Call-by-value non-determinism in a linear logic type discipline

Call-by-value non-determinism
in a linear logic type discipline

Alejandro Díaz-Caro Giulio Manzonetto
Université Paris-Ouest & INRIA LIPN, Université Paris 13

Michele Pagani
LIPN, Université Paris 13

Symposium on Logical Foundations of Computer Science
San Diego, California, U.S.A., January 6–8, 2013

Supported by the DIGITEO project 2011-070D “ALAL”

Intersection types discipline [Coppo-Dezani’78]

M :α∩β
M enjoys both properties α and β

With this idea in mind intersection is idempotent α ∩ α = α.

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M :α∩β

Used to capture various notions of termination:
Head, Weak and Strong normalisation [Coppo-Dezani’78, Sallé’80]

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M :α∩β


Resource-aware intersection types [De Carvalho’07]
Let us change point of view:
M :α∩β
M will be called once as data of type α and once as data of type β
Hence α ∩ α = α =⇒ Multisets
Used to capture quantitative properties of programs, e.g.:
CBN λ-calculus: number of linear head-reduction steps [De Carvalho’07]
CBV λ-calculus: number of weak head-reduction steps [Ehrhard’12]

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M :α∩β


Resource-aware intersection types [De Carvalho’07]
Let us change point of view:
M :α∩β
M will be called once as data of type α and once as data of type β
Hence α ∩ α = α =⇒ Multisets
Used to capture quantitative properties of programs, e.g.:
CBN λ-calculus: number of linear head-reduction steps [De Carvalho’07]
CBV λ-calculus: number of weak head-reduction steps [Ehrhard’12]

Our goal: extend Ehrhard’s system with non-determinism
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May/Must-convergent non-determinism

Consider the CBV λ-calculus extended with. . .

Non-deterministic choice
M +N The machine choses either M or N

Parallel composition
M N The machine interleaves reductions in M and in N

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May/Must-convergent non-determinism

Consider the CBV λ-calculus extended with. . .

Non-deterministic choice
M +N The machine choses either M or N

The non-deterministic choice M + N is may-convergent:
it converges if either M or N converges

Parallel composition
M N The machine interleaves reductions in M and in N

The parallel composition M N is must-convergent:
it converges if both M and N do

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Λ+ : Its syntax and operational semantics

Grammar of Λ+ terms
Terms: M, N, P, Q ::= V | MN | M + N | M N
Values: V ::= x | λx.M

Reduction semantics
βv -reduction +-reductions -reductions
M +N →M (M N)P → MP NP
(λx.M)V → M[V /x]
M +N →N V (M N) → VM VN

+ Contextual rules selecting the head redex. . .

The reduction is lazy (it does not reduce under λ-abstractions)

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Λ+ : Its syntax and operational semantics

Grammar of Λ+ terms
Terms: M, N, P, Q ::= V | MN | M + N | M N
Values: V ::= x | λx.M

Reduction semantics
βv -reduction +-reductions -reductions
M +N →M (M N)P → MP NP
(λx.M)V → M[V /x]
M +N →N V (M N) → VM VN

+ Contextual rules selecting the head redex. . .

The reduction is lazy (it does not reduce under λ-abstractions)

Convergence
M converges ⇔ M →∗ V1 ··· Vn

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Examples and remarks

Application is bilinear
op
(M + M )(N + N ) ≡ MN + MN + M N + M N

. . . but λ-abstraction is not
op
λx.(M + N) ≡ λx.M + λx.N

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Examples and remarks

Application is bilinear
op
(M + M )(N + N ) ≡ MN + MN + M N + M N

. . . but λ-abstraction is not
op
λx.(M + N) ≡ λx.M + λx.N

Example of parallel composition and non-deterministic choice

(λx.(x x))(V + V ) converges to either V V or V V

(λx.(x + x))(V V ) converges to V V only

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Linear logic based type system
Translation: Intuitionistic Logic → Polarized fragment of LL

ιv = ι, (α → β)v = αc β , αc = !αv , α =?αc

Based on [Ehrhard’12], based on second Girard’s translation.
Intuitions from the relational semantics of LL
The type for computations (·)c is a multiset [α1 , . . . , αn ] of value
v v
v
types (representing n calls to a single value of type αi ),
The type of parallel compositions (·) is another multiset
c c
[α1 , . . . , αn ] of types of each term in the composition,
The type for values (·)v are either basic types or functional types,
A functional type in this system is a pair (αc , [α1 , . . . , αn ]).
c c

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Linear logic based type system
Translation: Intuitionistic Logic → Polarized fragment of LL

ιv = ι, (α → β)v = αc β , αc = !αv , α =?αc

Based on [Ehrhard’12], based on second Girard’s translation.
Intuitions from the relational semantics of LL
The type for computations (·)c is a multiset [α1 , . . . , αn ] of value
v v
v
types (representing n calls to a single value of type αi ),
The type of parallel compositions (·) is another multiset
c c
[α1 , . . . , αn ] of types of each term in the composition,
The type for values (·)v are either basic types or functional types,
A functional type in this system is a pair (αc , [α1 , . . . , αn ]).
c c

Notation
First multiset layer −→ ⊗
Second multiset layer −→ `
Functional type (αc , [α1 , . . . , αn ])
c c
−→ αc α1 ` · · · ` αn
c c

Empty computational multiset −→ 1
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Linear logic based type system (cont.)

Grammar of Types:

parallel-types: α, β ::= α ` β | τ
computational-types: τ, ρ ::= 1 | τ ⊗ ρ | τ α

⊗ tensor product
associative and commutative
` par
1 neutral element of ⊗

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Grammar of Types:

parallel-types: α, β ::= α ` β | τ
computational-types: τ, ρ ::= 1 | τ ⊗ ρ | τ α

⊗ tensor product
associative and commutative
` par
1 neutral element of ⊗

Type environments:
Γ = x1 : τ1 , . . . , xn : τn represents the map

τi if y = xi ,
Γ(y ) =
1 otherwise.

Tensor is extended to environments pointwise (Γ ⊗ ∆)(x) = Γ(x) ⊗ ∆(x).

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Type inference rules
∆ M:α ∆ N:α + is may-convergent, so it
+ +r is enough that one term is
∆ M +N :α ∆ M +N :α typable

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∆ M : α1 Γ N : α2 is must-convergent, so both
I
∆⊗Γ M N : α1 ` α2 components must be typable

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I

¸
k ni ¸
ni
∆ M: (τij αij ) Γi N: τij 1≤i ≤k
i=1 j=1 j=1 k ≥1
E
k ¸¸
k ni ni ≥ 1
∆⊗ Γi MN : αij
i=1 i=1 j=1
It reﬂects the distribution of the parallel operator over the application

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I

¸
k ni ¸
ni
i=1 j=1 j=1 k ≥1
E
k ¸¸
k ni ni ≥ 1
i=1 i=1 j=1
It reﬂects the distribution of the parallel operator over the application
∆i , x : τ i M : αi 1≤i ≤n
n n I n≥0
ax
x :τ x :τ ∆i λx.M : (τi αi )
i=1 i=1
The axiom and the intersection type for values respectively
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Examples

∆ = x : (τ1 α1 ) ⊗ (τ2 α2 ) Γ = y : τ1 , y : τ2

∆ x : (τ1 α1 ) ⊗ (τ2 α2 ) Γ y y : τ1 ` τ2
E
∆⊗Γ x(y y ) : α1 ` α2

x(y y ) → xy xy

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Examples

∆ = x : (τ1 α1 ) ⊗ (τ2 α2 ) Γ = y : τ1 , y : τ2

∆ x : (τ1 α1 ) ⊗ (τ2 α2 ) Γ y y : τ1 ` τ2
E
∆⊗Γ x(y y ) : α1 ` α2

x(y y ) → xy xy

∆ = x : (τ1 α3 ) ⊗ (τ2 α4 )

∆⊗∆ x x : ((τ1 α1 ) ⊗ (τ2 α2 )) ` ((τ1 α3 ) ⊗ (τ2 α4 ))
Γ y y : τ1 ` τ2 Γ y y : τ1 ` τ2
E
∆⊗∆ ⊗Γ⊗Γ (x x )(y y ) : α1 ` α2 ` α3 ` α4

(x x )(y y ) →∗ xy xy xy xy

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Measuring derivation trees
π= ax |π| = 0
S
π1 · · · πn n
π= I |π| = i=1 |πi |
S
π1 π2
π= I |π| = |π1 | + |π2 |
S

π0 π1 . . . πk k k
π= E ni ≥ 1 |π| = i=0 |πi | + ( i=1 2ni ) − 1
S
π π
π= + or π = +r |π| = |π | + 1
S S
βv and redexes are typed by E
Only E, + and +r type redexes
+ redexes by + and +r
Each + and +r counts for 1 because a +-red. does not create new rules
in the derivation typing the contractum
E counts the number of “active” connectives in the principal premise
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Measuring derivation trees (cont.)

¸
k ni ¸
ni
i=1 j=1 j=1
E
k ¸¸
k ni
i=1 i=1 j=1

k k k
ni + (ni − 1) + (k − 1) = ( 2ni ) − 1
i=1 i=1 i=1
`’s
’s ⊗’s

The -reduction creates two new E rules in the derivation typing the
contractum

The measure decreases because the sum of their weights is less than the
weight of the eliminated rule

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Properties of the type system
Our system enjoys a quantitative version of standard properties.

Subject reduction
Let π = ∆ M:α
If M → N without +-red. then ∃π = ∆ N:α
If M → N1 and M → N2 with +-red.
then ∃π = ∆ N1 : α or π = ∆ N2 : α
In both cases, |π | = |π| − 1

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Subject reduction
Let π = ∆ M:α
then ∃π = ∆ N1 : α or π = ∆ N2 : α

Subject expansion
If M→N and π=∆ N:α
then ∃π = ∆ M : α s.t. |π | = |π| + 1

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Subject reduction
Let π = ∆ M:α
then ∃π = ∆ N1 : α or π = ∆ N2 : α

Subject expansion
If M→N and π=∆ N:α
then ∃π = ∆ M : α s.t. |π | = |π| + 1

Characterization of convergence
Let M closed. M typable ⇔ M converges
Can we say anything more quantitative?
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Combinatorial proof of normalization

Measure
Let M be a closed term. If π is a derivation of

M : α,

then |π| gives a bound on the number of steps M converges.

More precisely. . .

Exact bound
Let M be a closed term. If π is a derivation of

M : 1 ` · · · ` 1,

then M reaches its normal form in exactly |π| steps

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Properties of the underlying relational model

Let M, N and P be closed terms.

Deﬁnitions
A closed term M is interpreted by M = {α | M : α}
M N iﬀ ∀P M P converges ⇒ N P converges

As a corollary of the Convergence Theorem we get:

Adequacy
M ⊆ N implies M N

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Lack of full abstraction
M N does not imply M ⊆ N

CBV λ-calculus admits the creation of an ogre

Y = ∆ ∆ where ∆ = λxy .xx.

Remark: The ogre Y is a top of :

Y V V → (λy .Y )V V → Y V → · · · → Y .
n
All types of Y have shape α = i=0 (1 αi ).

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M N does not imply M ⊆ N

CBV λ-calculus admits the creation of an ogre

Y = ∆ ∆ where ∆ = λxy .xx.

Remark: The ogre Y is a top of :

Y V V → (λy .Y )V V → Y V → · · · → Y .
n
All types of Y have shape α = i=0 (1 αi ).

Counterexample (independent from + and )
Let I = λx.x, then
I Y , while I ⊆ Y
since (1 1) (1 1) ∈ I − Y

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Summarising

We introduced a call-by-value non-deterministic λ-calculus with a
type system ensuring convergence

The type system gives a bound of the length of the lazy cbv
reduction sequences (exact when the typing is minimal)

We show an adequate (but not fully abstract) model capturing the
type system

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Call-by-value non-determinism in a linear logic type discipline

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