Sendika tipi olan lamda terimlerinin karakterizasyonu

Birçok ders kitabı lambda-matemindeki kesişim tiplerini kapsar. Kavşak için yazım kuralları aşağıdaki gibi tanımlanabilir (basit bir şekilde alt tipleme ile yazılan lambda hesabının üstüne):

\frac{Γ ⊢ M : T_{1} Γ ⊢ M : T_{2}}{Γ ⊢ M : T_{1} \land T_{2}} (\land I) \frac{}{Γ ⊢ M : ⊤} (⊤ I)

$\dfrac{\Gamma \vdash M : T_1 \quad \Gamma \vdash M : T_2} {\Gamma \vdash M : T_1 \wedge T_2} (\wedge I) \qquad\qquad \dfrac{} {\Gamma \vdash M : \top} (\top I)$

Kavşak tipleri normalleşme açısından ilginç özelliklere sahiptir:

Bir lambda vadeli kullanmadan yazılabilir $\top I$ kuvvetle normale IFF kural.
Bir lambda süreli bir tür içermeyen kabul $\top$ normal bir formu vardır IFF.

Ya kavşak eklemek yerine sendika eklersek?

\frac{Γ ⊢ M : T_{1}}{Γ ⊢ M : T_{1} \lor T_{2}} (\lor I_{1}) \frac{Γ ⊢ M : T_{2}}{Γ ⊢ M : T_{1} \lor T_{2}} (\lor I_{2})

$\dfrac{\Gamma \vdash M : T_1} {\Gamma \vdash M : T_1 \vee T_2} (\vee I_1) \qquad\qquad \dfrac{\Gamma \vdash M : T_2} {\Gamma \vdash M : T_1 \vee T_2} (\vee I_2)$

Basit türleri, alt tipleri ve sendikaları olan lambda calculus'un benzer bir özelliği var mı? Sendika ile yazılabilir terimler nasıl tanımlanabilir?

lambda-calculus type-theory logic

— Gilles 'SO- stop being evil'
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Interesting question. Could you say that interfaces from OOP correspond to this?

— Raphael

Maybe you could be interested in this cs.cmu.edu/~rwh/courses/refinements/papers/Barbaneraetal95/…

— Fabio F.

Yanıtlar:

İlk sistemde alt tip olarak adlandırdığınız şey şu iki kuraldır:

\frac{Γ, x : T_{1} ⊢ M : S}{Γ, x : T_{1} \land T_{2} ⊢ M : S} (\land E_{1}) \frac{Γ, x : T_{2} ⊢ M : S}{Γ, x : T_{1} \land T_{2} ⊢ M : S} (\land E_{2})

$\dfrac{Γ, x:T_1 \vdash M:S}{Γ, x:T_1 ∧ T_2 \vdash M:S}(∧E_1)\quad\dfrac{Γ, x:T_2 \vdash M:S}{Γ, x:T_1 ∧ T_2 \vdash M:S}(∧E_2)$

They correspond to elimination rules for $∧$ ; without them the connective $∧$ is more or less useless.

In the second system (with connectives $∨$ and $→$ , to which we could also add a $⊥$ ), the above subtyping rules are irrelevant, and I think the accompanying rules you had in mind are the following:

\frac{Γ, x : T_{1} ⊢ M : S Γ, x : T_{2} ⊢ M : S}{Γ, x : T_{1} \lor T_{2} ⊢ M : S} (\lor E) \frac{}{Γ, x : ⊥ ⊢ M : S} (⊥ E)

$\dfrac{Γ, x: T_1 \vdash M:S\quad Γ, x:T_2 \vdash M:S}{Γ, x:T_1 ∨ T_2 \vdash M:S}(∨E)\quad\dfrac{}{Γ, x: {⊥} \vdash M:S}({⊥}E)$

For what it's worth, this system allows to type $(λx. I)Ω:A→A$ (using the ${⊥}E$ rule), which cannot be typed with just simple types, which has a normal form, but is not strongly normalizing.

Random thoughts: (maybe this is worth asking on TCS)

This leads me to conjecture that the related properties are something like:

a λ-term $M$ admits a type not containing $⊥$ iff $MN$ has a normal form for all $N$ which has a normal form. ( $δ$ fails both tests, but the above λ-term pass them)
a λ-term $M$ can be typed without using the ${⊥}E$ rule iff $MN$ is strongly normalizing for all strongly normalizing $N$ .

Exercise: prove me wrong.

Also it seems to be a degenerated case, maybe we should consider adding this guy into the picture. As far as I remember, it would allow to obtain $A ∨ (A → {⊥})$ ?

— Stéphane Gimenez
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Good point about the subtyping rules, they show that union types aren't nearly as natural as intersections (which get typed orthogonally to arrows). About the second part I need to think some more.

— Gilles 'SO- stop being evil'

I think

M = (λ x . x x) (λ y . y)

$M=(\lambda x.xx)(\lambda y.y)$ answers the exercise, if you are talking about union types.

— jmad

About call/cc: it needs more than just lambda-terms (like lambda-mu-terms or another framework) but type systems are more complex, logic systems, in which union types may be irrelevant.

— jmad

@jmad: Indeed, intersection types are needed to type this term :-( Maybe considering unions and intersections together would be interesting?

— Stéphane Gimenez

I would be interested in a λ-term one can type with union types (rs. with intersection types) but not with simple types (rs. with intersection types).

— jmad

I just want to explain why intersection types are well-suited to characterize classes of normalization (strong, head or weak), whereas other type systems can not. (simply-typed or system F).

The key difference is that you have to say: "if I can type $M_2$ and $M_1→M_2$ then I can type $M_1$ ". This is often not true in non-intersection types because a term can be duplicated:

(λ x . M x x) N \to M N N

$(\lambda x.Mxx)N → MNN$

and then typing $MNN$ means that you can type both occurrences of $N$ but not with the same type, for example

M : T_{1} \to T_{2} \to T_{3} N : T_{1} N : T_{2}

$M:T_1→T_2→T_3 \qquad N:T_1 \qquad N:T_2$ With intersection types you can transform this into:

M : T_{1} \land T_{2} \to T_{1} \land T_{2} \to T_{3} N : T_{1} \land T_{2}

$M:T_1\wedge T_2→T_1\wedge T_2→T_3 \qquad N:T_1\wedge T_2$ and then the crucial step is now really easy:

(λ x . M x x) : T_{1} \land T_{2} \to T_{3} N : T_{1} \land T_{2}

$(\lambda x.Mxx):T_1\wedge T_2→T_3 \qquad N:T_1\wedge T_2$ so

(λ x . M x x) N

$(\lambda x.Mxx)N$ can by typed with intersection types.

Now about union types: suppose you can type $(\lambda x.xx)(\lambda y.y)$ with some union type, then you can also type $\lambda x.xx$ and then get for some types $S, T_1, \dots$

x : T_{1} \lor T_{2} \lor \dots \lor T_{n} ⊢ x x : S

$x : T_1\vee T_2 \vee \dots \vee T_n ⊢xx:S$ But you still have to prove that for every

i

$i$ ,

x : T_{i} ⊢ x x : S

$x:T_i⊢xx:S$ which seems impossible even is

S

$S$ is an union type.

This is why I don't think there is an easy characterization about normalization for union types.

— jmad
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