Kleene realizability and negative translations

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Q E I U G I C Kleene realizability and negative translations Alexandre Miquel O P. D E. L Ō A U D E L A R April 21th, IMERL

Plan 1 Kleene realizability 2 Gödel-Gentzen negative translation 3 Lafont-Reus-Streicher negative translation

Plan 1 Kleene realizability 2 Gödel-Gentzen negative translation 3 Lafont-Reus-Streicher negative translation

The language of realizers (recall) Terms of PCF Terms t, u ::= x λx. t tu pair fst snd 0 S rec (= λ-calculus + primitive pairs & integers) Syntactic worship: Free & bound variables. Renaming. Work up to α-conversion. Set of free variables: FV (t). Capture-avoiding substitution: t{x := u} Notations: t 1, t 2 := pair t 1 t 2, n := S n 0 (n N) Reduction rules (λx. t) u t{x := u} fst t 1, t 2 t 1 rec t 0 t 1 0 t 0 snd t 1, t 2 t 2 rec t 0 t 1 (S u) t 1 u (rec t 0 t 1 u) Grand reduction written t u (reflexive, transitive, context-closed)

Definition of the relation t A (recall) Recall: For each closed FO-term e, we write e N its denotation in N Definition of the realizability relation t A (t, A closed) t e 1 = e 2 e1 N = e2 N t 0 t t t 0 t A B u (u A tu B) t A B t 1 t 2 (t t 1, t 2 t 1 A t 2 B) t A B u ((t 0, u u A) (t 1, u u B)) t x A(x) n (t n A(n)) t x A(x) n u (t n, u u A(n)) Lemma (closure under anti-evaluation) If t t and t A, then t A

The main Theorem (recall) Lemma (Adequacy) Let d : (A 1,..., A n B) be a derivation in NJ. Then: for all valuations ρ, for all realizers t 1 A 1 [ρ],..., t n A n [ρ], we have: d [ρ]{z 1 := t 1,..., z n := t n } B[ρ] Lemma All axioms of HA are realized Theorem (Soundness) If HA A, then t A for some closed PCF-term t

Harrop formulas (1/2) The class of Harrop formulas Harrop formulas H ::= e 1 = e 2 H 1 H 2 A H x H Intuition: Harrop formulas do not contain the two problematic constructions and, except on the left-hand side of implications Therefore, Harrop formulas are classical: Proposition For each Harrop formula H( x): HA x (H( x) H( x)) Proof. By structural induction on H( x).

Harrop formulas (2/2) To each (possibly open) Harrop formula H, we associate a closed PCF-term t H that is computationally trivial: τ H := 0 (H atomic) τ A H = λ. τ H τ H1 H 2 = τ H1, τ H2 τ x H = λ. τ H Theorem For all closed Harrop formulas H: If H is realized, then τ H H Moreover, all realizers of H are computationally equivalent to τ H Intuition: Harrop formulas have computationally irrelevant realizers, that can be replaced by the trivial realizers τ H Useful for optimizing extracted programs (e.g. Fermat s last theorem) But shows that Harrop formulas are computationally irrelevant

Plan 1 Kleene realizability 2 Gödel-Gentzen negative translation 3 Lafont-Reus-Streicher negative translation

How to cope with classical logic? Kleene realizability is definitely incompatible with classical logic: x (Halt(x) Halt(x)) any term x (Halt(x) Halt(x)) (The same holds for all variants of Kleene realizability) Two possible solutions: 1 Compose Kleene realizability with a negative translation from classical logic (LK) to intuitionistic logic (LJ) (next slide) 2 Reformulate the principles of realizability to make them compatible with classical logic: Krivine classical realizability (next talk)

The Gödel-Gentzen negative translation Idea: Turn positive constructions (atomic formulas,, ) into negative constructions (,,,, ) using De Morgan laws Every formula A is translated into a formula A G defined by: G := G := (A B) G := A G B G (e 1 = e 2 ) G := (e 1 = e 2 ) (A B) G := A G B G (A B) G := ( A G B G ) ( x A) G := x A G ( x A) G := x A G writing: A := A Theorem (Soundness) 1 LK A G A 2 If PA A, then HA A G

Realizing translated formulas Strategy: 1 Build a derivation d of A (in PA) 2 Turn it into a derivation d G of A G (in HA) 3 Turn d G into a Kleene realizer (program extraction) Does not work! Failure comes from: Proposition (Realizability collapse) For every closed formula A: 1 A G is a Harrop formula (computationally irrelevant) 2 Kleene s semantics for A G mimics Tarski s semantics for A: A G is realized iff τ A G A G iff N = A Proof. By structural induction on A. Conclusion: Kleene Gödel-Gentzen = Tarski

Friedman s R-translation (called A-translation by Friedman) Principle: In Gödel-Gentzen translation, replace each occurrence of (absurdity) by a fixed formula R, called the return formula Every formula A is translated into a formula A F defined by: F := F := R (A B) F := A F B F (e 1 = e 2 ) F := R R (e 1 = e 2 ) (A B) F := A F B F (A B) F := R ( R A F R B F ) ( x A) F := x A F ( x A) F := R x R A F writing: R A := A R Theorem (Soundness) If PA A, then HA A F (independently from the formula R) Beware! The formulas A and A F are no more classically equivalent (in general)

Friedman s trick Theorem (Kreisel-Friedman) PA conservatively extends HA over Π 0 2 -formulas: If PA x y f (x, y) = 0, then HA x y f (x, y) = 0 Proof. Assume that PA x y f (x, y) = 0. We have: HA x R y R R R f (x, y) = 0 (by R-translation) HA x R y R f (x, y) = 0 (since R R R R ) HA R y R f (x 0, y) = 0 ( -elim, x 0 fresh) HA y (f (x 0, y) = 0 R) R (def. of R ) We now let: R := y 0 f (x 0, y 0 ) = 0 (Friedman s trick!) From the def. of R: HA y (f (x 0, y) = 0 y 0 f (x 0, y 0 ) = 0) y 0 f (x 0, y 0 ) = 0 But the premise of the above implication is provable HA y (f (x 0, y) = 0 y 0 f (x 0, y 0 ) = 0) ( -intro) hence we get HA y 0 f (x 0, y 0 ) = 0 (modus ponens) HA x 0 y 0 f (x 0, y 0 ) = 0 ( -intro)

Realizing translated formulas, again Strategy: 1 Build a derivation d of a Π 0 2-formula A (in PA) 2 Turn it into a derivation F-trick(d F ) of A (in HA) 3 Turn F-trick(d F ) into a Kleene realizer of A (program extraction) This technique perfectly works in practice. However: The formula A F is not a Harrop formula (in general), even when A is. Possible fix: Introduce specific optimization techniques, e.g.: Refined Program Extraction [Berger et al. 2001] The translation A A F completely changes the structure of the underlying proof. Possible fix: cf next part

Plan 1 Kleene realizability 2 Gödel-Gentzen negative translation 3 Lafont-Reus-Streicher negative translation

The Lafont-Reus-Streicher negative translation Idea: Translate each formula A into the (relative) negation of a formula A already representing the negation of A: A LRS := R A A R (A defined by induction on A) (Again, this translation is parameterized by a return formula R) To every predicate symbol p (source language) we associate a predicate symbol p representing its negation (target language) Definition of the translations A A and A A LRS : (p(e 1,..., e k )) := p(e 1,..., e k ) := (A B) := A LRS B ( x A) := x A A LRS := R A A R Theorem (Soundness) 1 When R : LK A A and LK A LRS A 2 If LK A, then LJ A LRS (independently from the formula R)

Computational interpretation Intuition: The translated formula A represents the type of stacks opposing (classical) terms of type A: (A 1 A n B) A LRS 1 A LRS n B (A 1 A n B) A LRS 1 A LRS n B To analyze the computational contents of the LRS-translation, we need to work across to λ-calculi: A source calculus to represent classical proofs: λ source = λ + cc : ((A B) A) A (Peirce s law) (Polymorphic constant cc introduces classical reasoning) An intuitionistic target calculus to represent translated proofs: λ target = λ, (In this calculus, pairs are used to represent stacks)

The source λ-calculus ({,, }-fragment of LK) Syntax Types A, B ::= p(e 1,..., e k ) A B x A Proof-terms t, u ::= z λz. t tu cc Classical logic obtained by introducing an inert constant cc (call/cc) for Peirce s law (taken as an axiom) Constructions,,, encoded using De Morgan laws (= full LK) Typing rules Γ z : A (z:a) Γ Γ, z : A t : B Γ λz. t : A B Γ t : A Γ t : x A x / FV (Γ) Γ Γ cc : ((A B) A) A Γ t : A B Γ u : A Γ tu : B t : x A Γ t : A{x := e} Γ t : Γ t : A Note: is treated uniformly: x A(x) x A(x) (no function argument!)

The target λ-calculus ({,,, }-fragment of LJ) Syntax Types A, B ::= p(e 1,..., e k ) A B A B x A Proof-terms t, u ::= z λz. t tu t, u fst(t) snd(t) + usual reduction rules for proof-terms Typing rules Γ z : A (z:a) Γ Γ, z : A t : B Γ λz. t : A B Γ t : A Γ t : B Γ t, u : A B Γ t : A{x := e} Γ t : x A Γ t : FV (t) dom(γ) Γ t : A B Γ u : A Γ tu : B Γ t : A B Γ fst(t) : A Γ t : ( x A) B Γ t : x (A B) Γ t : A B Γ snd(t) : B x / FV (B) Note: treated uniformly: x A(x) x A(x) (no witness!)

Remark: Uniform vs non-uniform quantifiers (1/2) In the Curry-Howard correspondence (and in realizability), there are two different ways to interpret quantifiers: Quantifier x A(x) x A(x) Uniform (ML/Haskell style) A(x) x D (intersection type) A(x) x D (union type) Non-uniform (Type Theory style) A(x) x D (type of dependent functions) A(x) x D (type of dependent pairs) Remark: Tarski/Kripke/Heyting/Cohen models do not distinguish the two interpretations: difference only appears in realizability

Remark: Uniform vs non-uniform quantifiers (2/2) 1st-, 2nd- and higher-order logic support both interpretations (But uniform interpretation is more concise & natural) The same holds for impredicative set theories: ZF, IZF C, IZF R Arithmetic (PA/HA) only supports the non-uniform interpretation (due to the induction principle) But in all cases, the non-uniform interpretation can be encoded from the uniform interpretation, using a relativization: (non-uniform) x A(x) := (uniform) x (x D A(x) ) }{{} type of functions (non-uniform) x A(x) := (uniform) x (x D A(x) ) }{{} type of pairs This is why we shall prefer the uniform interpretation (in what follows)

The Lafont-Reus-Streicher logical translation The logical translation A A LRS (p(e 1,..., e k )) := p(e 1,..., e k ) := (A B) := A LRS B ( x A) := x A A LRS := R A corresponds to a program transformation on untyped proof terms, called a continuation-passing style (CPS) translation: (z) LRS := λs. z s (λz. t) LRS := λ z, s. t LRS (tu) LRS := λs. t LRS u LRS, s (cc) LRS := λ z, s. z k s, s where k s := λ z,. z s Note: λ z, s. t defined as λz 0. (λzs. t) (fst(z 0 )) (snd(z 0 )) Theorem (Soundness) If Γ t : A (in the source λ-calculus) then Γ LRS t LRS : A LRS (in the target λ-calculus)

Towards the Krivine abstract machine From the computational behavior of translated proof terms t LRS... (λz. t) LRS @ u, s t LRS {z := u} @ s (tu) LRS @ s t LRS @ u LRS, s (cc) LRS @ u, s u @ k s, s k s @ u, s u @ s... we deduce evaluation rules for classical proof terms: Krivine Abstract Machine (KAM) Grab λz. t u π t{z := u} π Push tu π t u π Save cc u π u k π π Restore k π u π u π Reformulating Kleene realizability through the LRS translation (and its CPS), we get Krivine classical realizability (cf next talk)

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