.g.

Introduction !-Logic Interpretation Example Summary Natural Deduction System Build logic from formulae using sequents: X 1 , X 2 , . . . , X n ⊢ Y . Based on positive intuitionistic logic. e.g. (Ident) X ⊢ X

Introduction !-Logic Interpretation Example Summary Natural Deduction System Build logic from formulae using sequents: X 1 , X 2 , . . . , X n ⊢ Y . Based on positive intuitionistic logic. e.g. Γ ⊢ X ∆ ⊢ Y (Ident) ( ∧ I ) X ⊢ X Γ , ∆ ⊢ X ∧ Y

Introduction !-Logic Interpretation Example Summary Natural Deduction System Build logic from formulae using sequents: X 1 , X 2 , . . . , X n ⊢ Y . Based on positive intuitionistic logic. e.g. Γ ⊢ X ∆ ⊢ Y (Ident) ( ∧ I ) X ⊢ X Γ , ∆ ⊢ X ∧ Y Γ , X ⊢ Y ( → I ) Γ ⊢ X → Y

Introduction !-Logic Interpretation Example Summary Natural Deduction System Build logic from formulae using sequents: X 1 , X 2 , . . . , X n ⊢ Y . Based on positive intuitionistic logic. e.g. Γ ⊢ X ∆ ⊢ Y (Ident) ( ∧ I ) X ⊢ X Γ , ∆ ⊢ X ∧ Y Γ , X ⊢ Y Γ ⊢ X ∆ , X ⊢ Y ( → I ) (Cut) Γ ⊢ X → Y Γ , ∆ ⊢ Y

Introduction !-Logic Interpretation Example Summary !-Box Rules • Add quantifier intro/elim: Γ ⊢ ∀ A . X Γ ⊢ X ′ ( ∀ E ) ( ∀ I ) Γ ⊢ X ′ Γ ⊢ ∀ A . X Where X ′ is X with the component containing A renamed (to names not present in Γ ):

Introduction !-Logic Interpretation Example Summary !-Box Rules • Add quantifier intro/elim: Γ ⊢ ∀ A . X Γ ⊢ X ′ ( ∀ E ) ( ∀ I ) Γ ⊢ X ′ 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 Γ ⊢ ∀ A . X (Kill B ) (Exp B ) Γ ⊢ Kill B ( X ) Γ ⊢ Exp B ( X ) where B is equal to or nested in A .

Introduction !-Logic Interpretation Example Summary Induction in !L Γ ⊢ Kill A ( X ) ∆ , X ⊢ ∀ B 1 . . . . ∀ B n . Exp A ( X ) (Induct) Γ , ∆ ⊢ X where B 1 to B n are the fresh names of children of A

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even.

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm )

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 m �→ 2

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 m �→ 2 P ( 4 )

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 m �→ 2 n �→ 5 m �→ 3 P ( 4 ) P ( 15 )

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 n �→ 2 m �→ 4 m �→ 2 n �→ 5 m �→ 3 P ( 4 ) P ( 15 ) P ( 8 )

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 n �→ 2 m �→ 4 m �→ 2 n �→ 5 m �→ 3 P ( 4 ) P ( 15 ) P ( 8 ) � P ( nm ) � =

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 n �→ 2 m �→ 4 m �→ 2 n �→ 5 m �→ 3 P ( 4 ) P ( 15 ) P ( 8 ) � P ( nm ) � = { n �→ 1 m �→ 2 , . . . } n �→ 1 n �→ 2 n �→ 1 n �→ 2 m �→ 1 , m �→ 2 , m �→ 1 , m �→ 3 ,

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 n �→ 2 m �→ 4 m �→ 2 n �→ 5 m �→ 3 P ( 4 ) P ( 15 ) P ( 8 ) � P ( nm ) � = { n �→ 1 m �→ 2 , . . . } n �→ 1 n �→ 2 n �→ 1 n �→ 2 m �→ 1 , m �→ 2 , m �→ 1 , m �→ 3 , � ∀ m P ( nm ) � =

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 n �→ 2 m �→ 4 m �→ 2 n �→ 5 m �→ 3 P ( 4 ) P ( 15 ) P ( 8 ) � P ( nm ) � = { n �→ 1 m �→ 2 , . . . } n �→ 1 n �→ 2 n �→ 1 n �→ 2 m �→ 1 , m �→ 2 , m �→ 1 , m �→ 3 , � ∀ m P ( nm ) � = { n �→ 1 , n �→ 2 , n �→ 3 , . . . }

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 n �→ 2 m �→ 4 m �→ 2 n �→ 5 m �→ 3 P ( 4 ) P ( 15 ) P ( 8 ) � P ( nm ) � = { n �→ 1 m �→ 2 , . . . } n �→ 1 n �→ 2 n �→ 1 n �→ 2 m �→ 1 , m �→ 2 , m �→ 1 , m �→ 3 , � ∀ m P ( nm ) � = { n �→ 1 , n �→ 2 , n �→ 3 , . . . } � ∀ n ( ∀ m P ( nm )) � = ∅

Introduction !-Logic Interpretation Example Summary Semantics for predicate logic Let P ( n ) be the proposition that n is even. P ( nm ) n �→ 2 n �→ 2 m �→ 4 m �→ 2 n �→ 5 m �→ 3 P ( 4 ) P ( 15 ) P ( 8 ) � P ( nm ) � = { n �→ 1 m �→ 2 , . . . } n �→ 1 n �→ 2 n �→ 1 n �→ 2 m �→ 1 , m �→ 2 , m �→ 1 , m �→ 3 , � ∀ m P ( nm ) � = { n �→ 1 , n �→ 2 , n �→ 3 , . . . } � ∀ n ( ∀ m P ( nm )) � = ∅ =: F

Introduction !-Logic Interpretation Example Summary Valuation � − � C Σ

Introduction !-Logic Interpretation Example Summary Valuation � − � C Σ Diagram (Σ)

Introduction !-Logic Interpretation Example Summary Valuation � − � C Σ � − � Diagram (Σ)

Introduction !-Logic Interpretation Example Summary Valuation � − � C Σ � − � Diagram (Σ) For G = H a concrete equation: � if � G � = � H � T � G = H � := (1) F otherwise

Introduction !-Logic Interpretation Example Summary XOR Example � � Given: Σ = × 0 , 1 ,

Introduction !-Logic Interpretation Example Summary XOR Example � � Given: Σ = × 0 , 1 , � � := 0 0

Introduction !-Logic Interpretation Example Summary XOR Example � � Given: Σ = × 0 , 1 , � � � � := 0 := 1 0 1

Introduction !-Logic Interpretation Example Summary XOR Example � � Given: Σ = × 0 , 1 , � � � � � � � ( 0 , 0 ) �→ 0 ( 0 , 1 ) �→ 1 := 0 := 1 := × ( 1 , 0 ) �→ 1 0 1 ( 1 , 1 ) �→ 0

Introduction !-Logic Interpretation Example Summary XOR Example � � Given: Σ = × 0 , 1 , � � � � � � � ( 0 , 0 ) �→ 0 ( 0 , 1 ) �→ 1 := 0 := 1 := × ( 1 , 0 ) �→ 1 0 1 ( 1 , 1 ) �→ 0 � � So: = = T × 0 1 1

Introduction !-Logic Interpretation Example Summary XOR Example � � Given: Σ = × 0 , 1 , � � � � � � � ( 0 , 0 ) �→ 0 ( 0 , 1 ) �→ 1 := 0 := 1 := × ( 1 , 0 ) �→ 1 0 1 ( 1 , 1 ) �→ 0 � � � � So: = = T = = F × × 0 0 1 1 0 1

Introduction !-Logic Interpretation Example Summary XOR Example � � Given: Σ = × 0 , 1 , � � � � � � � ( 0 , 0 ) �→ 0 ( 0 , 1 ) �→ 1 := 0 := 1 := × ( 1 , 0 ) �→ 1 0 1 ( 1 , 1 ) �→ 0 � � � � So: = = T = = F × × 0 0 1 1 0 1 × := × × . . . . . .

Introduction !-Logic Interpretation Example Summary Semantics for XOR × = 0 A 1

Introduction !-Logic Interpretation Example Summary Semantics for XOR × = 0 A 1 Exp A Kill A × = 0 1

Introduction !-Logic Interpretation Example Summary Semantics for XOR × = 0 A 1 Exp A Kill A Exp A Exp A Kill A × × = = 0 0 1 1 1

Introduction !-Logic Interpretation Example Summary Semantics for XOR × = 0 A 1 Exp A Exp A Exp A Exp A Kill A Exp A Exp A Kill A Exp A Kill A × × × = = = 0 0 0 1 1 1 1 1 1 1

Introduction !-Logic Interpretation Example Summary Semantics for XOR × = 0 A 1 Exp A Exp A Exp A Exp A Kill A Exp A Exp A Kill A Exp A Kill A × × × = = = 0 0 0 1 1 1 1 1 1 1 � � � � Exp A Exp A Exp A Exp A Exp A × = Kill A , Exp A = Exp A Exp A Kill A , , , , . . . Exp A 0 Exp A Kill A A Kill A 1 Kill A

Introduction !-Logic Interpretation Example Summary Semantics for XOR × = 0 A 1 Exp A Exp A Exp A Exp A Kill A Exp A Exp A Kill A Exp A Kill A × × × = = = 0 0 0 1 1 1 1 1 1 1 � � � � Exp A Exp A Exp A Exp A Exp A × = Kill A , Exp A = Exp A Exp A Kill A , , , , . . . Exp A 0 Exp A Kill A A Kill A 1 Kill A � �   × ∀ A . = ∅ =   0 A 1

Introduction !-Logic Interpretation Example Summary Semantics for XOR × = 0 A 1 Exp A Exp A Exp A Exp A Kill A Exp A Exp A Kill A Exp A Kill A × × × = = = 0 0 0 1 1 1 1 1 1 1 � � � � Exp A Exp A Exp A Exp A Exp A × = Kill A , Exp A = Exp A Exp A Kill A , , , , . . . Exp A 0 Exp A Kill A A Kill A 1 Kill A � �   × ∀ A . = ∅ = F =   0 A 1

Introduction !-Logic Interpretation Example Summary Semantics for Copy A A =