Constructive embeddings of intermediate

Sara Negri

University of Helsinki

Symposium on Constructivity and Computability 9-10 June 2011, Uppsala The G¨odel-Tarski-McKinsey embedding of intuitionistic into S4

∗ 1. G¨odel(1933): `Int A → `S4 A soundness ∗ 2. McKinsey & Tarski (1948): `S4 A → `Int A faithfulness ∗ 3. Dummett & Lemmon (1959): `Int+Ax A iff `S4+Ax∗ A A M is a modal companion of a superintuitionistic ∗ logic L if `L A iff `M A . So S4 is a modal companion of Int, S4+Ax∗ is a modal companion of Int+Ax. 2. and 3. are proved semantically. We look closer at McKinsey & Tarski (1948):

Proof by McKinsey & Tarski (1948) uses:

I 1. Completeness of wrt Heyting algebras (Brouwerian algebras) and of S4 wrt topological Boolean algebras (closure algebras)

I 2. Representation of Heyting algebras as the opens of topological Boolean algebras.

I 3. The proof is indirect because of 1. and non-constructive because of 2. (Uses Stone representation of distributive lattices, in particular Zorn’s lemma) The result was generalized to intermediate logics by Dummett and Lemmon (1959). No syntactic proof of faithfulness in the literature except the complex proof of the embedding of Int into S4 in Troelstra & Schwichtenberg (1996). Goal: give a direct, constructive, syntactic, and uniform proof. Background on labelled sequent systems Method for formulating systems of contraction-free sequent calculus for basic modal logic and its extensions and for systems of non-classical logics in Negri (2005). General ideas for basic modal logic K and other systems of normal modal logics. Syntax embodies the possible world semantics:

I Rules for 2, 3 through meaning explanation and inversion principles.

I Frame properties in the form of “non-logical rules” added to the basic sequent calculus. Extensions are modular. Relation of this approach to the extensive literature on labelled deductive systems of various forms is detailed in Negri (2011). Here explicit labelling and contraction-free systems. Sequent calculus for the basic modal logic K

I Start with the calculus G3c for classical propositional logic

I Enrich the language: Sequents Γ ⇒ ∆ where Γ and ∆ consist of expressions xRy and x : A (corresponding to x A of Kripke models), with x, y, z, ... ranging in a set W and with A any formula in the language of propositional logic extended with the modal operators of necessity and possibility, 2 and 3.

I Rules for basic modal logic obtained from the inductive definition of validity in a Kripke frame. From x : 2A iff for all y, xRy implies y : A obtain the rules

I If y : A can be derived for an arbitrary y accessible from x, then x : 2A can be derived xRy, Γ ⇒ ∆, y : A R2 Γ ⇒ ∆, x : 2A

arbitrariness of y becomes the variable condition y not (free) in Γ, ∆

I If x : 2A and y is accessible from x, then y : A y : A, x : 2A, xRy, Γ ⇒ ∆ L2 x : 2A, xRy, Γ ⇒ ∆ The rules for 3 are obtained similarly from the semantical explanation

x : 3A iff for some y, xRy and y : A

I If x : 3A, there exists y is accessible from x such that y : A xRy, y : A, Γ ⇒ ∆ L3 x : 3A, Γ ⇒ ∆ variable condition: y not (free) in Γ, ∆

I If y is accessible from x and y : A, then x : 3A gives the rule

xRy, Γ ⇒ ∆, x : 3A, y : A R3 xRy, Γ ⇒ ∆, x : 3A Initial sequents: x : P, Γ ⇒ ∆, x : P Propositional rules: x : A, x : B, Γ ⇒ ∆ Γ ⇒ ∆, x : A Γ ⇒ ∆, x : B L& R& x : A&B, Γ ⇒ ∆ Γ ⇒ ∆, x : A&B x : A, Γ ⇒ ∆ x : B, Γ ⇒ ∆ Γ ⇒ ∆, x : A, x : B L∨ R∨ x : A ∨ B, Γ ⇒ ∆ Γ ⇒ ∆, x : A ∨ B Γ ⇒ ∆, x : A x : B, Γ ⇒ ∆ x : A, Γ ⇒ ∆, x : B L⊃ R⊃ x : A ⊃ B, Γ ⇒ ∆ Γ ⇒ ∆, x : A ⊃ B L⊥ x :⊥, Γ ⇒ ∆ Modal rules: y : A, x : 2A, xRy, Γ ⇒ ∆ xRy, Γ ⇒ ∆, y : A L2 R2 x : 2A, xRy, Γ ⇒ ∆ Γ ⇒ ∆, x : 2A xRy, y : A, Γ ⇒ ∆ xRy, Γ ⇒ ∆, x : 3A, y : A L3 R3 x : 3A, Γ ⇒ ∆ xRy, Γ ⇒ ∆, x : 3A Modal systems and frame properties

I Modal logic K characterized by arbitrary frames.

I Restrictions of the class of frames amount to adding certain frame properties to the calculus.

I In Kripke frames for S4 the accessibility relation is reflexive ∀x.xRx and transitive ∀x∀y∀z((xRy & yRz) ⊃ xRz)

I For S4.2 also directed ∀xyz(xRy & xRz ⊃ ∃w(yRw & zRw)) Axioms as rules (cont.)

I We use the method developed in NvP (1998) and in N (2003)

I Universal axioms are transformed into conjunctive normal form, that is, conjunctions of formulas of the form P1& ... &Pm ⊃ Q1 ∨ · · · ∨ Qn

I Each conjunct is then converted into the regular rule,

Q1, P1,..., Pm, Γ ⇒ ∆ ... Qn, P1,..., Pm, Γ ⇒ ∆ Reg P1,..., Pm, Γ ⇒ ∆

I Other rules may be added by the closure condition. Those that are instances of contraction are admissible. Examples of universal axioms Axiom Frame property T 2A ⊃ A ∀x xRx reflexivity 4 2A ⊃ 22A ∀xyz(xRy & yRz ⊃ xRz) transitivity E 3A ⊃ 23A ∀xyz(xRy & xRz ⊃ yRz) euclideanness B A ⊃ 23A ∀xy(xRy ⊃ yRx) symmetry 3 2(2A ⊃ B) ∨ 2(2B ⊃ A) ∀xyz(xRy & xRz ⊃ yRz ∨ zRy) connectedness D 2A ⊃ 3A ∀x∃y xRy seriality 2 32A ⊃ 23A ∀xyz(xRy & xRz ⊃ ∃w(yRw & zRw)) directedness Frame property Rule xRx, Γ ⇒ ∆ T ∀x xRx reflexivity Γ ⇒ ∆ xRz, xRy, yRz, Γ ⇒ ∆ 4 ∀xyz(xRy & yRz ⊃ xRz) trans. xRy, yRz, Γ ⇒ ∆ xRz, xRy, yRz, Γ ⇒ ∆ E ∀xyz(xRy & xRz ⊃ yRz) euclid. xRy, xRz, Γ ⇒ ∆ yRx, xRy, Γ ⇒ ∆ B ∀xy(xRy ⊃ yRx) symmetry xRy, Γ ⇒ ∆ 3 ∀xyz(xRy & xRz ⊃ yRz ∨ zRy) yRz, xRy, xRz, Γ ⇒ ∆ zRy, xRy, xRz, Γ ⇒ ∆ connectedness xRy, xRz, Γ ⇒ ∆ D ∀x∃y xRy seriality 2 ∀xyz(xRy & xRz ⊃ ∃w(yRw & zRw)) Axioms as rules (cont.)

I Method extended (N2003) to geometric theories, that is, theories axiomatized by formulas

∀z(A ⊃ B)

where A and B do not contain ⊃ or ∀.

I These can be reduced to conjunctions of

∀z(P1& ... &Pm ⊃ ∃x1M1 ∨ · · · ∨ ∃xnMn)

where Mj is conjunction of atoms Qj

I and turned into the geometric rule, with the yi ’s not in the conclusion

Q1(y1/x1), P, Γ ⇒ ∆ ... Qn(yn/xn), P, Γ ⇒ ∆ GR P, Γ ⇒ ∆ Examples of geometric implications Axiom Frame property T 2A ⊃ A ∀x xRx reflexivity 4 2A ⊃ 22A ∀xyz(xRy & yRz ⊃ xRz) transitivity E 3A ⊃ 23A ∀xyz(xRy & xRz ⊃ yRz) euclideanness B A ⊃ 23A ∀xy(xRy ⊃ yRx) symmetry 3 2(2A ⊃ B) ∨ 2(2B ⊃ A) ∀xyz(xRy & xRz ⊃ yRz ∨ zRy) connectedness D 2A ⊃ 3A ∀x∃y xRy seriality 2 32A ⊃ 23A ∀xyz(xRy & xRz ⊃ ∃w(yRw & zRw)) directedness Frame property Rule xRx, Γ ⇒ ∆ T ∀x xRx reflexivity Γ ⇒ ∆ xRz, xRy, yRz, Γ ⇒ ∆ 4 ∀xyz(xRy & yRz ⊃ xRz) trans. xRy, yRz, Γ ⇒ ∆ yRz, xRy, xRz, Γ ⇒ ∆ E ∀xyz(xRy & xRz ⊃ yRz) euclid. xRy, xRz, Γ ⇒ ∆ yRx, xRy, Γ ⇒ ∆ B ∀xy(xRy ⊃ yRx) symmetry xRy, Γ ⇒ ∆ yRz, xRy, xRz, Γ ⇒ ∆ zRy, xRy, xRz, Γ ⇒ ∆ 3 ∀xyz(xRy & xRz ⊃ yRz ∨ zRy) xRy, xRz, Γ ⇒ ∆ xRy, Γ ⇒ ∆ y D ∀x∃y xRy seriality Γ ⇒ ∆ yRw, zRw, xRy, xRz, Γ ⇒ ∆ w 2 ∀xyz(xRy & xRz ⊃ ∃w(yRw & zRw)) xRy, xRz, Γ ⇒ ∆ Structural properties of the basic system

I Theorem The structural rules are admissible in extensions of G3c with regular or geometric rules satisfying the closure condition. Weakening and contraction are height-preserving admissible. All the rules are invertible.

I The same structural properties hold in G3K, in addition substitution of labels is height-preserving admissible. Results Let G3K* be any extension of G3K with universal or geometric rules rules for the accessibility relation.

I All the structural rules–weakening, contraction, and cut–are admissible in the system G3K*.

I The characteristic axioms are derivable.

I The necessitation rule is admissible.

I Indirect completeness, through equivalence with the corresponding axiomatic system.

I Direct completeness: proof search in the system either gives a derivation or a Kripke countemodel.

I Answers to questions of undefinability through conservativity theorems.

I Answers to decidability questions through algorithms of terminating proof search. Intuitionistic logic for intuitionistic logic was inspired by the modal embedding. We use the semantics to get a syntactic proof of the embedding. The accessibility relation is a pre-order

∀x.x 6 x (Ref) ∀xyz(x 6 y& y 6 z ⊃ x 6 z) (Trans) Forcing relation as in basic modal logic except for implication:

x A ⊃ B iff ∀y(x 6 y & y A ⊃ y B) Gives the rules: x y, x : A ⊃ B, Γ ⇒ ∆, y : A x y, x : A ⊃ B, y : B, Γ ⇒ ∆ 6 6 L⊃ x 6 y, x : A ⊃ B, Γ ⇒ ∆ x y, y : A, Γ ⇒ ∆, y : B 6 R ⊃ Γ ⇒ ∆, x : A ⊃ B Rule R ⊃ has the condition that y is not in the conclusion. Intuitionistic logic

Initial sequents of G3K modified to have monotonicity of the forcing relation. Enough to have monotonicity with respect to atomic formulas

x 6 y, x : P, Γ ⇒ ∆, y : P

Monotonicity w.r.t. arbitrary formulas admissible.

The mathematical rules for the accessibility relation 6 are the rules Ref and Trans. The system G3I

Initial sequents: x 6 y, x : P, Γ ⇒ ∆, y : P Logical rules: As in G3K for &, ∨, ⊥, x y, x : A ⊃ B, Γ ⇒ y : A, ∆, x y, x : A ⊃ B, y : B, Γ ⇒ ∆ 6 6 L⊃ x 6 y, x : A ⊃ B, Γ ⇒ ∆ x y, y : A, Γ ⇒ ∆, y : B 6 R⊃ Γ ⇒ ∆, x : A ⊃ B Order rules:

x 6 x, Γ ⇒ ∆ x 6 z, x 6 y, y 6 z, Γ ⇒ ∆ Ref Trans Γ ⇒ ∆ x 6 y, y 6 z, Γ ⇒ ∆

Rule R⊃ has the condition that y must not be in Γ, ∆. Intermediate logics (joint work with Roy Dyckhoff) G3I can be extended with rules expressing additional properties of

the pre-order 6 exactly as done for modal logic. For example, G¨odel-Dummettlogic has a strongly connected accessibility relation

∀xyz((x 6 y & x 6 z) ⊃ (y 6 z ∨ z 6 y)). This becomes the rule x y, x z, y z, Γ ⇒ ∆ x y, x z, z y, Γ ⇒ ∆ 6 6 6 6 6 6 GD x 6 y, x 6 z, Γ ⇒ ∆

Add the rule to G3I and obtain a (labelled) sequent system for G¨odel-Dummettlogic. Denote by G3I* any extension of G3I with rules following the geometric rule scheme. Below more examples of intermediate logics. Structural properties of G3I* All sequents of the following form are derivable in G3I*:

1. x 6 y, x : A, Γ ⇒ ∆, y : A 2. x : A, Γ ⇒ ∆, x : A The substitution rule

Γ ⇒ ∆ (y/x) Γ(y/x) ⇒ ∆(y/x)

is hp-admissible in G3I*. The rules of Weakening

Γ ⇒ ∆ Γ ⇒ ∆ Γ ⇒ ∆ LW LW RW 6 x : A, Γ ⇒ ∆ Γ ⇒ ∆, x : A x 6 y, Γ ⇒ ∆

are hp-admissible in G3I*. Structural properties of G3I* (cont.) All the rules of G3I* are hp-invertible. The rules of Contraction x : A, x : A, Γ ⇒ ∆ Γ ⇒ ∆, x : A, x : A L-Ctr R-Ctr x : A, Γ ⇒ ∆ Γ ⇒ ∆, x : A

x 6 y, x 6 y, Γ ⇒ ∆ L-Ctr 6 x 6 y, Γ ⇒ ∆ are hp-admissible in G3I*. The Cut rule Γ ⇒ ∆, x : A x : A, Γ0 ⇒ ∆0 Cut Γ, Γ0 ⇒ ∆, ∆0

is admissible in G3I*. The rule ⇒ x :A ⊃ B ⇒ x :A MP ⇒ x :B is admissible in G3I*. The axioms corresponding to the frame properties are derivable in G3I*. Each system in G3I* is equivalent to the obtained by adding to Int the axiom(s) that correspond((s)) to the frame property(ies)

∗ `Int+Ax A iff G3IAx `⇒ x : A Interpolable logics: axioms, frame conditions, and rules

I Jan Jankov-De Morgan Logic: The relation 6 is directed or convergent, i.e.

∀xyz((x 6 y & x 6 z) ⊃ ∃w(y 6 w & z 6 w)).

Known as KC or as “logic of weak excluded middle”. Axiomatised by ¬A ∨ ¬¬A or ¬(A&B) ⊃ (¬A ∨ ¬B).

y w, z w, x y, x z, Γ ⇒ ∆ 6 6 6 6 Jan x 6 y, x 6 z, Γ ⇒ ∆ (w fresh)

I Bd2: Bounded depth at most 2

∀xyz((x 6y 6z) ⊃ (y 6 x ∨ z 6 y)). Axiomatised by A ∨ (A ⊃ (B ∨ ¬B)).

y 6 x, x 6 y, y 6 z, Γ ⇒ ∆ x 6 y, y 6 z, z 6 y, Γ ⇒ ∆ Bd2 x 6 y, y 6 z, Γ ⇒ ∆ I GSc:Depth at most 2 and at most 2 final elements

∀xyz∃v((x 6 v & y 6 v) ∨ (y 6 v & z 6 v) ∨ (x 6 v & z 6 v)). This logic is axiomatised by, for example, (A ⊃ B) ∨ (B ⊃ A) ∨ ((A ⊃ ¬B)& (¬B ⊃ A)) and A ∨ (A ⊃ (B ∨ ¬B)). Rules are the rule for Bd2 and (with v fresh) x v, y v, Γ ⇒ ∆ y v, z v, Γ ⇒ ∆ x v, z v, Γ ⇒ ∆ 6 6 6 6 6 6 F Γ ⇒ ∆ 2

I Sm: Smetanich logic, also known as LC2 or HT, the “logic of here and there”, or as G¨odel’s3-valued logic. The accessibility relation is strongly connected and has depth at most 2, i.e.

∀xyz((x 6 y & y 6 z) ⊃ (y 6 x ∨ z 6 y))

Axiomatised by the GD axiom plus the Bd2 axiom, or, equivalently, by (¬B ⊃ A) ⊃ (((A ⊃ B) ⊃ A) ⊃ A). Rules as rules for GD and Bd2 above. I Cl Classical logic: The accessibility relation is symmetric,

∀xy(x 6y ⊃ y 6 x).

Axiomatised by A ∨ ¬A or by ¬¬A ⊃ A. The rule is x y, y x, Γ ⇒ ∆ 6 6 Cl x 6 y, Γ ⇒ ∆ Expressive power Not only the interpolable ones can be treated by this method: Several variants of these logics are non-interpolable but still have geometric frame conditions: Bdn for n > 2 (“Bounded depth n”) and btwn for n > 2 (approximately, “bounded top-width” n). For n = 3, frame condition is For n = 3, for example, the latter’s characteristic frame condition is the geometric implication

3 ^ _ ∀x0x1x2x3( x0Rxi ⊃ ∃y( xi Ry & xj Ry)). i=1 i6=j

a geometric implication Displayable logics Properly displayable logics are captured by the extension with rules for geometric implications. By Kracht’s results, displayable extensions of basic modal logic are characterized by primitive frame conditions, of the form

(∀R )(∃R )A

∀R ...... ∀u(wRu ⊃ Au) ∃R ...... ∃u(wRu & Au) A is built from conjunction and disjunctions of w = u, wRu, wR−1u where w and u not both in the scope of an ∃ Displayable logics (cont.) Through standard conversions of first order logic, primitive frame conditions convert to the form of a geometric implication:

∀w1(At1(w1) ⊃ (∀w2At2(w1, w2) ⊃ ... ∃u1(Bt1(u1)&(∃u2Bt2(u2)& ... )))))

;

∀w1∀w2 ... ∀wn(At1(w1)&At2(w1, w2) ⊃ ∃u1∃u2Bt1(u1)&Bt2(u1, u2)....))))

Not every geometric implication satisfies the additional conditions on variables, but those that are needed in our context do: the existential label licences additional steps if related to a universal label. If both labels in an atom were bound by the existential quantifier they would be both fresh in the geometric rule scheme and thus useless. Analyticity

I Rules not analytic in a strong sense (each expression in a premiss is a subexpression of the conclusion).

I Less suffices to ensure the consequences of strong analiticity.

I We can transform derivations so that they satisfy the Subterm property: All terms (variables, worlds) in a derivation are either eigenvariables or terms (variables, worlds) in the conclusion.

I Weak subformula property: All formulas in a derivation are either subformulas of (formulas in) the endsequent or atomic formulas of the form xRy.

I Analyticity properties + structural constraints (e.g., no duplications) permit proofs of decidability by terminating proof search algorithms.

I Completeness proof (Negri 2009) gives a method for either finding a proof or generating a countermodel. Given an extension G3I* of G3I with rules for 6 , we denote by G3S4* the corresponding extension of G3S4. Soundness If G3I* ` Γ ⇒ ∆ then G3S4* ` Γ2 ⇒ ∆2. Proof: By induction on the structure of the derivation. x 6y, Γ, x :P ⇒ y :P, ∆ ; Ax ..., Γ2, x :2P, z :P ⇒ z :P, ∆2 2 2 L2 x 6y, y 6z, x 6z, Γ , x :2P ⇒ z :P, ∆ 2 2 Trans x 6y, y 6z, Γ , x :2P ⇒ z :P, ∆ 2 2 R2 x 6y, Γ , x :2P ⇒ y :2P, ∆ x y, Γ2, y :A2 ⇒ y :B2, ∆2 6 R ⊃ x y, Γ, y :A ⇒ y :B, ∆ x y, Γ2 ⇒ y :A2 ⊃ B2, ∆2 6 R ⊃ 6 R2 Γ ⇒ x :A ⊃ B, ∆ ; Γ2 ⇒ x :2(A2 ⊃ B2), ∆2

L⊃ similar; conjunction, disjunction and absurdity routine. Frame rules identical in the two systems, so nothing to prove for them. Faithfulness If G3S4* ` Γ2 ⇒ ∆2 then G3I* ` Γ ⇒ ∆. Follows as a special case from: If Γ, ∆ are multisets of labelled formulas (with relational atoms also possibly in Γ) and Γ0, ∆0 are multisets of labelled atomic formulas, and G3S4* ` Γ2, Γ0 ⇒ ∆2, ∆0, then G3I* ` Γ, Γ0 ⇒ ∆, ∆0. Proof: By induction on the derivation. By induction on the derivation of Γ2, Γ0 ⇒ ∆2, ∆0. If it is an initial sequent, then some atom x :P is in Γ0 and in ∆0; the conclusion then follows in G3I* by Refl from the initial sequent 0 0 x 6 x, Γ, Γ ⇒ ∆, ∆ . If it is a conclusion of L⊥, so also is Γ, Γ0 ⇒ ∆, ∆0. If it is derived by a rule for & or for ∨, the inductive hypothesis applies to the premisses and then the corresponding rule in G3I* gives the conclusion. If it is derived by a modal rule, the principal formula, being a translated formula, can only be of the form 2P or of the form 2(A2 ⊃ B2). There are four cases:

I 1. With 2P principal on the left, the step 002 0 2 0 x 6 y, y :P, x :2P, Γ , Γ ⇒ ∆ , ∆ 002 0 2 0 L2 x 6 y, x :2P, Γ , Γ ⇒ ∆ , ∆ is translated to the admissible G3I* step 00 0 0 x 6 y, y :P, x :P, Γ , Γ ⇒ ∆, ∆ 00 0 0 x 6 y, x :P, Γ , Γ ⇒ ∆, ∆

I 2. With 2P principal on the right, the step (with y fresh) x y, Γ2, Γ0 ⇒ ∆002, ∆0, y :P 6 R2 Γ2, Γ0 ⇒ ∆002, ∆0, x :2P is translated (using admissibility of substitution) to the G3I* steps x y, Γ, Γ0 ⇒ ∆00, ∆0, y :P 6 (x/y) x x, Γ, Γ0 ⇒ ∆00, ∆0, x :P 6 Refl Γ, Γ0 ⇒ ∆00, ∆0, x :P 2 2 I 3. With 2(A ⊃ B ) principal on the left, the step

2 2 2 2 002 0 2 0 x 6 y, x :2(A ⊃ B ), y :A ⊃ B , Γ , Γ ⇒ ∆ , ∆ 2 2 002 0 2 0 L2 x 6 y, x :2(A ⊃ B ), Γ , Γ ⇒ ∆ , ∆

Observe that A2 ⊃ B2 is not a translated formula, nor an atomic one. By hp-invertibility of L⊃ in G3S4* we have

2 2 002 0 2 0 2 x 6 y, x :2(A ⊃ B ), Γ , Γ ⇒ ∆ , ∆ , y :A

and

2 2 2 002 0 2 0 x 6 y, x :2(A ⊃ B ), y :B , Γ , Γ ⇒ ∆ , ∆

Now the inductive hypothesis applies. We therefore have the derivation in G3I*

I.H. I.H. 00 0 0 00 0 0 x 6 y, x :A ⊃ B, Γ , Γ ⇒∆, ∆ , y :A x 6 y, x :A ⊃ B, y :B, Γ , Γ ⇒∆, ∆ 00 0 0 L ⊃ x 6 y, x :A ⊃ B, Γ , Γ ⇒ ∆, ∆ 2 2 I 4. If 2(A ⊃ B ) is principal on the right, the step is

x y, Γ2, Γ0 ⇒ ∆002, ∆0, y :A2 ⊃ B2 6 R2 Γ2, Γ0 ⇒ ∆002, ∆0, x :2(A2 ⊃ B2)

from which, by hp-invertibility of R ⊃ in G3S4*, we have a derivation in G3S4* of

2 2 0 002 0 2 x 6 y, y :A , Γ , Γ ⇒ ∆ , ∆ , y :B

to which the inductive hypothesis applies. An R ⊃ step in G3I* gives us the desired conclusion. QED The proof is direct, constructive, syntactic, and uniform for all the intermediate logics obtained as geometric extensions of intuitionistic logic References

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