A bridge between semirings

A bridge between semirings

Flavien BREUVART

PPS, Paris Denis Diderot & LIPN, Paris Nord

march 11th-26th 2014

Introduction

Semirings S , R for resource management

Many papers are using semirings for modelizing quantitative resources.

Bounded logics B

LL [Ghica&Al13,Brunel&Al13]

Generalisation of previous works that decompose LL exponential along semirings.

No generic concrete model.

Model Rel

of LL [CarraroEhrhardSalibra09]

The relational model endowed with a non free exponential using semiring-indexed multisets.

General theorem

Every bounded logic B

LL has as model REL

.

Table of content

Logic

•

Ordered semirings

•

Curry Howard

•

The logic B

LL

•

Examples

Categorical semantic

•

Ghica&Smith vs Brunel&Al

•

Refining models of LL

•

Brunel&Al’s

Concrete semantic

•

Rel

•

Our Rel

•

Carraro&All’s Rel

•

Universality

Table of content

Logic

•

Ordered semirings

•

Curry Howard

•

The logic B

LL

•

Examples

Categorical semantic

•

Ghica&Smith vs Brunel&Al

•

Refining models of LL

•

Brunel&Al’s

Concrete semantic

•

Rel

•

Our Rel

•

Carraro&All’s Rel

•

Universality

Definition of ordered semirings

Ordered semiring S

An ordered semiring is a structure S = (|S|, +, 0, ∗, 1, ≤) where (|S|, ≤) is a poset, 0, 1 ∈ |S| and where +, · : |S|

→ S such that:

•

+ and ∗ are associative,

•

+ is commutative and distribute over ∗,

•

0 is neutral for + and 1 is neutral for ∗,

•

if J ≤ J

and K ≤ K

then J ∗ J

≤ K ∗ K

and J+J

≤ K +K

.

Examples of ordered semirings

•

1 : the trivial Semiring with just one element.

•

P oly : the polynomials with their usual sum, product and order.

•

T rop = ( N , max, −∞, +, 0, ≤) and A rt = ( ¯ N , min, ∞, +, 0, ≤),

•

A ut

: for any commutative monoid ( M , +, 0), the automorphisms

(Mon( M , M ), +, cst

, ◦, id

, id

)

Grammar of B S LL

Formulas

(formulas) A, B , C := α | A ⊗ B | A ( B | !

A ,∀K ∈ S This is only IMB

LL and can be generalised.

Sequents Γ ` A

IMLL plus:

Γ ` B (!w) Γ, !

A ` B

Γ, A ` B Γ, !

A ` B (!d)

Γ, !

A, !

A ` B Γ, !

A ` B (!c)

!

A

, · · · , !

A

` B

!

A

, · · ·!

A

`!

B (!s)

Γ, !

A ` B K ≥ J

(Sw )

Γ, !

A ` B

Curry-Howard: Λ _S

Grammar of Λ

Given a semiring S we define typed lambda calculus:

(terms) Λ M, N ::= x | λx.M | M N

(types) T

θ := α | K ·θ ( θ ,∀K ∈ S

Some examples (with S = Hom( N ))

λxy .y: 0·θ ( 1·θ

( θ

λxy.x(xy ): 4·(3·θ ( θ) ( 9·θ ( θ λxy.xyy: 1·(2·θ ( 3·θ ( θ

) ( 5·θ ( θ

λxyz.x (yz ): 1·(f ·θ ( θ) ( f ·(g ·θ ( θ) ( f ◦g ·θ ( θ

λxyz .y (xz ): g ·(f ·θ ( θ) ( 1·(g ·θ ( θ) ( g ◦f ·θ ( θ

Typing judgements

Ghica’s linear type system

x:1·θ ` x:θ Id Γ ` M :θ Γ, x :0·θ

` M:θ Weak

Γ, x:K ·θ ` M:θ

Γ ` λx .M:K ·θ ( θ

Abs Γ ` M:K ·θ ( θ

Γ

` N:θ |Γ| = |Γ

|

Γ+ K ∗Γ

` M N : θ

App

where |Γ| = |Γ

| means the equality of contexts except for the multiplicity of the types:

|(x

:K

·θ

)

| = |(y

:L

·θ

)

| ⇔ (k = k

∧ x

= y

∧ θ

= θ

) and where Γ+ K ·Γ

is the context obtain by applying addition and multiplication to the semiring:

(x

:K

·θ

)

+(x

:L

·θ

)

:= (x

:(K +L)·θ

)

K ∗(x

:L·θ

)

:= (x

:K ∗L·θ

)

Examples and interest

Counting resources

N : if M : n·A ( U then the KAM on M N will evaluate at most n times the N.

P oly : if M is typable then M N has a polynomial head reduction on the size on N.

A rt: with streams of type Str and the operations tl : 1·Str ( Str and hd : 1·Str ( U , if M : n·Str ( A, its argument will be explored at depth at most n.

A more complicated example: Ghica&Smith’s

[ ]+[ , ] = [ , , ]

[ ] ∗ [ , ] = [ , ]

This computes the sequentially of an execution.

Table of content

Logic

•

Ordered semirings

•

Curry Howard

•

The logic B

LL

•

Examples

Categorical semantic

•

Ghica&Smith vs Brunel&Al

•

Refining models of LL

•

Brunel&Al’s

Concrete semantic

•

Rel

•

Our Rel

•

Carraro&All’s Rel

•

Universality

Categorical semantic

There is two choices of categorical semantic in the literature:

Brunel&al’s categorical semantic

– Difficult to understand – Over 20 commutative

diagrams

+ Accept any model of LL as a degenerative model + The syntax itself is a model

GhicaSmith’s categorical semantic

+ Quite simple to understand and realise

– The only accepted model of LL is the singleton (up-to iso)

– The syntax is not a model

Restricting models of LL into models of B

LL

LL model + π

:!A →!

A + ι,

:!

A →!A + 5 diagrams = B

LL

model

Brunel&al’s categorical semantic

Ordered semirings as categories

Any ordered semiring can be seen as bimonoidal category whose objects are the elements of the semirings and whose homesets S(J, K ) are either singleton (when J ≤ K ) or empty.

Bounded exponential situation

A bounded exponential situation is the given of:

•

a symmetric monoidal category (model of MLL) (A, ⊗, ( , 1),

•

a bifunctor: • : S × A → A

•

6 naturals transformations:

δ

: (J∗K )•A = ⇒ J •K •A

: 1•A = ⇒ A c

: (J+K )•A = ⇒ J•A ⊗ K •A w

: 0•A = ⇒ 1 m

: J•A ⊗ J•B = ⇒ J •(A ⊗ B) n

: 1 = ⇒ J•1

•

and more than 20 commutative diagrams.

An example of wanted diagram

(J+K )•A ⊗ (J +K )•B

((J •A) ⊗ (K •A)) ⊗ ((J•B) ⊗ (K •B )) (J+K )•(A ⊗ B)

((J•A) ⊗ (J •B)) ⊗ (K •A ⊗ K •B) J•(A ⊗ B) ⊗ K •(A ⊗ B)

c

⊗ c

m

α

c

m

⊗ m

Obtaining bounded exponential as retractions of usual exponential

Bounded exponential situation by retraction-embedding

A bounded exponential situation by retraction-embedding is the given of:

•

a model of MELL (A, ⊗, !, 1, ( , δ, , c, w , m, n),

•

a bifunctor: • : S × A → A

•

2 naturals transformations:

ι : J •A = ⇒!A π :!A = ⇒ J•A

•

and 5 commutative diagrams.

The retraction-embedding diagram

The equation ι

; π

= id

is not a priori needed, but is true in every

interesting considered model.

An examples and interest of wanted diagrams

!A !A⊗!A !A⊗!B !(A ⊗ B)

(J +K )•A J • A ⊗ K •A J•A ⊗ J•B J •(A ⊗ B)

!A !A⊗!A !A⊗!B !(A ⊗ B)

c

π π ⊗ π

ι π

c

m

π π ⊗ π

ι π

m

An examples and interest of wanted diagrams

!A !A⊗!A !A⊗!B !(A ⊗ B)

(J +K )•A J • A ⊗ K •A J•A ⊗ J•B J •(A ⊗ B)

def .c

def .m

!A !A⊗!A !A⊗!B !(A ⊗ B)

c

π π ⊗ π

c

π π ⊗ π

c

ι π

c

m

π π ⊗ π

m

π π ⊗ π

m

ι π

m

An examples and interest of wanted diagrams

!A !A⊗!A !A⊗!B !(A ⊗ B)

C M

(J +K )•A J • A ⊗ K •A J•A ⊗ J•B J •(A ⊗ B)

!A !A⊗!A !A⊗!B !(A ⊗ B)

c

π π ⊗ π

c

ι π

c

ι π

c

m

π π ⊗ π

m

ι π

m

ι π

m

Theorem

Theorem:

LL model + π

:!A → J•A + ι,

: J•A →!A + 5 diagrams = B

LL model

δ

= ι

; δ

; !π

; π

= ι

;

c

= ι

; c

; (π

⊗ π

) w

= ι

; w

m

= (ι

⊗ ι

); m

; π

n

= n; π

An example of wanted diagram

(J+K )•A ⊗ (J +K )•B

((J •A) ⊗ (K •A)) ⊗ ((J•B) ⊗ (K •B )) (J+K )•(A ⊗ B)

LL

((J•A) ⊗ (J •B)) ⊗ (K •A ⊗ K •B) J•(A ⊗ B) ⊗ K •(A ⊗ B) def c

⊗ def c

def m

nat α

C

M ⊗ M

c

⊗ c

m

α

c

m

⊗ m

ι ⊗ ι

c ⊗ c

(π ⊗ π) ⊗ (π ⊗ π) m π

α

(π ⊗ π) ⊗ (π ⊗ π)

c

π ⊗ π

m ⊗ m

Table of content

Logic

•

Ordered semirings

•

Curry Howard

•

The logic B

LL

•

Examples

Categorical semantic

•

Ghica&Smith vs Brunel&Al

•

Refining models of LL

•

Brunel&Al’s

Concrete semantic

•

Rel

•

Our Rel

•

Carraro&All’s Rel

•

Universality

The model Rel of LL [folklore]

The category Rel : a model of MLL

Objects: Sets Morphisms: Relations Multiplicatifs: A ⊗ B = A ( B = A × B and 1 = {∗}

Rel

: a model of MELL

Exponential: !A = M

(A)

δ

= {(u, V ) | V ∈!!A, u = ΣV }

= {([a], a) | a ∈ A}

c

= {(u, (v, w )) | u = v +w } w

= {([], ∗)} n = {(∗, [∗, ..., ∗])}

m

= {((u, v ), w ) | ∀a, u(a) = Σ

w (a, b), ∀b, v (b) = Σ

w (a, b)}

Hypothesis: multiplicity [Carraro&Al]

A semiring R has multiplicities if

(MS1) it is positive: m+n = 0 ⇒ m = n = 0 (MS2) it is discreet: m+n = 1 ⇒ m = 0 or n = 0 (MS3) it has additive splitting properties.

m

+n

= m

+n

⇒ ∃(p

)

, m

= p

+p

, n

= p

+p

(MS4) it has multiplicative k-ary splitting property:

n

+n

= pm ⇒ ∃k, p

, p

, m

, m

, m

, m

, p = X

j≤k

p

,

n

= X

j≤k

p

m

, ∀j ≤ k , m = m

+mj 2

Examples

•

N

•

N ¯

•

N × N

•

M

( N )

R morally behave like N

•

R contain N as sub semiring

•

The sum in R is the one of N

The model Rel ^R of LL [Carraro&Al]

Definition of R

hAi

Given a semiring R and a set A, the space of functions f : A → R of finite support is denoted R

hAi. It is a semimodule over R with:

•

A commutative sum: (f +g )(x) = f (x)+g (x )

•

An external product: (K ·f )(x ) = K ·f (x )

Rel

: a model of MELL

Exponential: !A = R

hAi

δ

= {(u, V ) | V ∈!!A, u = Σ

V (v)·v }

= {([1·a], a) | a ∈ A}

c

= {(u, (v , w ) | u = v+w ∈!A} w

= {([], ∗)} n = {(∗, [J·∗])|J ∈R}

m

= {((u, v ), w ) | ∀a, u(a) = Σ

w (a, b), ∀b, v (b) = Σ

w (a, b)}

It is a generalisation of Rel

Indeed M

(A) is an other notation for N hAi.

The model Rel ^{(R,˜ ∈,S)} of B S LL

Semiring refinement (R, ∈, ˜ S)

•

R is a semiring with multiplicities,

•

S is an ordered semiring,

•

∈ ˜ is a relation between those,

•

1

∈1 ˜

and 0

∈0 ˜

,

•

n ∈J ˜ and m ∈K ˜ imply n+m ∈ ˜ J+K and n∗m ∈ ˜ J∗K ,

•

J ≤ K iff J ⊆K ˜ .

Rel

: a model of B

LL

Bounded exponential: J•A = {u ∈ R

hAi | Σ

u(a)˜ ∈J}

projection and injection:: π

= ι

= {(u, u) | u ∈ J•A}

Since !A = RhAi, we have that i•A ⊆!A

Some examples

R S ∈ ˜ 0•A 1•A

( B ool, id ) {(0, 0)} ∪ {(n + 1, 1)} {[]} M

(A)−{[]}

( B ool, 0≤1) {(0, 0)} ∪ {(n, 1)} {[]} M

(A) ( B ool, 1≤0) {(n, 0), (n+1, 1)} M

(A) M

(A)−{[]}

( B ool, id ) R × B ool − {(0, 1), (1, 0)} M

(A)−{[a]} M

(A)−{[]}

N ( Z /2 Z , id ) {(2n, 0), (2n+1, 1)} even size odd size ( N , id ) {(n, n) | n ∈ N } n•A = {[a

, ..., a

]}

( N , ≤

) {(m, n) | m ≤ n} n•A = {[a

, ..., a

] | m ≤ n}

( ¯ N , ≤

) {(m, n) | m ≤ n} n•A : idem ω•A = M

(A)

(P

( N ), ⊆) {(n, U ) | n ∈ U } U•A = {[a

, ..., a

] | n ∈ U }

The semiring M _f ( M )

M

( M )

Given a monoid M , M

( M ) is a semiring with:

•

A commutative sum:

(f +g )(x ) = f (x)+g (x)

•

A product (Dirichlet convolution):

(f ·g )(x) = X

x=yy,z·z

f (y)·g (z )

Example of such semirings

•

N = M

( B ool)

•

Ghica’s M

(Aff

)

•

P oly = M

( N

)

M

( M ) has multiplicities M

(S

) refines S

[K

, ..., K

] ˜ ∈

J iff Σ

K

≤ J

The two level of semirings

R: semantic semiring

•

R represent the actual resources.

•

0 : the absence of resources.

•

1 : the unitary amount of resources.

•

Any bag of resources can be separated and reformed

S : syntactical semiring

•

S represent information on resources.

•

0: you may not have resources.

•

Sames goes for 1.

•

≤

is the accuracy of the information.

S as a quotient

With R = M

(S

) where you forgot

“paralleled history”.

S as non-determinism

S = (P

(R), ⊆) can encode non deterministic operators.

S as a limit

S = (P (R), ⊆) or

S = R can encode

fixpoints.

Extensions

Does S have to be a semiring?

Remark: only 0 ∈J ˜ is necessary to perform weakening in J and 1 ∈J ˜ to perform dereliction.

One can replace 0 and 1 by sets Z and U such that:

•

if J ∈ Z then J +K ≥ K

•

if J ∈ U then J ∗ K ≥ K

•

if J ∈ Z then J ∗ K ∈ Z

Alex’s intersection type

[[J

·A

· · · J

·A

]] = {σ

u

∈![[A]] | ∀i ≤ n, u

∈ [[J

·A

]]}

Where A is the simple type obtained by forgetting resource annotation in

any of the A

.

Theorem and conclusion

General theorem

Any bounded logic B

LL has as model REL

. but it may has other more accurate relational models Current works:

•

Identifying the categorical sens of the relation ˜ ∈.

•

Investigate the Locals of 1.

•

Identifying the sens of the natural transformations ι and π.

•

Creating non degenerated models via double gluing.

•

Find models when S is a bimonoidal category and not a semiring.

•

Generalising all this to resource-dependants logics.

Forgetful adjunction

If we denotes RpoSemirings the category of ordered semirings and relation (preserving the structure) and RpoMonoids the same category for monoids:

RpoSemirings > RpoMonoids ( )

M

( )

Indeed ( )

is the forgetful functor and M

( ) the free functor generating the free semimodule over the multiplicative monoid M .

We have that ˜ ∈

= der (M

(( )

))

: (M

(()S

) → S)