A direct proof that intuitionistic predicate calculus is complete with respect to presheaves of classical models

A direct proof that

intuitionistic predicate calculus is complete with respect to presheaves of classical models

Ivano Ciardelli (ILLC, Universiteit van Amsterdam) Christian Retor ´e (LIRMM, Universit ´e de Montpellier)

Topology and languages, Toulouse, May 22–24

Remarks

This direct completeness proof is essentially due to Ivano Ciardelli (in TACL 2011 cf. reference at the end).

Thanks to Guillaume Bonfante for inviting me.

Thanks for Jacques van de Wiele

who introduced me to (pre)sheaf semantics in his 1986-1987 lecture (Paris 7).

Thanks to the Topos & Logic group

(Jean Malgoire, Nicolas Saby, David Theret)

of the Institut Montpelli ´erain Alexander Grothendieck (+ Abdelkader Gouaich, LIRMM)

A Logic?

formulas

proofs ! interpretations

A.1. Logic

Logic: language (trees with binding relations)

whose expressions can be true (or not):

wellformed expressions of a logical language have a meaning.

A.2. Intutionnistic logic vs. classical logic (the usual logic of mathematics)

Absence of Tertium no Datur,A∨ ¬Adoes not always hold.

Disjunctive statemeents are stronger.

Existential statements are stronger.

Proof have a constructive meaning, algorithms can be extracted from proofs.

A.3. Rules of intuitionist logic: structures

Structural rules Γ,A,B, ∆`C

E_g Γ,B,A, ∆`C

∆`C A<sub>g</sub> A, ∆`C

Γ,A,A, ∆`C C<sub>g</sub> Γ,A, ∆`C

A.4. Rules of intuitionistic logic: connectives

Axioms areA`A(ifAthenA...) for every A.

Negation¬Ais just a short hand forA⇒ ⊥.

Θ`(A∧B)

∧_e Θ`A

Θ`(A∧B)

∧_e Θ`B

Θ`A ∆`B

∧_d Θ, ∆`(A∧B)

Θ`(A∨B) A, Γ`C B, ∆`C

∨_e Θ, Γ, ∆`C

Θ`A

∨_d Θ`(A∨B)

Θ`B

∨_d Θ`(A∨B)

Θ`A Γ`A⇒B

⇒_e Γ, Θ`B

Γ,A`B

⇒_d Γ`(A⇒B)

Γ` ⊥

⊥_e Γ`C

A.5. Differences

A∨ ¬Adoes not hold for anyA.

¬¬B does not entailB.

However¬¬(C∨ ¬C)hods for anyC.

¬∀x.¬P(x)does not entail∃xP(x).

An example, in the language of rings: ∀x.((x=0)∨ ¬(x=0))is not provable and their are concrete counter models.

[¬∀x.((x =0)∨ ¬(x =0))]→[¬∀x.¬¬((x=0)∨ ¬(x =0))]

is also non provable.

B Topological models

B.1. Usual FOL models

One is given a language,

e.g. constants (0,1), functions (+,∗), and predicates (6).

One is given a set|M|.

Constants are interpreted by elements of|M|,

n-ary functions symbols by n-ary applications from |M|ⁿ to |M|, and n-ary predicates by parts of|M|ⁿ.

Logical connectives and quantifiers are interpreted intuitively (Tarskian truth: ”∧” means ”and”, ”∀” means ”for all” etc.).

B.2. Soundness

any provable formula is true for every interpretation

or:

when T entails F then any model that satisfies T satisfies F

B.3. Completeness

Completeness (a word that often encompass soundness):

a formula that is true in every interpretation is derivable

or

a formula F that is true in every model of T is a logical consequence of T

e.g.

a formula F of ring theory is true in any ring if and only if

F is provable from the axioms of ring theory Soundness, completeness (and compacity)

are typical fo first order logic (as opposed to higher order logic).

B.4. Presheaves

A pre sheaf can de defined as a contravariant functorF

• from open subsets of a topological set (this partial order can be viewed as a category)

• to a category (e.g. sets, groups, rings):

Contravariant functor: whenU⊂V there is a restriction mapρ_V,U fromF(V)to F(U)andρ_U3,U2◦ρ_U1,U2 =ρ_U1,U3 whenever is makes sense, i.e. U₃⊂U₂⊂U₁.

Example of pre-sheaf on the topological spaceR: U7→F(U)the ring of bounded functions fromU toR.

B.5. Sheaves

The presheaf (resp separated presheaf) is said to be a sheaf if every family of compatible elements has unique glueing:

given a coverU_i of an open setU,

with for everyi an elementc_i ∈F(U_i)such that for every pairi,j ρ_Ui,Uj(c_i) =ρ_Uj,Ui(c_j)

there is a unique(resp. at most one) c inF(u) such that c_i =ρ_U,Ui(c).

Example of pre-sheaf on the topological spaceR:

U7→C(U,R)the ring of continuous functions from U toR.

B.6. Presheaf semantics

Grothendieck generalized the notion of topological space,using coverings.

A site is a category with every object is provided with various cov- ering.

A covering of an objectϕ consists in a set of arrowsf_i,i ∈I with codomain ϕ_i — when the category is a preorder it is enough to know the domain of everyf_i: there is at most one arrow fromϕ_i to ϕ.

1. ϕ{ϕ};

2. ifψ 6ϕ andϕ{ϕ_i|i ∈I}thenψ{ψ∧ϕ_i|i ∈I};

3. if ϕ{ϕ_i|i ∈I} and if for each i ∈I, ϕ_i {ψ_i,k|k ∈Ki}, thenϕ{ψ_i,k|i ∈I,k∈Ki}.

Who, when? 70’s Joyal, Lawvere, Lambek,...

B.7. Presheaf semantics: models

Apresheaf modelMforL is a presheaf of first-orderL−structures over a Grothendieck site(C,):

• for any objectu a first-order modelM_u

• for any arrowf :v →ua homomorphismf:M_u→M_v satisfying the following extra conditions.

Separateness For any elementsa,bofM_u, if there is a coveru{f_i :u_i →u|i ∈I} such that for alli ∈I we haveaf_i=bf_i, thena=b.

Local character of atoms For anyn−ary relation symbolR, for any tuple(a₁, ... ,a_n)fromM_u

if there is a coveru{f_i :u_i →u|i ∈I}

such that for alli ∈I we have(a₁f_i, ... ,a_nf_i)∈R_ui, then(a₁, ... ,a_n)∈R_u.

B.8. Presheaf semantics: Kripke-Joyal forcing — 1/3 atoms and conjunction

Formulas of L can be inductively interpreted on an object u of a given presheaf modelM (ν: assignment intoM_u):

• uν R(t₁, ... ,t_n)iff([t₁]_ν, ... ,[t_n]_ν)∈R_u.

• uν t₁=t₂iff [t₁]_ν = [t₂]_ν.

• uν ⊥iffu/0(M_/0⊥)

• uν ϕ∧ψ iffuν ϕ anduν ψ.

B.9. Presheaf semantics: Kripke-Joyal forcing — 2/3 disjunction and existential

Formulas of L can be inductively interpreted on an object u of a given presheaf modelM (ν: assignment intoMu):

• uνϕ∨ψ iff there is a covering family{f_i:u_i →u|i∈I}such that for anyi ∈I we haveu_i ν ϕ oru_i ν ψ.

• uν ∃xϕ ⇐⇒ there exist a covering family{f_i :u_i→u|i∈ I} and elements a_i ∈ |Mu_i| for i ∈I such that u_i ν[x7→ai]ϕ for any indexi.

B.10. Presheaf semantics: Kripke-Joyal forcing — 3/3 implication and universal

Formulas of L can be inductively interpreted on an object u of a given presheaf modelM (ν: assignment intoM_u):

• uϕ→ψ iff for allf :v →u, ifv ϕ thenv ψ.

• u¬ϕ iff for allf :v →u, withv 6= /0,v 6ϕ.

• uν ∀xϕ iff for all f :v →u and alla∈M_v,v ν[x7→a]ϕ.

Notice that the usual Kripke semantics is obtained as a particular case when the underlying Grothendieck site is a poset equipped with the trivial coveringuF ⇐⇒u∈F.

B.11. Properties of Kripke-Joyal forcing

Fonctoriality of:

iff_i :U_i →U_j andU_jF(t₁, ... ,t_n)thenU_i F(t₁ⁱ, ... ,t_nⁱ)wheret_k^j is simply the restriction oft_k toU_i.

Locality of validity:

we asked for the validity of atoms to be local, but Krike-Joyal forc- ing propagates this property to all formulae:

If there exist a covering ofU by f_i :U_i →U and if for alli one has U_i F(t₁ⁱ, ... ,t_nⁱ)thenUF(t₁, ... ,t_n)

Given a closed term (no variables)ϕν,x7→t^ϕ ψ(x)iffϕ νψ(t).

B.12. Soundness

Whenever`F in IQC then any presheaf semantics satisfiesF.

WheneverΓ`F in IQC then any presheaf semantics that satisfies ΓsatisfiesF as well.

The theory of rings, whose language has two binary functions (+, .) two constants 0,1 and equality, can be interpreted in the presheaf on the topological space R which maps U to the ring C_U,R of continuous functions from the open setU toR.

In this model, both[¬∀x.((x =0)∨ ¬(x=0))]

and[∀x¬¬((x =0)∨ ¬(x =0))]are both valid.

B.13. Soundness proof

Induction on the proof height, looking at every possible last rule, e.g. in natural deduction.

C The completeness part of

completeness

C.1. Completeness for presheaf semantics

If every presheaf models satisfiesϕ thenϕ is provable inintuitionisticlogic.

Usually established by:

• equivalence withΩ−models;

• construction of a canonical Kripke model.

C.2. Canonical model construction:

the underlying site

Canonical site:

• Category: we take the Lindenbaum-Tarski algebraL – Objects: classes of provably equivalent formulasϕ.

– Arrows: ϕ≤ψ ⇐⇒ ϕ `ψ

• Grothendieck topology: ϕ{ψ_i}_i∈I whenever

∀χ

ϕ`χ iff (∀i ∈I ψi `χ)

Think of the last line asϕ=^W_iψ_i

(incorrect, because FOL formulae are finite!)

C.3. Properties of this site

The proposed site is actually a site i.e. it enjoys the three properties.

1. ϕ{ϕ};

2. ifψ `ϕ andϕ{ϕ_i|i ∈I}thenψ{ψ∧ϕ_i|i ∈I};

3. if ϕ{ϕ_i|i ∈I} and if for each i ∈I, ϕ_i {ψ_i,k|k ∈Ki}, thenϕ{ψ_i,k|i ∈I,k∈Ki}.

C.4. Canonical model construction: the presheaf

• Putt≡_ϕ t⁰in caseϕ `t =t⁰.

• Denote byt^ϕ the equivalence class oft modulo≡_ϕ.

Canonical presheaf:

• ModelM_ϕ:

1. Universe|M_ϕ|:

set of equivalence classest^ϕ of closed terms;

2. Function symbols: f_ϕ(~t^ϕ) =f(~t)^ϕ;

3. Relation symbols:~t^ϕ ∈R_ϕ ⇐⇒ ϕ`R(~t).

• Restriction. Ift^ψ ∈M_ψ andϕ≤ψ, putt^ψϕ=t^ϕ.

C.5. The canonical presheaf is well defined

The canonical presheaf is separated. If two elements have the same restictions on each part of a cover, then they are equal.

The interpretation of atomic formulas is local. If an atomic formula holds on each part of a cover ofU then it holds onU.

Observe that it is not a sheaf

(the glueing of compatible elements may not exist).

C.6. Method for the proof of completeness

∀ψ

∀ϕ [if ϕψ then ϕ`ψ]

By induction on the formulaψ.

It is also possible to prove directly:

∀ψ

∀ϕ[ϕψ iff ϕ `ψ]

What is fun is thatsoundnessmainly usesintroductionrules whilecompletenessmainly useseliminationrules.

C.7. Variants

The method and the construction can be parametrised by a con- textΓfor obtaining what is calledstrong completeness:

The quotient on formula is not really needed.

Equality is not mandatory but pleasant.

Terms and constants can be eliminated in FOL with equality.

If there are no constants, one should add a denumerable set of constants. Indeed some proof steps need afresh constant.

C.8. Future work

Can we construct a canonical sheaf and not just separated presheaf e.g. with the sheaf completion method that basically simply for- mally adds the missing global sections?

Does the proof works as well?

Initially the idea was to define couple models of first order linear logic, following some hints by Giovani Sambin, but nowadays not so many people are interested in such questions.

Thank you for your attention Reference:

A Canonical Model for Presheaf Semantics Ivano CiardelliTopol- ogy, Algebra and Categories in Logic (TACL) 2011, Jul 2011, Mar- seille.https://hal.inria.fr/inria-00618862/fr/