On the Computational Representation of Classical Logical Connectives

On the Computational Representation of Classical Logical Connectives
Jayshan Raghunandan and Alexander Summers Department of Computing Imperial College London Set context of work/intro

Introduction Curry-Howard Correspondence for Classical Logic
Originally: notice calculi have a correspondence Recently: design calculi to correspond to a logic “Inhabitation” of the proof rules Term assignments for Classical Sequent Calculi Different logical connectives may be chosen Implication is most common Conjunction, disjunction, negation How easy is it to add and remove connectives? Are there any which are not understood computationally? What we’re going to talk about the effect of different connectives on term calculi Write this at the end, when all other slides are done.#

Overview Sequent Calculi & Inhabitation Binary Boolean Connectives
E.g. the X calculus (van Bakel, Lengrand, Lescanne) Binary Boolean Connectives Identify related classes of connectives “Once you know one, you know them all..” ↔ is not well-known computationally Develop a term calculus based on ↔ What can be expressed computationally? What we’re going to talk about the effect of different connectives on term calculi Write this at the end, when all other slides are done.#

A Sequent Calculus for Implication
Annotation process The X-calculus Curry howard correspondance

(Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ Annotation process The X-calculus Curry howard correspondance

P : . Γ ⊢ Δ, α:A Q : . x:A, Γ ⊢ Δ (Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ (cut) Pα̂†x̂Q : . Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance

P : . Γ ⊢ Δ, α:A Q : . x:A, Γ ⊢ Δ (Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ (cut) Pα̂†x̂Q : . Γ ⊢ Δ P : . Γ, x:A ⊢ α:B , Δ Annotation process The X-calculus Curry howard correspondance (→R) x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ

P : . Γ ⊢ Δ, α:A Q : . x:A, Γ ⊢ Δ (Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ (cut) Pα̂†x̂Q : . Γ ⊢ Δ P : . Γ, x:A ⊢ α:B , Δ P : . Γ ⊢ Δ, α:A Q : . x:B, Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance (→R) (→L) x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ Pα̂[z]x̂Q : . z:A→B, Γ ⊢ Δ

Inhabitation P : . Γ ⊢ Δ, α:A Q : . x:A, Γ ⊢ Δ
(Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ (cut) Pα̂†x̂Q : . Γ ⊢ Δ P : . Γ, x:A ⊢ α:B , Δ P : . Γ ⊢ Δ, α:A Q : . x:B, Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance (→R) (→L) x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ Pα̂[z]x̂Q : . z:A→B, Γ ⊢ Δ

Inhabitation x, y, … INPUTS P : . Γ ⊢ Δ, α:A Q : . x:A, Γ ⊢ Δ
(Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ (cut) Pα̂†x̂Q : . Γ ⊢ Δ P : . Γ, x:A ⊢ α:B , Δ P : . Γ ⊢ Δ, α:A Q : . x:B, Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance (→R) (→L) x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ Pα̂[z]x̂Q : . z:A→B, Γ ⊢ Δ

Inhabitation x, y, … INPUTS α, δ, … OUTPUTS P : . Γ ⊢ Δ, α:A
Q : . x:A, Γ ⊢ Δ (Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ (cut) Pα̂†x̂Q : . Γ ⊢ Δ P : . Γ, x:A ⊢ α:B , Δ P : . Γ ⊢ Δ, α:A Q : . x:B, Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance (→R) (→L) x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ Pα̂[z]x̂Q : . z:A→B, Γ ⊢ Δ

Q : . x:A, Γ ⊢ Δ (Ax) (cut) ‹x·α› : . Γ, x:A ⊢ α:A, Δ Pα̂†x̂Q : . Γ ⊢ Δ P : . Γ, x:A ⊢ α:B , Δ P : . Γ ⊢ Δ, α:A Q : . x:B, Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance (→R) (→L) x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ Pα̂[z]x̂Q : . z:A→B, Γ ⊢ Δ

Q : . x:A, Γ ⊢ Δ (Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ (cut) Pα̂†x̂Q : . Γ ⊢ Δ P : . Γ, x:A ⊢ α:B , Δ P : . Γ ⊢ Δ, α:A Q : . x:B, Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance (→R) (→L) x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ Pα̂[z]x̂Q : . z:A→B, Γ ⊢ Δ

Inhabitation x, y, … INPUTS P, Q, … TERMS α, δ, … OUTPUTS

Inhabitation x, y, … INPUTS P, Q, … TERMS α, δ, … OUTPUTS ^ BINDER

(cut) P : . Γ ⊢ Δ, α:A Q : . x:A, Γ ⊢ Δ Pα̂†x̂Q : . Γ ⊢ Δ (Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ P : . Γ, x:A ⊢ α:B , Δ P : . Γ ⊢ Δ, α:A Q : . x:B, Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance (→R) (→L) x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ Pα̂[z]x̂Q : . z:A→B, Γ ⊢ Δ

(cut) P : . Γ ⊢ Δ, α:A Q : . x:A, Γ ⊢ Δ Pα̂†x̂Q : . Γ ⊢ Δ (Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ (→R) P : . Γ, x:A ⊢ α:B , Δ x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ P : . Γ ⊢ Δ, α:A Q : . x:B, Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance (→L) Pα̂[z]x̂Q : . z:A→B, Γ ⊢ Δ

Inhabitation (X Calculus)
x, y, … INPUTS P, Q, … TERMS α, δ, … OUTPUTS ^ BINDER (cut) P : . Γ ⊢ Δ, α:A Q : . x:A, Γ ⊢ Δ Pα̂†x̂Q : . Γ ⊢ Δ (Ax) ‹x·α› : . Γ, x:A ⊢ α:A, Δ (→R) P : . Γ, x:A ⊢ α:B , Δ x̂Pα̂·δ : . Γ ⊢ δ:A→B, Δ P : . Γ ⊢ Δ, α:A Q : . x:B, Γ ⊢ Δ Annotation process The X-calculus Curry howard correspondance (→L) Pα̂[z]x̂Q : . z:A→B, Γ ⊢ Δ

Sequent-style term calculi
Symmetry: outputs are treated as explicitly as inputs Basic building blocks: one input and one output In X, these are capsules, ‹x·α› Redexes are explicitly represented by cuts, Connect output of one term to input of another In X, these are written as Pα̂†x̂Q c.f. applicative style: redexes defined by pattern matching - Inputs and outputs

Sequent-style term calculi
Remaining syntax constructs come in pairs One describes the most general situation for using the other e.g. build functions and ‘function contexts’ In X: functions are built with exports: x̂Pα̂·δ ‘contexts’ are built with mediators: Pα̂[z]x̂Q For each logical connective, one pair is required Corresponds to left and right introduction rules Might not be obvious what the sequent rules for a particular connective is (tho it usu is)

Cut-Elimination Only one reduction rule is significant per connective
Defines how the two syntactic constructs interact For example: implication Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

(ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q)
Cut-Elimination Only one reduction rule is significant per connective Defines how the two syntactic constructs interact For example: implication (ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q) Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

(ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q)
Cut-Elimination Only one reduction rule is significant per connective Defines how the two syntactic constructs interact For example: implication (ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q) Pα̂ [ z ] x̂Q Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

(ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q)
Cut-Elimination Only one reduction rule is significant per connective Defines how the two syntactic constructs interact For example: implication (ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q) Pα̂ [ z ] x̂Q ŷRγ̂·δ Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

(ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q)
Cut-Elimination Only one reduction rule is significant per connective Defines how the two syntactic constructs interact For example: implication (ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q) Pα̂ [ ] x̂Q ŷRγ̂·δ Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

(ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q)
Cut-Elimination Only one reduction rule is significant per connective Defines how the two syntactic constructs interact For example: implication (ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q) Pα̂ [ ] x̂Q ŷRγ̂ Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

(ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q)
Cut-Elimination Only one reduction rule is significant per connective Defines how the two syntactic constructs interact For example: implication (ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q) Pα̂ † [ † ] x̂Q ŷRγ̂ Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

(ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q)
Cut-Elimination Only one reduction rule is significant per connective Defines how the two syntactic constructs interact For example: implication (ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q) Pα̂†ŷRγ̂†x̂Q Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

(ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q)
Cut-Elimination Only one reduction rule is significant per connective Defines how the two syntactic constructs interact For example: implication We call this the principal logical rule for the connective (ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q) Pα̂†ŷRγ̂†x̂Q Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

(ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q)
Cut-Elimination Only one reduction rule is significant per connective Defines how the two syntactic constructs interact For example: implication We call this the principal logical rule for the connective Only rule which varies substantially for different connectives (ŷRγ̂·δ) δ̂†ẑ (Pα̂[z]x̂Q) Pα̂†ŷRγ̂†x̂Q Make terms appear one at a time Explain what the exp-med rule does when its onscreen Bracketing Once u know syntax and principal rule, you know EVERYTHING

Computational Representation of a Connective
To include a particular logical connective, we need: The sequent rules Corresponding term syntax Principal logical rule There are an infinite number of logical connectives! 22n connectives of arity n We limit our investigations to those of arity 2. Common choice in the literature Smallest arity which can express everything We can vary the primitive connectives in the underlying logic How can we ‘synthesise’ a corresponding term calculus?

Binary Logical Connectives
FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B A⋁B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B A⋁B A↑B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B A⋁B A↑B A↓B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B A⋁B A→B A↑B A↓B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B A⋁B A→B A↑B A↓B A-B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B A⋁B A→B A↑B A↓B B→A A-B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B A↔B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B A↔B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram A⊗B

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B ⊤ A↔B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B ⊤ A↔B ID A FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B ⊤ A↔B ID A FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B ¬A

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B ⊤ A↔B ID A ID B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B ¬A ¬B

Relating Logical Connectives
Let ● and ○ be arbitrary binary connectives We use the following relations: Duality: ● is dual to ○ iff A●B ≡ ¬(¬A○¬B) Negation: ● is the negation of ○ iff A●B ≡ ¬(A○B) Reversal: ● is the reverse of ○ iff A●B ≡ B○A Flipping Inputs: ● is obtained from ○ by flipping an input iff either A●B ≡ ¬A○B or A●B ≡ A○¬B We write ≡ for logical equivalence ● ○

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B ⊤ A↔B ID A ID B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B ¬A ¬B

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B ⊤ A↔B ID A ID B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B ¬A ¬B DUALITY

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B ⊤ A↔B ID A ID B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B ¬A ¬B DUALITY NEGATION

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B ⊤ A↔B ID A ID B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B ¬A ¬B DUALITY NEGATION REVERSAL

A⋀B A⋁B B-A A→B A↑B A↓B B→A A-B ⊤ A↔B ID A ID B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B ¬A ¬B DUALITY NEGATION REVERSAL Reduce modulo REVERSAL

⋀ ⋁ B-A A→B ↑ ↓ B→A A-B ⊤ A↔B ID A ID B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ A⊗B ¬A ¬B DUALITY NEGATION REVERSAL Reduce modulo REVERSAL

⋀ ⋁ B-A A→B ↑ ↓ B→A A-B ⊤ ↔ ID A ID B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ ⊗ ¬A ¬B DUALITY NEGATION REVERSAL Reduce modulo REVERSAL

⋀ ⋁ B-A A→B ↑ ↓ B→A A-B ⊤ ↔ ID A ID B FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ ⊗ DUALITY NEGATION REVERSAL Reduce modulo REVERSAL

⋀ ⋁ B-A A→B ↑ ↓ B→A A-B ⊤ ↔ ID FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ ⊗ DUALITY NEGATION REVERSAL Reduce modulo REVERSAL

⋀ ⋁ B-A A→B A-B ↑ ↓ B→A ⊤ ↔ ID FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ ⊗ DUALITY NEGATION REVERSAL Reduce modulo REVERSAL

⋀ ⋁ B-A A-B A→B ↑ ↓ B→A ⊤ ↔ ID FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ ⊗ DUALITY NEGATION REVERSAL Reduce modulo REVERSAL

⋀ ⋁ - - → ↑ ↓ B→A ⊤ ↔ ID FIXME: Draw all the arrows Can be simplified If we allow reversals, then can reduce the diagram ⊥ ⊗ DUALITY NEGATION REVERSAL Reduce modulo REVERSAL

- ⋀ ⋁ ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION REVERSAL Reduce modulo REVERSAL

- ⋀ ⋁ ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION

- - ⋀ ⋀ ⋁ ⋁ ↑ ↑ ↓ ↓ → → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION

- - ⋀ ⋁ ↑ ↑ ↓ ↓ → → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION

- ⋀ ⋁ ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

- ⋀ ⋁ A → B ≡ ¬A ⋁ B ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

- ⋀ ⋁ A → B ≡ ¬A ⋁ B A → B ≡ ¬(A ⋀ ¬B) ≡ A ↑ ¬B ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

- ⋀ ⋁ ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

All relationships involve negation - ⋀ ⋁ ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

All relationships involve negation - ⋀ ⋁ Γ, A ⊢ Δ (¬R) Γ ⊢ ¬A, Δ ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

All relationships involve negation - ⋀ ⋁ Γ, A ⊢ Δ (¬R) Γ ⊢ ¬A, Δ ↑ ↓ → Γ ⊢ A, Δ ⊤ ↔ ID (¬L) Γ, ¬A ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

All relationships involve negation - ⋀ ⋁ Γ, A ⊢ Δ (¬R) Γ ⊢ ¬A, Δ d. x̂P.α ↑ ↓ → Γ ⊢ A, Δ ⊤ ↔ ID (¬L) Γ, ¬A ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

All relationships involve negation - ⋀ ⋁ Γ, A ⊢ Δ (¬R) Γ ⊢ ¬A, Δ d. x̂P.α ↑ ↓ → Γ ⊢ A, Δ ⊤ ↔ ID (¬L) Γ, ¬A ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Pα ̂ ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

All relationships involve negation - ⋀ ⋁ Γ, A ⊢ Δ (¬R) Γ ⊢ ¬A, Δ d. x̂P.α ↑ ↓ → Γ ⊢ A, Δ ⊤ ↔ ID (¬L) Γ, ¬A ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Pα ̂ ⊥ ⊗ Negation swaps inputs with outputs! DUALITY NEGATION FLIPPING INPUTS

⋀ - ⋁ ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

⋀ - ⋁ (⋀R) Γ ⊢ A, Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ ↑ ↓ → ⊤ ↔ ID AND SEQUENT RULES ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

⋀ - ⋁ (⋀R) Γ ⊢ A, Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ ↑ ↓ → (⋀L) Γ, A, B ⊢ Δ Γ, A⋀B ⊢ Δ ⊤ ↔ ID AND SEQUENT RULES ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

⋀ - ⋁ (⋀R) Γ ⊢ A, Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ d.‹Pα̂, Qγ̂›.μ ↑ ↓ → (⋀L) Γ, A, B ⊢ Δ Γ, A⋀B ⊢ Δ ⊤ ↔ ID AND SEQUENT RULES ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

⋀ - ⋁ (⋀R) Γ ⊢ A, Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ d.‹Pα̂, Qγ̂›.μ ↑ ↓ → (⋀L) Γ, A, B ⊢ Δ Γ, A⋀B ⊢ Δ ⊤ ↔ ID AND SEQUENT RULES x.ŷẑR ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

⋀ - ⋁ (⋀R) Γ ⊢ A, Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ d.‹Pα̂, Qγ̂›.μ ↑ ↓ → (⋀L) Γ, A, B ⊢ Δ Γ, A⋀B ⊢ Δ ⊤ ↔ ID AND SEQUENT RULES x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS

⋀ - ⋁ (⋀R) Γ ⊢ A, Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ d.‹Pα̂, Qγ̂›.μ ↑ ↓ → (⋀L) Γ, A, B ⊢ Δ Γ, A⋀B ⊢ Δ ⊤ ↔ ID AND SEQUENT RULES x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

Relating Logical Connectives (Duality)
⋀ - ⋁ (⋀R) Γ ⊢ A, Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ d.‹Pα̂, Qγ̂›.μ ↑ ↓ → (⋀L) Γ, A, B ⊢ Δ Γ, A⋀B ⊢ Δ ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - (⋀R) Γ ⊢ A, Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ d.‹Pα̂, Qγ̂›.μ ↑ ↓ → (⋀L) Γ, A, B ⊢ Δ Γ, A⋀B ⊢ Δ ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ ⊢ A, Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ d.‹Pα̂, Qγ̂›.μ ↑ ↓ → Γ, A, B ⊢ Δ (⋀L) ⊤ ↔ ID Γ, A⋀B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ ⊢ B, Δ Γ ⊢ A⋀B, Δ d.‹âP, Qγ̂›.μ ↑ ↓ → Γ, A, B ⊢ Δ (⋀L) ⊤ ↔ ID Γ, A⋀B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ, B ⊢ Δ Γ ⊢ A⋀B, Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ, A, B ⊢ Δ (⋀L) ⊤ ↔ ID Γ, A⋀B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ, B ⊢ Δ Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ, A, B ⊢ Δ (⋀L) ⊤ ↔ ID Γ, A⋀B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ, A, B ⊢ Δ (⋀L) ⊤ ↔ ID Γ, A⋀B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ, A, B ⊢ Δ ⊤ ↔ ID Γ, A⋀B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷẑR ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ ⊢ A, B, Δ ⊤ ↔ ID Γ, A⋀B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ ⊢ A, B, Δ ⊤ ↔ ID Γ ⊢ A⋁B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ ⊢ A, B, Δ (⋁R) ⊤ ↔ ID Γ ⊢ A⋁B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (‹Pα̂, Qγ̂›.μ) μ̂†x̂ (x.ŷẑR) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ ⊢ A, B, Δ (⋁R) ⊤ ↔ ID Γ ⊢ A⋁B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (Rπ̂σ̂.τ) τ̂†d ̂(d.‹âP, ĝQ›) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

⋀ ⋁ - Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ ⊢ A, B, Δ (⋁R) ⊤ ↔ ID Γ ⊢ A⋁B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (Rπ̂σ̂.τ) τ̂†d ̂(d.‹âP, ĝQ›) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

⋁ - ⋀ Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ ⊢ A, B, Δ (⋁R) ⊤ ↔ ID Γ ⊢ A⋁B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (Rπ̂σ̂.τ) τ̂†d ̂(d.‹âP, ĝQ›) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

Relating Logical Connectives (Negation)
⋁ - ⋀ Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ ⊢ A, B, Δ (⋁R) ⊤ ↔ ID Γ ⊢ A⋁B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (Rπ̂σ̂.τ) τ̂†d ̂(d.‹âP, ĝQ›) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

⋁ - ⋀ Γ, A ⊢ Δ Γ, B ⊢ Δ (⋁L) Γ, A⋁B ⊢ Δ d.‹âP, ĝQ›.μ ↓ ↑ → Γ ⊢ A, B, Δ (⋁R) ⊤ ↔ ID Γ ⊢ A⋁B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (Rπ̂σ̂.τ) τ̂†d ̂(d.‹âP, ĝQ›) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

⋁ - ⋀ Γ, A ⊢ Δ Γ, B ⊢ Δ (↓R) Γ ⊢ A↓B, Δ d.‹âP, ĝQ›.μ ↓ ↑ → Γ ⊢ A, B, Δ (⋁R) ⊤ ↔ ID Γ ⊢ A⋁B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (Rπ̂σ̂.τ) τ̂†d ̂(d.‹âP, ĝQ›) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

⋁ - ⋀ Γ, A ⊢ Δ Γ, B ⊢ Δ (↓R) Γ ⊢ A↓B, Δ d.‹âP, ĝQ›.μ ↓ ↑ → Γ ⊢ A, B, Δ (↓L) ⊤ ↔ ID Γ, A↓B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (Rπ̂σ̂.τ) τ̂†d ̂(d.‹âP, ĝQ›) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

⋁ - ⋀ Γ, A ⊢ Δ Γ, B ⊢ Δ (↓R) Γ ⊢ A↓B, Δ d.‹âP, ĝQ›.μ ↓ ↑ → Γ ⊢ A, B, Δ (↓L) ⊤ ↔ ID Γ, A↓B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (‹âP, ĝQ›.μ) μ̂†x̂ (x.Rπ̂σ̂) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

- ⋀ ⋁ Γ, A ⊢ Δ Γ, B ⊢ Δ (↓R) Γ ⊢ A↓B, Δ d.‹âP, ĝQ›.μ ↓ ↑ → Γ ⊢ A, B, Δ (↓L) ⊤ ↔ ID Γ, A↓B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (‹âP, ĝQ›.μ) μ̂†x̂ (x.Rπ̂σ̂) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

- ⋀ ⋁ Γ, A ⊢ Δ Γ, B ⊢ Δ (↓R) Γ ⊢ A↓B, Δ d.‹âP, ĝQ›.μ ↑ ↓ → Γ ⊢ A, B, Δ (↓L) ⊤ ↔ ID Γ, A↓B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (‹âP, ĝQ›.μ) μ̂†x̂ (x.Rπ̂σ̂) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

- ⋀ ⋁ Γ ⊢ A, Δ Γ ⊢ B, Δ Γ, A↑B ⊢ Δ d.‹Pα̂, Qγ̂›.μ (↑L) ↑ ↓ → Γ ⊢ A, B, Δ (↓L) ⊤ ↔ ID Γ, A↓B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.Rπ̂σ̂.τ ⊥ ⊗ (‹âP, ĝQ›.μ) μ̂†x̂ (x.Rπ̂σ̂) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

- ⋀ ⋁ Γ ⊢ A, Δ Γ ⊢ B, Δ Γ, A↑B ⊢ Δ d.‹Pα̂, Qγ̂›.μ (↑L) ↑ ↓ → Γ, A, B ⊢ Δ (↑R) ⊤ ↔ ID Γ ⊢ A↑B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷẑR.τ ⊥ ⊗ (‹âP, ĝQ›.μ) μ̂†x̂ (x.Rπ̂σ̂) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

- ⋀ ⋁ Γ ⊢ A, Δ Γ ⊢ B, Δ Γ, A↑B ⊢ Δ d.‹Pα̂, Qγ̂›.μ (↑L) ↑ ↓ → Γ, A, B ⊢ Δ (↑R) ⊤ ↔ ID Γ ⊢ A↑B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷẑR.τ ⊥ ⊗ (ŷẑR.τ) τ̂†d ̂(d.‹Pα̂, Qγ̂›) DUALITY NEGATION FLIPPING INPUTS (Rπ̂†ĝQ)σ̂†âP

- ⋀ ⋁ Γ ⊢ A, Δ Γ ⊢ B, Δ Γ, A↑B ⊢ Δ d.‹Pα̂, Qγ̂›.μ (↑L) ↑ ↓ → Γ, A, B ⊢ Δ (↑R) ⊤ ↔ ID Γ ⊢ A↑B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷẑR.τ ⊥ ⊗ (ŷẑR.τ) τ̂†d ̂(d.‹Pα̂, Qγ̂›) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

Relating Logical Connectives (Flipping Inputs)
- ⋀ ⋁ Γ ⊢ A, Δ Γ ⊢ B, Δ (↑L) Γ, A↑B ⊢ Δ d.‹Pα̂, Qγ̂›.μ ↑ → ↓ Γ, A, B ⊢ Δ (↑R) ⊤ ↔ ID Γ ⊢ A↑B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷẑR.τ ⊥ ⊗ (ŷẑR.τ) τ̂†d ̂(d.‹Pα̂, Qγ̂›) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

- ⋀ ⋁ Γ ⊢ A, Δ Γ, B ⊢ Δ (→L) Γ, A→B ⊢ Δ d.‹Pα̂, ẑQ›.μ ↑ → ↓ Γ, A, B ⊢ Δ (↑R) ⊤ ↔ ID Γ ⊢ A↑B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷẑR.τ ⊥ ⊗ (ŷẑR.τ) τ̂†d ̂(d.‹Pα̂, Qγ̂›) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

- ⋀ ⋁ Γ ⊢ A, Δ Γ, B ⊢ Δ (→L) Γ, A→B ⊢ Δ ↑ → ↓ Pα̂ [d] ẑQ Γ, A, B ⊢ Δ (↑R) ⊤ ↔ ID Γ ⊢ A↑B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷẑR.τ ⊥ ⊗ (ŷẑR.τ) τ̂†d ̂(d.‹Pα̂, Qγ̂›) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

- ⋀ ⋁ Γ ⊢ A, Δ Γ, B ⊢ Δ (→L) Γ, A→B ⊢ Δ ↑ → ↓ Pα̂ [d] ẑQ Γ, A ⊢ B, Δ (→R) ⊤ ↔ ID Γ ⊢ A→B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷRγ̂.τ ⊥ ⊗ (ŷẑR.τ) τ̂†d ̂(d.‹Pα̂, Qγ̂›) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

- ⋀ ⋁ Γ ⊢ A, Δ Γ, B ⊢ Δ (→L) Γ, A→B ⊢ Δ ↑ → ↓ Pα̂ [d] ẑQ Γ, A ⊢ B, Δ (→R) ⊤ ↔ ID Γ ⊢ A→B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷRγ̂.τ ⊥ ⊗ (ŷRγ̂.τ) τ̂†d ̂(Pα̂ [d] ẑQ) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Qγ̂†ẑR)

- ⋀ ⋁ Γ ⊢ A, Δ Γ, B ⊢ Δ (→L) Γ, A→B ⊢ Δ ↑ → ↓ Pα̂ [d] ẑQ Γ, A ⊢ B, Δ (→R) ⊤ ↔ ID Γ ⊢ A→B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷRγ̂.τ ⊥ ⊗ (ŷRγ̂.τ) τ̂†d ̂(Pα̂ [d] ẑQ) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Rγ̂†ẑQ)

- ⋀ ⋁ Γ ⊢ A, Δ Γ, B ⊢ Δ (→L) Γ, A→B ⊢ Δ → ↑ ↓ Pα̂ [d] ẑQ Γ, A ⊢ B, Δ (→R) ⊤ ↔ ID Γ ⊢ A→B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷRγ̂.τ ⊥ ⊗ (ŷRγ̂.τ) τ̂†d ̂(Pα̂ [d] ẑQ) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Rγ̂†ẑQ)

- ⋀ ⋁ Γ ⊢ A, Δ Γ, B ⊢ Δ (-L) Γ ⊢ A-B, Δ → ↑ ↓ Pα̂ [μ] ẑQ Γ, A ⊢ B, Δ (→R) ⊤ ↔ ID Γ ⊢ A→B, Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x. ŷRγ̂.τ ⊥ ⊗ (ŷRγ̂.τ) τ̂†d ̂(Pα̂ [d] ẑQ) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Rγ̂†ẑQ)

- ⋀ ⋁ Γ ⊢ A, Δ Γ, B ⊢ Δ (-L) Γ ⊢ A-B, Δ → ↑ ↓ Pα̂ [μ] ẑQ Γ, A ⊢ B, Δ (-R) ⊤ ↔ ID Γ, A-B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷRγ̂ ⊥ ⊗ (ŷRγ̂.τ) τ̂†d ̂(Pα̂ [d] ẑQ) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Rγ̂†ẑQ)

- ⋀ ⋁ Γ ⊢ A, Δ Γ, B ⊢ Δ (-L) Γ ⊢ A-B, Δ → ↑ ↓ Pα̂ [μ] ẑQ Γ, A ⊢ B, Δ (-R) ⊤ ↔ ID Γ, A-B ⊢ Δ Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes x.ŷRγ̂ ⊥ ⊗ (Pα̂ [μ] ẑQ) μ̂†x̂ (x.ŷRγ) DUALITY NEGATION FLIPPING INPUTS Pα̂†ŷ(Rγ̂†ẑQ)

- ⋀ ⋁ Arrows define a “class” of related connectives Proof rules, syntax, reductions are similar for each “Once you know one, you know them all” Saves a lot of work Some are well-studied already ↑ → ↓ ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

- ⋀ ⋁ Arrows define a “class” of related connectives Proof rules, syntax, reductions are similar for each “Once you know one, you know them all” Saves a lot of work Some are well-studied already Same results for other “classes” ↑ ↓ → ⊤ ↔ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊥ ⊗ DUALITY NEGATION FLIPPING INPUTS

- ⋀ ⋁ Arrows define a “class” of related connectives Proof rules, syntax, reductions are similar for each “Once you know one, you know them all” Saves a lot of work Some are well-studied already Same results for other “classes” ↑ ↓ → ↔ ⊤ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊗ ⊥ DUALITY NEGATION FLIPPING INPUTS

- ⋀ ⋁ Arrows define a “class” of related connectives Proof rules, syntax, reductions are similar for each “Once you know one, you know them all” Saves a lot of work Some are well-studied already Same results for other “classes” If all of the class is unknown? ↑ ↓ → ↔ ⊤ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊗ ⊥ DUALITY NEGATION FLIPPING INPUTS

- ⋀ ⋁ Arrows define a “class” of related connectives Proof rules, syntax, reductions are similar for each “Once you know one, you know them all” Saves a lot of work Some are well-studied already Same results for other “classes” If all of the class is unknown? Work on one member (↔) ↑ ↓ → ↔ ⊤ ID Reduce to smaller diagram Introduce the notion of `pairing’ connective, and point out they are all related + essentially the same We know syntax for implication, by following arrows, we can get the syntax n rules for other connectives within each block On rhs – highlight bits of diagram while we cycle through the nodes ⊗ ⊥ DUALITY NEGATION FLIPPING INPUTS

Interpreting ↔ Say what it is, as a connective
XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Interpreting ↔ (proof rules)
Find an equivalent formula, in terms of connectives we know Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Find an equivalent formula, in terms of connectives we know A ↔ B ≡ ¬(A⋁B)⋁(A⋀B) Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. A ↔ B ≡ ¬(A⋁B)⋁(A⋀B) Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Γ ⊢ A, B, Δ (⋁R) Γ ⊢ A⋁B, Δ Γ, A, B ⊢ Δ (¬L) (⋀L) Γ, ¬(A⋁B) ⊢ Δ Γ, A⋀B ⊢ Δ (⋁L) Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Γ ⊢ A, B, Δ (⋁R) Γ ⊢ A⋁B, Δ Γ, A, B ⊢ Δ (¬L) (⋀L) Γ, ¬(A⋁B) ⊢ Δ Γ, A⋀B ⊢ Δ (⋁L) Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Γ ⊢ A, B, Δ Γ, A, B ⊢ Δ Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Γ ⊢ A, B, Δ Γ, A, B ⊢ Δ Γ, A↔B ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Γ ⊢ A, B, Δ Γ, A, B ⊢ Δ (↔L) Γ, A↔B ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Similarly, for right introduction Γ ⊢ A, B, Δ Γ, A, B ⊢ Δ (↔L) Γ, A↔B ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Similarly, for right introduction Γ ⊢ A, B, Δ Γ, A, B ⊢ Δ (↔L) Γ, A↔B ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution Γ, A ⊢ B, Δ Γ, B ⊢ A, Δ (↔R) Γ ⊢ A↔B, Δ

Interpreting ↔ (syntax)
Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Similarly, for right introduction Inhabit with syntax, as usual Γ ⊢ A, B, Δ Γ, A, B ⊢ Δ (↔L) Γ, A↔B ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution Γ, A ⊢ B, Δ Γ, B ⊢ A, Δ (↔R) Γ ⊢ A↔B, Δ

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Similarly, for right introduction Inhabit with syntax, as usual Mμ̂σ̂[z]îĵN Γ ⊢ A, B, Δ Γ, A, B ⊢ Δ (↔L) Γ, A↔B ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution Γ, A ⊢ B, Δ Γ, B ⊢ A, Δ (↔R) Γ ⊢ A↔B, Δ

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Similarly, for right introduction Inhabit with syntax, as usual Mμ̂σ̂[z]îĵN Γ ⊢ A, B, Δ Γ, A, B ⊢ Δ (↔L) Γ, A↔B ⊢ Δ [x̂Pα̂,ẑQδ̂]·γ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution Γ, A ⊢ B, Δ Γ, B ⊢ A, Δ (↔R) Γ ⊢ A↔B, Δ

Find an equivalent formula, in terms of connectives we know Consider a “general” derivation of this formula on the left of a sequent.. Derive a suitable left introduction rule.. Similarly, for right introduction Inhabit with syntax, as usual Note similarities with implication Mμ̂σ̂[z]îĵN Γ ⊢ A, B, Δ Γ, A, B ⊢ Δ (↔L) Γ, A↔B ⊢ Δ [x̂Pα̂,ẑQδ̂]·γ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution Γ, A ⊢ B, Δ Γ, B ⊢ A, Δ (↔R) Γ ⊢ A↔B, Δ

Interpreting ↔ (reduction rule)
Mμ̂σ̂[z]îĵN [x̂Pα̂,ẑQδ̂]·γ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? Mμ̂σ̂[z]îĵN [x̂Pα̂,ẑQδ̂]·γ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? [x̂Pα̂,ẑQδ̂]·γ Mμ̂σ̂[z]îĵN Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? ([x̂Pα̂,ẑQδ̂]·γ) (Mμ̂σ̂[z]îĵN) γ̂†ẑ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? Go back to the derivations.. ([x̂Pα̂,ẑQδ̂]·γ) (Mμ̂σ̂[z]îĵN) γ̂†ẑ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? Go back to the derivations.. ([x̂Pα̂,ẑQδ̂]·γ) (Mμ̂σ̂[z]îĵN) γ̂†ẑ M N Γ ⊢ A, B, Δ (⋁R) Γ, A, B ⊢ Δ Γ ⊢ A⋁B, Δ (¬L) (⋀L) Γ, ¬(A⋁B) ⊢ Δ Γ, A⋀B ⊢ Δ (⋁L) Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? Go back to the derivations.. ([x̂Pα̂,ẑQδ̂]·γ) (Mμ̂σ̂[z]îĵN) γ̂†ẑ P Γ, B ⊢ A, Δ Γ, A ⊢ A, Δ Q M (Ax) Γ, A ⊢ B, Δ Γ, B ⊢ B, Δ (Ax) (⋁L) (⋁L) N Γ, A⋁B ⊢ B, Δ Γ, A⋁B ⊢ A, Δ Γ ⊢ A, B, Δ (⋀R) (⋁R) Γ, A⋁B ⊢ A⋀B, Δ Γ, A, B ⊢ Δ Γ ⊢ A⋁B, Δ (¬R) (¬L) (⋀L) Γ ⊢ ¬(A⋁B), A⋀B, Δ Γ, ¬(A⋁B) ⊢ Δ Γ, A⋀B ⊢ Δ (⋁R) (⋁L) Γ ⊢ ¬(A⋁B)⋁(A⋀B), Δ Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? Go back to the derivations.. ([x̂Pα̂,ẑQδ̂]·γ) (Mμ̂σ̂[z]îĵN) γ̂†ẑ Γ ⊢ ¬(A⋁B)⋁(A⋀B), Δ Γ ⊢ ¬(A⋁B), A⋀B, Δ Γ, A⋁B ⊢ A⋀B, Δ Γ, A⋁B ⊢ B, Δ Γ, A⋁B ⊢ A, Δ Γ, B ⊢ B, Δ Γ, A ⊢ B, Δ Γ, B ⊢ A, Δ Γ, A ⊢ A, Δ (Ax) (⋁L) (⋀R) (¬R) (⋁R) Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Γ, ¬(A⋁B) ⊢ Δ Γ, A⋀B ⊢ Δ Γ, A, B ⊢ Δ Γ ⊢ A⋁B, Δ Γ ⊢ A, B, Δ (⋀L) (¬L) Γ ⊢ Δ (cut) P Q M N Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? Go back to the derivations.. Apply cut elimination for this derivation… ([x̂Pα̂,ẑQδ̂]·γ) (Mμ̂σ̂[z]îĵN) γ̂†ẑ Γ ⊢ ¬(A⋁B)⋁(A⋀B), Δ Γ ⊢ ¬(A⋁B), A⋀B, Δ Γ, A⋁B ⊢ A⋀B, Δ Γ, A⋁B ⊢ B, Δ Γ, A⋁B ⊢ A, Δ Γ, B ⊢ B, Δ Γ, A ⊢ B, Δ Γ, B ⊢ A, Δ Γ, A ⊢ A, Δ (Ax) (⋁L) (⋀R) (¬R) (⋁R) Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Γ, ¬(A⋁B) ⊢ Δ Γ, A⋀B ⊢ Δ Γ, A, B ⊢ Δ Γ ⊢ A⋁B, Δ Γ ⊢ A, B, Δ (⋀L) (¬L) Γ ⊢ Δ (cut) P Q M N Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? Go back to the derivations.. Apply cut elimination for this derivation… We get two possible reducts: ([x̂Pα̂,ẑQδ̂]·γ) (Mμ̂σ̂[z]îĵN) γ̂†ẑ Γ ⊢ ¬(A⋁B), A⋀B, Δ Γ, B ⊢ A, Δ Γ, A ⊢ A, Δ P Q M N Γ ⊢ ¬(A⋁B)⋁(A⋀B), Δ Γ, A⋁B ⊢ A⋀B, Δ Γ, A⋁B ⊢ B, Δ Γ, A⋁B ⊢ A, Δ Γ, B ⊢ B, Δ Γ, A ⊢ B, Δ (Ax) (⋁L) (⋀R) (¬R) (⋁R) Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Γ, ¬(A⋁B) ⊢ Δ Γ, A⋀B ⊢ Δ Γ, A, B ⊢ Δ Γ ⊢ A⋁B, Δ Γ ⊢ A, B, Δ (⋀L) (¬L) Γ ⊢ Δ (cut) Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? Go back to the derivations.. Apply cut elimination for this derivation… We get two possible reducts: ((Mμ̂†x̂P)σ̂†û‹u.α›)α̂†ĵ (((Mσ̂†ẑQ)μ̂†ŵ̂‹w.δ›)δ̂†îN) ([x̂Pα̂,ẑQδ̂]·γ) (Mμ̂σ̂[z]îĵN) γ̂†ẑ Γ ⊢ ¬(A⋁B), A⋀B, Δ Γ, B ⊢ A, Δ Γ, A ⊢ A, Δ P Q M N Γ ⊢ ¬(A⋁B)⋁(A⋀B), Δ Γ, A⋁B ⊢ A⋀B, Δ Γ, A⋁B ⊢ B, Δ Γ, A⋁B ⊢ A, Δ Γ, B ⊢ B, Δ Γ, A ⊢ B, Δ (Ax) (⋁L) (⋀R) (¬R) (⋁R) Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Γ, ¬(A⋁B) ⊢ Δ Γ, A⋀B ⊢ Δ Γ, A, B ⊢ Δ Γ ⊢ A⋁B, Δ Γ ⊢ A, B, Δ (⋀L) (¬L) Γ ⊢ Δ (cut) Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

What is a suitable reduction rule? Go back to the derivations.. Apply cut elimination for this derivation… We get two possible reducts: ((Mμ̂†x̂P)σ̂†û‹u.α›)α̂†ĵ (((Mσ̂†ẑQ)μ̂†ŵ̂‹w.δ›)δ̂†îN) or (Mσ̂†ẑ(‹z.τ›τ̂†ĵ(Qδ̂†îN)))μ̂†x̂(‹x.π›π̂†î(Pα̂†ĵN)) ([x̂Pα̂,ẑQδ̂]·γ) (Mμ̂σ̂[z]îĵN) γ̂†ẑ Γ ⊢ ¬(A⋁B), A⋀B, Δ Γ, B ⊢ A, Δ Γ, A ⊢ A, Δ P Q M N Γ ⊢ ¬(A⋁B)⋁(A⋀B), Δ Γ, A⋁B ⊢ A⋀B, Δ Γ, A⋁B ⊢ B, Δ Γ, A⋁B ⊢ A, Δ Γ, B ⊢ B, Δ Γ, A ⊢ B, Δ (Ax) (⋁L) (⋀R) (¬R) (⋁R) Γ, ¬(A⋁B)⋁(A⋀B) ⊢ Δ Γ, ¬(A⋁B) ⊢ Δ Γ, A⋀B ⊢ Δ Γ, A, B ⊢ Δ Γ ⊢ A⋁B, Δ Γ ⊢ A, B, Δ (⋀L) (¬L) Γ ⊢ Δ (cut) Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Interpreting ↔ (reduct diagrams)
Paths from each output of M to each input of N This is a requirement in the general case N M μ σ α̂†j ̂ δ̂†i ̂ σ̂†ẑ μ̂†ŵ μ̂†x̂ σ̂†k̂ ‹k·α› ‹w·δ› i j Q z δ P x α Discuss what happens: each output of M is connected to each input of N. To do this, two copies of one or other are made. Etc. etc.

Interpreting ↔ (reduct diagrams)
Copying is undesirable Can we find a ‘better’ reduction rule? α̂†j ̂ π̂†i ̂ σ̂†ẑ μ̂†x̂ ‹x·π› M σ μ N i j Q z δ P x α δ̂†i ̂ ‹z·τ› τ̂†j ̂ Discuss what happens: each output of M is connected to each input of N. To do this, two copies of one or other are made. Etc. etc.

Interpreting ↔ (connection diagrams)
Abstract diagrams showing just the paths For these paths, crossings are necessary Too many connections to just M (or N) Idea: Share the connections more evenly… P P x α x α μ j M N μ j M N σ i σ i z Q δ Q z δ This is a simplified version of the previous one. We spot a different way we could connect them, that requires fewer ‘crossings’. Turns out we can achieve it.

Interpreting ↔ (connection diagrams)
Idea: Share the connections more evenly… We can now find reducts without copying P P x α x α μ j M μ j N M N σ i σ i z Q δ z Q δ This is a simplified version of the previous one. We spot a different way we could connect them, that requires fewer ‘crossings’. Turns out we can achieve it.

The X↔ calculus ‹x·α› Pα̂†x̂Q [x̂Pα̂,ẑQδ̂]·γ Mμ̂σ̂[z]îĵN
Calculus based only on the ↔ connective Syntax: ‹x·α› Pα̂†x̂Q [x̂Pα̂,ẑQδ̂]·γ Mμ̂σ̂[z]îĵN Principal reduction rule: ([x̂Pα̂,ẑQδ̂]·γ)γ̂†ẑ(Mμ̂σ̂[z]îĵN) → ((Mμ̂†x̂P)σ̂†û‹u.α›)α̂†ẑ(‹z.π›π̂†ĵ(Qδ̂†îN)) or ((Mσ̂†ẑQ)μ̂†û‹u.δ›)δ̂†x̂(‹x.π›π̂†î(Pα̂†ĵN)) Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Computational expressivity
The ↔ connective cannot logically express much. Only the connectives ⊤ and ID Surprisingly, the X ↔ calculus can simulate the reductions of the X calculus. We say ↔ can computationally express → ↔ can computationally express →, ⋀, ↑, ¬, ⊤, ID Similarly, ⊗ can logically express only ⊥ and ID But ⊗ can computationally express −, ⋁, ↓, ¬, ⊥, ID Computational expressivity ⊃ logical expressivity Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

Future work What does ↔ express in itself?
We know it has powerful simulation properties Further investigation in terms of “moving connectors” e.g. a derived calculus contains non-terminating terms if and only if it contains a connective which swaps an input for an output. Formalisation of results Some simulation results hold only up to permutations of proof structure. To make these formal, we might want proof nets. Say what it is, as a connective XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

The End Thank you for listening! Say what it is, as a connective
XOR is closely-related (if we know one, we know the other) Derive sequent calculus rules Derive syntax Derive cut-elimination rule This gives the copying solution

On the Computational Representation of Classical Logical Connectives

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On the Computational Representation of Classical Logical Connectives

Similar presentations

Presentation on theme: "On the Computational Representation of Classical Logical Connectives"— Presentation transcript:

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About project

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