Double Negation Translations as Morphisms Olivier Hermant CRI, MINES ParisTech December 12, 2014 Deducteam Seminar, INRIA O. Hermant (Mines) Double Negations December 12, 2014 1 / 31
Double-Negation Translations Double-Negation translations: ◮ a shallow way to encode classical logic into intuitionistic ◮ Zenon’s backend for Dedukti ◮ existing translations: Kolmogorov’s (1925), Gentzen-Gödel’s (1933), Kuroda’s (1951), Krivine’s (1990), · · · Minimizing the translations: ◮ turns more formulæ into themselves; ◮ shifts a classical proof into an intuitionistic proof of the same formula. O. Hermant (Mines) Double Negations December 12, 2014 2 / 31
Morphisms ◮ A morphism preserves the operations between two structures: ( R ∗ , ∗ , 1 ) ( Z , + , 0 ) �→ h ( 0 ) Group morphism: → 1 h ( a + b ) → h ( a ) ∗ h ( b ) ◮ a translation that is a morphism: h ( P ) = P h ( A ∧ B ) = h ( A ) ∧ h ( B ) h ( A ∨ B ) = h ( A ) ∨ h ( B ) h ( A ⇒ B ) = h ( A ) ⇒ h ( B ) h ( ∀ xA ) = ∀ x h ( A ) h ( ∃ xA ) = ∃ x h ( A ) (of course this is the identity) O. Hermant (Mines) Double Negations December 12, 2014 3 / 31
Morphisms ◮ A morphism preserves the operations between two structures: ( R ∗ , ∗ , 1 ) ( Z , + , 0 ) �→ h ( 0 ) Group morphism: → 1 h ( a + b ) → h ( a ) ∗ h ( b ) ◮ a more interesting translation that is a morphism: h ( P ) = P h ( A ∧ B ) = h ( A ) ∧ c h ( B ) h ( A ∨ B ) = h ( A ) ∨ c h ( B ) h ( A ⇒ B ) = h ( A ) ⇒ c h ( B ) h ( ∀ xA ) = ∀ c x h ( A ) h ( ∃ xA ) = ∃ c x h ( A ) two kinds of connectives: the classical and the intuitionistic ones. O. Hermant (Mines) Double Negations December 12, 2014 3 / 31
Morphisms ◮ A morphism preserves the operations between two structures: ( R ∗ , ∗ , 1 ) ( Z , + , 0 ) �→ h ( 0 ) Group morphism: → 1 h ( a + b ) → h ( a ) ∗ h ( b ) ◮ a more interesting translation that is a morphism: h ( P ) = P h ( A ∧ B ) = h ( A ) ∧ c h ( B ) h ( A ∨ B ) = h ( A ) ∨ c h ( B ) h ( A ⇒ B ) = h ( A ) ⇒ c h ( B ) h ( ∀ xA ) = ∀ c x h ( A ) h ( ∃ xA ) = ∃ c x h ( A ) two kinds of connectives: the classical and the intuitionistic ones. ◮ Design a unified logic, where we can reason both classically and intuitionistically: strange premises Γ ⊢ A Γ ⊢ A ∨ i B Γ ⊢ A ∨ c B O. Hermant (Mines) Double Negations December 12, 2014 3 / 31
Translations that are Morphisms ◮ None of the previous translations is a morphism. ◮ Dowek has shown one, it is very verbose. ◮ We make it lighter. Plan: Classical and Intuitionistic Logic 1 Sequent Calculus 2 Double Negation Translations 3 Morphisms 4 O. Hermant (Mines) Double Negations December 12, 2014 4 / 31
Classical vs. Intuitionistic ◮ The principle of excluded-middle. Should A ∨ ¬ A be provable ? Yes or no ? ◮ Yes. This is what is called classical logic. ◮ Wait a minute ! The Drinker’s Principle In a bar, there is somebody such that, if he drinks, then everybody drinks. Two Irrationals There exists i 1 , i 2 ∈ R \ Q such that i i 2 1 ∈ Q . IRL: A Manicchean World You are with us, or against us (G. W. Bush) Rashomon (A. Kurosawa) O. Hermant (Mines) Double Negations December 12, 2014 5 / 31
Classical vs. Intuitionistic ◮ The principle of excluded-middle. Should A ∨ ¬ A be provable ? Yes or no ? ◮ No. This is the constructivist school (Brouwer, Heyting, Kolmogorov). ◮ Intuitionistic logic is one of those branches. It features the BHK interpretation of proofs: Witness Property A proof of ∃ xA (in the empty context) gives a witness t for the property A . Disjunction Property A proof of A ∨ B (in the empty context) reduces eventually either to a proof of A , or to a proof of B . O. Hermant (Mines) Double Negations December 12, 2014 6 / 31
The Classical Sequent Calculus (LK) ax Γ , A ⊢ A , ∆ Γ , A , B ⊢ ∆ Γ ⊢ A , ∆ Γ ⊢ B , ∆ ∧ R ∧ L Γ , A ∧ B ⊢ ∆ Γ ⊢ A ∧ B , ∆ Γ , A ⊢ ∆ Γ , B ⊢ ∆ ∨ L Γ ⊢ A , B , ∆ ∨ R Γ , A ∨ B ⊢ ∆ Γ ⊢ A ∨ B , ∆ Γ ⊢ A , ∆ Γ , B ⊢ ∆ ⇒ L Γ , A ⊢ B , ∆ ⇒ R Γ , A ⇒ B ⊢ ∆ Γ ⊢ A ⇒ B , ∆ Γ ⊢ A , ∆ Γ , A ⊢ ∆ ¬ L ¬ R Γ , ¬ A ⊢ ∆ Γ ⊢ ¬ A , ∆ Γ , A [ c / x ] ⊢ ∆ ∃ L Γ ⊢ A [ t / x ] , ∆ ∃ R Γ , ∃ xA ⊢ ∆ Γ ⊢ ∃ xA , ∆ Γ , A [ t / x ] ⊢ ∆ ∀ L Γ ⊢ A [ c / x ] , ∆ ∀ R Γ , ∀ xA ⊢ ∆ Γ ⊢ ∀ xA , ∆ O. Hermant (Mines) Double Negations December 12, 2014 7 / 31
The Intuitionistic Sequent Calculus (LJ) ax Γ , A ⊢ A Γ , A , B ⊢ ∆ Γ ⊢ A Γ ⊢ B ∧ L ∧ R Γ , A ∧ B ⊢ ∆ Γ ⊢ A ∧ B Γ , A ⊢ ∆ Γ , B ⊢ ∆ ∨ L Γ ⊢ A Γ ⊢ B ∨ R 1 ∨ R 2 Γ ⊢ A ∨ B Γ ⊢ A ∨ B Γ , A ∨ B ⊢ ∆ Γ ⊢ A Γ , B ⊢ ∆ ⇒ L Γ , A ⊢ B ⇒ R Γ , A ⇒ B ⊢ ∆ Γ ⊢ A ⇒ B Γ ⊢ A Γ , A ⊢ ¬ L ¬ R Γ , ¬ A ⊢ ∆ Γ ⊢ ¬ A Γ , A [ c / x ] ⊢ ∆ ∃ L Γ ⊢ A [ t / x ] ∃ R Γ , ∃ xA ⊢ ∆ Γ ⊢ ∃ xA Γ , A [ t / x ] ⊢ ∆ ∀ L Γ ⊢ A [ c / x ] ∀ R Γ , ∀ xA ⊢ ∆ Γ ⊢ ∀ xA O. Hermant (Mines) Double Negations December 12, 2014 8 / 31
Note on Frameworks ◮ structural rules are not shown (contraction, weakening) ◮ left-rules seem very similar in both cases ◮ so, lhs formulæ can be translated by themselves ◮ this accounts for polarizing the translations ◮ another work [Boudard & H] : ⋆ does not behave well in presence of cuts ⋆ appeals to focusing techniques O. Hermant (Mines) Double Negations December 12, 2014 9 / 31
Examples ◮ proofs that behave identically in classical/intuitionistic logic: ax ax A , B ⊢ B A , B ⊢ A ∧ L ⇒ R A ∧ B ⊢ B ∨ R A ⊢ B ⇒ A A ∧ B ⊢ B ∨ C ◮ proof of the excluded-middle: Classical Logic Intuitionistic Logic ax A ⊢ A ¬ R ⊢ A , ¬ A ∨ R ⊢ A ∨ ¬ A O. Hermant (Mines) Double Negations December 12, 2014 10 / 31
Examples ◮ proofs that behave identically in classical/intuitionistic logic: ax ax A , B ⊢ B A , B ⊢ A ∧ L ⇒ R A ∧ B ⊢ B ∨ R A ⊢ B ⇒ A A ∧ B ⊢ B ∨ C ◮ proof of the excluded-middle: Classical Logic Intuitionistic Logic ax A ⊢ A ¬ R ?? ⊢ A , ¬ A ∨ R ⊢ A ∨ ¬ A ⊢ A ∨ ¬ A O. Hermant (Mines) Double Negations December 12, 2014 10 / 31
The Excluded-Middle in Intuitionistic Logic ◮ is not provable. However, its negation is inconsistent. ax A ⊢ A ∨ R 1 A ⊢ A ∨ ¬ A ¬ L ¬ ( A ∨ ¬ A ) , A ⊢ ¬ R ¬ ( A ∨ ¬ A ) ⊢ ¬ A ∨ R 2 ¬ ( A ∨ ¬ A ) ⊢ A ∨ ¬ A ¬ L ¬ ( A ∨ ¬ A ) , ¬ ( A ∨ ¬ A ) ⊢ contraction ¬ ( A ∨ ¬ A ) ⊢ O. Hermant (Mines) Double Negations December 12, 2014 11 / 31
The Excluded-Middle in Intuitionistic Logic ◮ is not provable. However, its negation is inconsistent. ◮ this suggests a scheme for a translation between int. and clas. logic: ax ax A ⊢ A A ⊢ A ∨ R 1 A ⊢ A ∨ ¬ A ¬ L ¬ ( A ∨ ¬ A ) , A ⊢ ¬ R ¬ R ¬ ( A ∨ ¬ A ) ⊢ ¬ A ⊢ A , ¬ A ∨ R 2 ¬ ( A ∨ ¬ A ) ⊢ A ∨ ¬ A ¬ L ¬ ( A ∨ ¬ A ) , ¬ ( A ∨ ¬ A ) ⊢ contr ∨ R ¬ ( A ∨ ¬ A ) ⊢ ⊢ A ∨ ¬ A ◮ need: ¬¬ everywhere in ∆ (and Γ ) ◮ the proof of the “negation of the excluded middle” requires duplication (contraction), which partly explain why we allow several formulæ on the rhs in LK. O. Hermant (Mines) Double Negations December 12, 2014 11 / 31
The Excluded-Middle in Intuitionistic Logic ◮ is not provable. However, its negation is inconsistent. ◮ this suggests a scheme for a translation between int. and clas. logic: ax ax A ⊢ A A ⊢ A ∨ R 1 A ⊢ A ∨ ¬ A ¬ L ¬ ( A ∨ ¬ A ) , A ⊢ ¬ R ¬ R ¬ ( A ∨ ¬ A ) ⊢ ¬ A ⊢ A , ¬ A ∨ R 2 ¬ ( A ∨ ¬ A ) ⊢ A ∨ ¬ A ¬ L ¬ ( A ∨ ¬ A ) , ¬ ( A ∨ ¬ A ) ⊢ contr ∨ R ¬ ( A ∨ ¬ A ) ⊢ ⊢ A ∨ ¬ A ◮ given a classical proof Γ ⊢ ∆ , store ∆ on the lhs, and translate: Clas. Int. Γ , ¬ A 1 , ¬ ∆ ⊢ Γ , ¬ A 2 , ¬ ∆ ⊢ ¬ R ¬ Γ ⊢ A 1 , ∆ Γ ⊢ A 2 , ∆ Γ , ¬ ∆ ⊢ ¬¬ A 1 Γ , ¬ ∆ ⊢ ¬¬ A 2 rule r rule r Γ ⊢ A , ∆ Γ , ¬ ∆ ⊢ A ¬ L Γ , ¬ ∆ , ¬ A ⊢ ◮ need: ¬¬ everywhere in ∆ (and Γ ) ◮ the proof of the “negation of the excluded middle” requires duplication (contraction), which partly explain why we allow several formulæ on the rhs in LK. O. Hermant (Mines) Double Negations December 12, 2014 11 / 31
Kolmogorov’s Translation Kolmogorov’s ¬¬ -translation introduces ¬¬ everywhere: B Ko = ¬¬ B (atoms) ( B ∧ C ) Ko = ¬¬ ( B Ko ∧ C Ko ) ( B ∨ C ) Ko = ¬¬ ( B Ko ∨ C Ko ) ( B ⇒ C ) Ko = ¬¬ ( B Ko ⇒ C Ko ) ( ∀ xA ) Ko = ¬¬ ( ∀ xA Ko ) ( ∃ xA ) Ko = ¬¬ ( ∃ xA Ko ) Theorem Γ ⊢ ∆ is provable in LK iff Γ Ko , � ∆ Ko ⊢ is provable in LJ. Antinegation � is an operator, such that: ◮ � ¬ A = A ; ◮ � B = ¬ B otherwise. O. Hermant (Mines) Double Negations December 12, 2014 12 / 31
Light Kolmogorov’s Translation Moving negation from connectives to formulæ [Dowek& Werner] : B K = B (atoms) = ( ¬¬ B K ∧ ¬¬ C K ) ( B ∧ C ) K = ( ¬¬ B K ∨ ¬¬ C K ) ( B ∨ C ) K = ( ¬¬ B K ⇒ ¬¬ C K ) ( B ⇒ C ) K ( ∀ xA ) K = ∀ x ¬¬ A K ( ∃ xA ) K = ∃ x ¬¬ A K Theorem Γ ⊢ ∆ is provable in LK iff Γ K , ¬ ∆ K ⊢ is provable in LJ. Correspondence A Ko = ¬¬ A K O. Hermant (Mines) Double Negations December 12, 2014 13 / 31
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