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 On Proof Interpretations and Linear Logic Overview -- ILω . . . interpretations IL IL = Intuitionistic predicate logic On Proof Interpretations and Linear Logic Overview -- ILω . . . interpretations IL ILLωp -ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Overview -. . . interpretations IL - ILω 6ILLωp -ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Overview -. . . interpretations IL - ILω 6?? ILLωp -ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Overview -. . . interpretations IL - ILω 6?? ILLωp -ILLp -- !A . . . !A IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Outline 1Proof Interpretations 23Linear Logic Unified Interpretation of Linear Logic On Proof Interpretations and Linear Logic Proof Interpretations Outline 1Proof Interpretations 23Linear Logic Unified Interpretation of Linear Logic On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A<ul style="display: flex;"><li style="flex:1">→</li><li style="flex:1">{π : π ` A} </li></ul>On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A<ul style="display: flex;"><li style="flex:1">→</li><li style="flex:1">{π : π ` A} </li></ul>E.g. the set of prime numbers is infinite On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A<ul style="display: flex;"><li style="flex:1">→</li><li style="flex:1">{π : π ` A} </li></ul>E.g. the set of prime numbers is infinite The book “Proofs from THE BOOK” contains six beautiful proofs of this On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A<ul style="display: flex;"><li style="flex:1">→</li><li style="flex:1">{π : π ` A} </li></ul>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<ul style="display: flex;"><li style="flex:1">→</li><li style="flex:1">{π : π ` A} </li></ul>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 Programs Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A→{p : p satisfies A} On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Programs Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A→{p : p satisfies A} 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 On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Programs Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A→{p : p satisfies A} 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 On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Programs Brouwer-Heyting-Kolmogorov Interpretation Formulas as specifications (set of programs) A→{p : p satisfies A} 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→{p : p satisfies A} 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 Types Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus <ul style="display: flex;"><li style="flex:1">π `ML tπ : A </li><li style="flex:1">A</li></ul>⇔On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Types Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus <ul style="display: flex;"><li style="flex:1">π `ML tπ : A </li><li style="flex:1">A</li></ul>⇔Extension 1: Heyting arith. + Gödel primitive rec. On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Types Curry-Howard Correspondence Minimal Logic + Simply-typed λ-calculus <ul style="display: flex;"><li style="flex:1">π `ML tπ : A </li><li style="flex:1">A</li></ul>⇔Extension 1: Heyting arith. + Gödel primitive rec. Extension 2: Classical logic + Continuation operator <ul style="display: flex;"><li style="flex:1">¬A </li><li style="flex:1">A</li></ul>·B ¬A ··· ··· ··⊥B<ul style="display: flex;"><li style="flex:1">A</li><li style="flex:1">B</li></ul>On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations <ul style="display: flex;"><li style="flex:1">Formula A </li><li style="flex:1">⇒ Set of functionals |A| </li></ul>On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations <ul style="display: flex;"><li style="flex:1">Formula A </li><li style="flex:1">⇒ Set of functionals |A| </li></ul><ul style="display: flex;"><li style="flex:1">Proof π </li><li style="flex:1">⇒ Functional fπ ∈ |A| </li></ul>On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Formula A Proof π ⇒ Set of functionals |A| ⇒ Functional fπ ∈ |A| S1 `π A ⇒ S2 ` fπ ∈ |A| On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Formula A ∀x∃y(y > x) Proof π ⇒ Set of functionals |A| {f : fx > x} ⇒ Functional fπ ∈ |A| S1 `π A ⇒ S2 ` fπ ∈ |A| On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Formula A ∀x∃y(y > x) Proof π . . . ⇒ Set of functionals |A| {f : fx > x} ⇒ Functional fπ ∈ |A| λx.x + 1 S1 `π A ⇒ S2 ` fπ ∈ |A| On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Dialectica (Go¨del 1958) Relative consistency of Peano arithmetic PA On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Dialectica (Go¨del 1958) Relative consistency of Peano arithmetic PA (Falsity interpreted as empty set (| ⊥ | ≡ ∅) PA `⊥ ⇒ T ` ∃f(f ∈ ∅) On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Dialectica (Go¨del 1958) Relative consistency of Peano arithmetic PA (Falsity interpreted as empty set (| ⊥ | ≡ ∅) PA `⊥ ⇒ T ` ∃f(f ∈ ∅) Modified realizability (Kreisel 1959) Independence results for Heyting arithmetic HA On Proof Interpretations and Linear Logic Proof Interpretations Functional Interpretations Dialectica (Go¨del 1958) Relative consistency of Peano arithmetic PA (Falsity interpreted as empty set (| ⊥ | ≡ ∅) PA `⊥ ⇒ T ` ∃f(f ∈ ∅) Modified realizability (Kreisel 1959) Independence results for Heyting arithmetic HA (|P| set of non-computable functionals HA ` P <ul style="display: flex;"><li style="flex:1">⇒</li><li style="flex:1">HAω ` ∃f(f ∈ |P|) </li></ul>

On Proof Interpretations and Linear Logic

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