Approximations and η-rule

Lambda-Calculus (III-6)

[email protected] Tsinghua University, September 17, 2010

Plan

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Bohm trees -- reminders

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Morris extensional equivalence

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Bohm trees and !-rule

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Observational equivalences

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Relation with Scott’s models

Bohm tree semantics - reminders

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Theorem [continuity] For allb∈N such thatb�C[M], thenb�C[a]

for somea∈N such thata�M.

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Theorem [monotony] M�NimpliesC[M]�C[N]

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Theorem [!-theory] M≡NimpliesC[M]≡C[N]

Proofs: easy consequences of previous proofs.

Approximations and

η-rule

Bohm tree and !-rule

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We need take !-rule into account ! How to mix !-rule and Bohm tree construction ?

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Functional extensionality has not been considered since we can have:

MP≡NP for allP, butM�≡N.

(TakeM=xandN=λy.xy)

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We take for granted that _ηand β,ηare confluent.

Moreover η strongly normalizes.

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The prefix ordering between approximants must be extended. For instance:

xΩ=ηλy.xΩy≤λy.xyy λx.yΩ≤λx.yx=ηy

Finite approximants

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Lemma : We have following commutation properties:

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We consider the set of !-normal forms of finite approximants with following relation:

N^e

a≤^eb iff a(≤ ∪=η)^∗b

η≤ ⊂ ≤ η

≤ η ⊂ η≤

η η ⊂ η η

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Corollary: a≤^eb iff a ηa^�≤b^� η b

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^Examples:

λy.xΩ ≤λy.xy ηx xΩ ηλy.xΩy ≤ λy.xyy

Extensional Bohm trees

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Definition : Let ω^e(M) be theη-normal form ofω(M).

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Definition : The extensional Bohm tree BT^e(M) ofMis defined by:

BT^e(M) ={a∈N^e | a ≤^e ω^e(N), M N}

M�^eN iff BT^e(M)⊂BT^e(N) M≡^eN iff BT^e(M) = BT^e(N)

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Definition : Extensional Bohm tree semantics

Extensional Bohm trees

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^{Theorem :} �^eis a monotonic semantics and≡^eforms aλ-theory.

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Theorem [Hyland, 1975]:

M�^eN iff for allC[ ],C[M] n.f. impliesC[N] n.f.

J.Morris extensional equivalence

MIT-1968

Closed Bohm trees

Finite Bohm trees revisited

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We just added !-rule to the standard Bohm tree construction with completion by ideals.

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However, we forgot slight difficulty: now, thanks to !-rule, finite Bohm tree now dominates an infinite number of other finite Bohm trees. See:

a0=Ω a1=λx1.xΩ a2=λx1.x(λx2.x1Ω) a3=λx1.x(λx2.x1(λx3.x2Ω))

I0={Ω}

I1={a∈N^e | a≤^ea1} I2={a∈N^e | a≤^ea2} I3={a∈N^e | a≤^ea3} I_∞={a∈N^e | a≤^ex}

�_∞

n=0In

a_∞=x

new point

added by ideal completion

Finite Bohm trees revisited

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There are two ways of completing finite Bohm trees.

standard completion by ideals (what we did)

completion with closed ideals (does not add new limit point) 1-

2-

BT^e_c(M) = cl(BT^e_c(M))

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We define the closure of directed sets as being the set with its already existing limit in N^e

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We therefore have:

I≡^eclJ where J=Y(λfxy.x(fy))

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and normal forms are no longer isolated points.

Finite Bohm trees revisited

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The equality between I and J is not so unnatural since one may emulate infinite !- expansion with the "-rule:

I=λx.x η λxx1.xx1

η λxx1.x(λx2.x1x2)

η λxx1.x(λx2.x1(λx3.x2x3))

η ...

J=Y(λfxx1.x(fx1)) =β λxx1.x(Jx1)

=β λxx1.x(λx2.x1(Jx2))

=β λxx1.x(λx2.x1(λx3.x2(Jx3)))

=β ...

.

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same phenomenon as for these two versions of identity on natural numbers:

I(n) ⇐ n

J(n) ⇐ ifn= 0then1else1 +J(n−1) .

Bohm trees and Scott’s models

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We have following correspondances:

1- 2-

M�^ecN iff M�D_∞N M�N iff M�T^ωN

(Scott’s model) (Plotkin’s model)

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One can also show that:

3- M�^e+N iff M�PωN (Scott’s model)

where�^e+ is Bohm tree construction from η≤ordering onN^e.

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finally, one may order Bohm trees with symmetrical:

4- M�^e−N

where�^e−is Bohm tree construction from≤ η ordering onN^e.

Observational equivalences

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To conclude we have the following results:

1- 2-

M�^ecN iff for all contextsC[ ], C[M] hnf implies C[N] hnf M�^eN iff for all contextsC[ ], C[M] nf implies C[N] nf

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Making observational equivalences with other Bohm tree semantics is more difficult since one has to fight with !-equality. Take for instanceλx.xx andλx.x(λy.xy)

Homeworks

Exercices

1-Show that ifM has no hnf, thenMis totally undefined.

2- ShowΩM≡Ωandλx.Ω≡Ω. Show thatM ωN, thenM≡N.

4- ShowM�≡λx.Mxwhenx�∈var(M). What ifM≡λx.M1?

5- LetY0=Y,Yn+1=Yn(λxy.y(xy)). Show thatY ≡Ynfor alln. However allYn

are pairwise non interconvertible.

6 If M ≤ P andN ≤ P (M andN are prefix compatible), then BT(M�N) = BT(M)∩BT(N). (Thus BT is stable in Berry’s sense, 1978). What if not com- patible ?

3- FindM andN such thatMP ≡NP for allP, butM�≡N. (Meaning that≡is not extensional)

Exercices

7- [Barendregt 1971]

A closed expressionM (i.e. var(M) =∅) is solvable iff:

∀P,∃N1,N2, ...Nnsuch thatMN1N2· · ·Nn=βP (in short:

∀P,∃N,� MN� =βP )

Show that for every closed termM, the following are equivalent:

1. M has a hnf

2. ∃N,� MN� has a normal form 3. ∃N,� MN� =βI

4. M is solvable 8- [Barendregt 1974]

Show that, in theλI-calculus, a termM is solvable iffit has a normal form.

Exercices

9- LetRbe a preorder onN (reflexive + transitive) compatible with its structure:

a1Rb1, ...anRbn implies xa1a2· · ·an

aRb implies λx.aRλx.b

LetM�RN iff ∀a∈BT(M), ∃b∈BT(N), aRb Show that whenM is a closed term, one has:

∀P�,M�P�RNP� iff ∀C[ ], C[M]�RC[N]

10- (cont’d 1) LetMRNbe “ifM has a normal form, thenNhas a normal form”

Give examples ofM andNsuch thatM�RNbutM��N.

(cont’d 2) LetMRNbe “ifMhas a hnf, thenN has a hnf”

Give examples ofMandN such thatM�RNbutM��N.

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12- (cont’d 2) LetMRN be “ifMhas a hnf, thenNhas a similar hnf”

Give examples of M andN such that M �R N butM �� N. (Hint: consider M=λx.xxandN=λx.x(λy.xy)) [Compare withHyland 1975])