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Computational Representation of Logical Connectives in Classical Logic

This study explores the computational aspect of classical logical connectives, focusing on implication, conjunction, disjunction, and negation. It delves into term assignments and the Curry-Howard Correspondence.

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Computational Representation of Logical Connectives in Classical Logic

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  1. On the Computational Representation of Classical Logical Connectives Jayshan Raghunandan and Alexander Summers Department of Computing Imperial College London

  2. 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?

  3. Overview • Sequent Calculi & Inhabitation • 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?

  4. A Sequent Calculus for Implication

  5. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ A Sequent Calculus for Implication

  6. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ A Sequent Calculus for Implication P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: .Γ⊢Δ

  7. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ A Sequent Calculus for Implication P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: .Γ⊢Δ P: .Γ,x:A ⊢α:B, Δ (→R) x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  8. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ A Sequent Calculus for Implication P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: .Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  9. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: .Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  10. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: .Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  11. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS α, δ, … OUTPUTS P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: .Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  12. Inhabitation x, y, … INPUTS α, δ, … OUTPUTS P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : . Γ, x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  13. Inhabitation x, y, … INPUTS α, δ, … OUTPUTS P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : . Γ, x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  14. Inhabitation x, y, … INPUTS α, δ, … OUTPUTS P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : . Γ, x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  15. Inhabitation x, y, … INPUTS α, δ, … OUTPUTS P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α›: . Γ, x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  16. Inhabitation x, y, … INPUTS α, δ, … OUTPUTS P: .Γ⊢Δ,α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α›: . Γ, x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  17. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  18. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  19. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  20. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  21. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: . x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  22. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: . x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  23. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: . x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  24. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: . x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  25. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: . x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  26. (Ax) ‹x·α› : .Γ,x:A⊢α:A, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS P: . Γ⊢Δ, α:A Q: . x:A, Γ⊢Δ (cut) Pα̂†x̂Q: . Γ⊢Δ Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: .Γ,x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  27. P: .Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : .Γ,x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: . Γ, x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  28. P: .Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : .Γ,x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: . Γ, x:A ⊢α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  29. P: .Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : .Γ,x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: . Γ, x:A ⊢ α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  30. P: .Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : .Γ,x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: . Γ, x:A ⊢ α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  31. P: .Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : .Γ,x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: . Γ, x:A ⊢ α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  32. P: .Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : .Γ,x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: . Γ, x:A ⊢ α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  33. P: .Γ⊢Δ, α:A Q: .x:A, Γ⊢Δ (Ax) (cut) ‹x·α› : .Γ,x:A⊢α:A, Δ Pα̂†x̂Q: .Γ⊢Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A P: . Γ, x:A ⊢ α:B, Δ (→L) (→R) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ x̂Pα̂·δ: .Γ⊢δ:A→B, Δ

  34. P: .Γ⊢Δ, α:A 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 (→R) x̂Pα̂·δ: .Γ⊢δ:A→B, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: . Γ⊢Δ, α:A (→L) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ

  35. P: .Γ⊢Δ, α:A 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 (→R) x̂Pα̂·δ: .Γ⊢δ:A→B, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: . Γ⊢Δ, α:A (→L) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ

  36. P: .Γ⊢Δ, α:A 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 (→R) x̂Pα̂·δ: .Γ⊢δ:A→B, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: . x:B, Γ⊢Δ P: . Γ⊢Δ, α:A (→L) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ

  37. P: .Γ⊢Δ, α:A 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 (→R) x̂Pα̂·δ: .Γ⊢δ:A→B, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: . x:B, Γ⊢Δ P: . Γ⊢Δ, α:A (→L) Pα̂[z]x̂Q : . z:A→B, Γ⊢ Δ

  38. P: .Γ⊢Δ, α:A 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 (→R) x̂Pα̂·δ: .Γ⊢δ:A→B, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: . x:B, Γ⊢Δ P: . Γ⊢Δ, α:A (→L) Pα̂[z]x̂Q : . z:A→B, Γ⊢ Δ

  39. P: .Γ⊢Δ, α:A 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 (→R) x̂Pα̂·δ: .Γ⊢δ:A→B, Δ Inhabitation x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: . x:B, Γ⊢Δ P: . Γ⊢Δ, α:A (→L) Pα̂[z]x̂Q : . z:A→B, Γ⊢ Δ

  40. P: .Γ⊢Δ, α:A 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 (→R) x̂Pα̂·δ: .Γ⊢δ:A→B, Δ Inhabitation (X Calculus) x, y, … INPUTS P, Q, … TERMS ^ BINDER α, δ, … OUTPUTS Q: .x:B, Γ⊢Δ P: .Γ⊢Δ,α:A (→L) Pα̂[z]x̂Q : .z:A→B, Γ⊢ Δ

  41. 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

  42. 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

  43. Cut-Elimination • Only one reduction rule is significant per connective • Defines how the two syntactic constructs interact • For example: implication

  44. 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)

  45. 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)

  46. 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)

  47. 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)

  48. 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) ] [ x̂Q Pα̂ z

  49. 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) ] [ x̂Q Pα̂ z ŷRγ̂·δ

  50. 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) ] [ x̂Q Pα̂ z ŷRγ̂·δ

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