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The Dialectica Translation of Type Theory Andrej Bauer Pierre-Marie Pdrot University of Ljubljana INRIA TYPES 24th May 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  1. The Dialectica Translation of Type Theory Andrej Bauer Pierre-Marie Pédrot University of Ljubljana INRIA TYPES 24th May 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 1 / 20

  2. Previously at TYPES... Analytical description of the TYPES 2013 social event (Toulouse) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 2 / 20

  3. Previously at TYPES... Analytical description of the TYPES 2013 social event (Toulouse) U. Kohlenbach C. Riba An analytical table An analytical bottle of wine Myself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 2 / 20

  4. Previously at TYPES... Analytical description of the TYPES 2013 social event (Toulouse) U. Kohlenbach C. Riba An analytical table An analytical bottle of wine Myself Dramatis Personae: Ulrich Kohlenbach , King of Dialectica Colin Riba , a Proof-Theory Gentleman Pierre-Marie Pédrot , a Novice PhD Student The Bottle of Wine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 2 / 20

  5. Previously at TYPES... The Bottle of Wine is almost empty. Colin , carried away by the enthusiasm of proof theory, begins to claim his love for the works of Gödel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 3 / 20

  6. Previously at TYPES... The Bottle of Wine is almost empty. Colin , carried away by the enthusiasm of proof theory, begins to claim his love for the works of Gödel . Colin O, Dialectica, the mysterious functional interpretation! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 3 / 20

  7. Previously at TYPES... The Bottle of Wine is almost empty. Colin , carried away by the enthusiasm of proof theory, begins to claim his love for the works of Gödel . Colin O, Dialectica, the mysterious functional interpretation! Ulrich nods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 3 / 20

  8. Previously at TYPES... The Bottle of Wine is almost empty. Colin , carried away by the enthusiasm of proof theory, begins to claim his love for the works of Gödel . Colin O, Dialectica, the mysterious functional interpretation! Ulrich nods . Colin For thou canst not be understood through Curry-Howard! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 3 / 20

  9. Previously at TYPES... The Bottle of Wine is almost empty. Colin , carried away by the enthusiasm of proof theory, begins to claim his love for the works of Gödel . Colin O, Dialectica, the mysterious functional interpretation! Ulrich nods . Colin For thou canst not be understood through Curry-Howard! Ulrich nods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 3 / 20

  10. Previously at TYPES... The Bottle of Wine is almost empty. Colin , carried away by the enthusiasm of proof theory, begins to claim his love for the works of Gödel . Colin O, Dialectica, the mysterious functional interpretation! Ulrich nods . Colin For thou canst not be understood through Curry-Howard! Ulrich nods . P.-M. That can’t be true! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 3 / 20

  11. Previously at TYPES... The Bottle of Wine is almost empty. Colin , carried away by the enthusiasm of proof theory, begins to claim his love for the works of Gödel . Colin O, Dialectica, the mysterious functional interpretation! Ulrich nods . Colin For thou canst not be understood through Curry-Howard! Ulrich nods . P.-M. That can’t be true! Ulrich nods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 3 / 20

  12. Previously at TYPES... The Bottle of Wine is almost empty. Colin , carried away by the enthusiasm of proof theory, begins to claim his love for the works of Gödel . Colin O, Dialectica, the mysterious functional interpretation! Ulrich nods . Colin For thou canst not be understood through Curry-Howard! Ulrich nods . P.-M. That can’t be true! Ulrich nods . The Bottle of Wine is empty. The characters disappear in a blurred mist. Noone can really recollect this dialogue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 3 / 20

  13. Previously at TYPES... The Bottle of Wine is almost empty. Colin , carried away by the enthusiasm of proof theory, begins to claim his love for the works of Gödel . Colin O, Dialectica, the mysterious functional interpretation! Ulrich nods . Colin For thou canst not be understood through Curry-Howard! Ulrich nods . P.-M. That can’t be true! Ulrich nods . The Bottle of Wine is empty. The characters disappear in a blurred mist. Noone can really recollect this dialogue. But I had found the matter for my PhD! (Morale: you definitely should attend social events.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 3 / 20

  14. IP MP HA A Quick Recap Dialectica is a logical translation due to Gödel Nowadays would be called a realizability intepretation { A λ -term π • : [ [ A ] ] ⊢ HA π : A ⇝ A logical property π • ⊩ A in the meta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 4 / 20

  15. A Quick Recap Dialectica is a logical translation due to Gödel Nowadays would be called a realizability intepretation { A λ -term π • : [ [ A ] ] ⊢ HA π : A ⇝ A logical property π • ⊩ A in the meta It preserves consistency, i.e. there is no π : [ [ ⊥ ] ] s.t. π ⊩ ⊥ It interprets strictly more than HA , namely: MP : ¬ ( ∀ x : N . ¬ P ) → ∃ x : N . P ( P decidable) IP : ( I → ∃ x : N . P ) → ∃ x : N . I → P ( I irrelevant) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 4 / 20

  16. Curry-Howard & Realizability “Realizability interpretations tend to hide a programming translation.” Logic Programming Kreisel modified realizability Identity translation Krivine classical realizability Lafont-Reus-Streicher CPS Gödel Dialectica realizability ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 5 / 20

  17. Curry-Howard & Realizability “Realizability interpretations tend to hide a programming translation.” Logic Programming Kreisel modified realizability Identity translation Krivine classical realizability Lafont-Reus-Streicher CPS Gödel Dialectica realizability A fancy one! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 5 / 20

  18. Curry-Howard & Realizability “Realizability interpretations tend to hide a programming translation.” Logic Programming Kreisel modified realizability Identity translation Krivine classical realizability Lafont-Reus-Streicher CPS Gödel Dialectica realizability A fancy one! Gives first-class status to stacks Features a computationally relevant substitution Mix of LRS with delimited continuations Requires computational (finite) multisets M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bauer & Pédrot (U. Ljubljana, INRIA) The Dialectica Translation of TT 24/05/2016 5 / 20

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