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On Proof Interpretations and Linear Logic On Proof Interpretations and Linear Logic On Proof Interpretations and Linear Logic Paulo Oliva Queen Mary, University of London, UK ([email protected]) Heyting Day Utrecht, 27 February 2015 To Anne Troelstra and Dick de Jongh 6 ?? - !A ::: - !A - ! ILLp ILLp ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations IL = Intuitionistic predicate logic 6 ?? - !A ::: - !A On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations - ! ILLp ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) ?? - !A ::: - !A On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations 6 - ! ILLp ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) - !A ::: - !A On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations 6 ?? - ! ILLp ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations 6 ?? - ! ILLp ILLp - !A ::: - !A IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Outline 1 Proof Interpretations 2 Linear Logic 3 Unified Interpretation of Linear Logic On Proof Interpretations and Linear Logic Proof Interpretations Outline 1 Proof Interpretations 2 Linear Logic 3 Unified Interpretation of Linear Logic E.g. the set of prime numbers is infinite The book \Proofs from THE BOOK" contains six beautiful proofs of this Non-theorems map to the empty set E.g. the set of even numbers is finite Need to fix the formal system On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A 7! fπ : π ` Ag The book \Proofs from THE BOOK" contains six beautiful proofs of this Non-theorems map to the empty set E.g. the set of even numbers is finite Need to fix the formal system On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A 7! fπ : π ` Ag E.g. the set of prime numbers is infinite Non-theorems map to the empty set E.g. the set of even numbers is finite Need to fix the formal system On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A 7! fπ : π ` Ag E.g. the set of prime numbers is infinite The book \Proofs from THE BOOK" contains six beautiful proofs of this Need to fix the formal system On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A 7! fπ : π ` Ag E.g. the set of prime numbers is infinite The book \Proofs from THE BOOK" contains six beautiful proofs of this Non-theorems map to the empty set E.g. the set of even numbers is finite On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A 7! fπ : π ` Ag E.g. the set of prime numbers is infinite The book \Proofs from THE BOOK" contains six beautiful proofs of this Non-theorems map to the empty set E.g. the set of even numbers is finite Need to fix the formal system E.g. the set of prime numbers is infinite Specifies a program that on input n: N outputs a prime number p ≥ n and a certificate that p is prime If A is provable then the set should be non-empty Non-theorems map to un-implementable specifications Need to fix formal system and programming language On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Programs Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A 7! fp : p satisfies Ag If A is provable then the set should be non-empty Non-theorems map to un-implementable specifications Need to fix formal system and programming language On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Programs Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A 7! fp : p satisfies Ag E.g. the set of prime numbers is infinite Specifies a program that on input n: N outputs a prime number p ≥ n and a certificate that p is prime Non-theorems map to un-implementable specifications Need to fix formal system and programming language On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Programs Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A 7! fp : p satisfies Ag E.g. the set of prime numbers is infinite Specifies a program that on input n: N outputs a prime number p ≥ n and a certificate that p is prime If A is provable then the set should be non-empty Need to fix formal system and programming language On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Programs Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A 7! fp : p satisfies Ag E.g. the set of prime numbers is infinite Specifies a program that on input n: N outputs a prime number p ≥ n and a certificate that p is prime If A is provable then the set should be non-empty Non-theorems map to un-implementable specifications On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Programs Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A 7! fp : p satisfies Ag E.g. the set of prime numbers is infinite Specifies a program that on input n: N outputs a prime number p ≥ n and a certificate that p is prime If A is provable then the set should be non-empty Non-theorems map to un-implementable specifications Need to fix formal system and programming language Extension 1: Heyting arith. + G¨odelprimitive rec. Extension 2: Classical logic + Continuation operator :A A :A · · · · · · · · · ? B B A B On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Types Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus π `ML A , tπ : A Extension 2: Classical logic + Continuation operator :A A :A · · · · · · · · · ? B B A B On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Types Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus π `ML A , tπ : A Extension 1: Heyting arith. + G¨odelprimitive rec. On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Types Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus π `ML A , tπ : A Extension 1: Heyting arith. + G¨odelprimitive rec. Extension 2: Classical logic + Continuation operator :A A :A · · · · · · · · · ? B B A B 8x9y(y > x) ff : fx > xg Proof π ) Functional fπ 2 jAj ::: λx.x + 1 S1 `π A ) S2 ` fπ 2 jAj On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Formula A ) Set of functionals jAj 8x9y(y > x) ff : fx > xg ::: λx.x + 1 S1 `π A ) S2 ` fπ 2 jAj On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Formula A ) Set of functionals jAj Proof π ) Functional fπ 2 jAj 8x9y(y > x) ff : fx > xg ::: λx.x + 1 On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Formula A ) Set of functionals jAj Proof π ) Functional fπ 2 jAj S1 `π A ) S2 ` fπ 2 jAj ::: λx.x + 1 On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Formula A ) Set of functionals jAj 8x9y(y > x) ff : fx > xg Proof π ) Functional fπ 2 jAj S1 `π A ) S2 ` fπ 2 jAj On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Formula A ) Set of functionals jAj 8x9y(y > x) ff : fx > xg Proof π ) Functional fπ 2 jAj ::: λx.x + 1 S1 `π A ) S2 ` fπ 2 jAj ( jP j set of non-computable functionals HA ` P ) HA! ` 9f(f 2 jP j) ( Falsity interpreted as empty set (j ? j ≡ ;) PA `? ) T ` 9f(f 2 ;) Modified realizability (Kreisel 1959) Independence results for Heyting arithmetic HA On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Dialectica (G¨odel1958) Relative consistency of Peano arithmetic PA ( jP j set of non-computable functionals HA ` P ) HA! ` 9f(f 2 jP j) Modified realizability (Kreisel 1959) Independence results for Heyting arithmetic HA On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Dialectica (G¨odel1958) Relative consistency of Peano arithmetic PA ( Falsity interpreted as empty set (j ? j ≡ ;) PA `? ) T ` 9f(f 2 ;) ( jP j set of non-computable functionals HA ` P ) HA! ` 9f(f 2 jP j) On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Dialectica (G¨odel1958) Relative consistency of Peano arithmetic PA ( Falsity interpreted as empty set (j ? j ≡ ;) PA `? ) T ` 9f(f 2 ;) Modified realizability (Kreisel 1959) Independence results for Heyting arithmetic HA On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Dialectica (G¨odel1958) Relative consistency of Peano arithmetic PA ( Falsity interpreted as empty set (j ? j ≡ ;) PA `? ) T ` 9f(f 2 ;) Modified realizability (Kreisel 1959) Independence results for Heyting arithmetic HA ( jP j set of non-computable functionals HA ` P ) HA! ` 9f(f 2 jP j) Conservation results S + P ` A ) S ` A Proof mining S + A ` B ) S + Aw ` Bs Build models M j= S On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Relative consistence S `? ) T `? Independence results S 6` P Proof mining S + A ` B ) S + Aw ` Bs Build models M j= S On Proof Interpretations and Linear Logic Proof Interpretations
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