musings around the geometry of interaction, and coherencegoubault/geocal2006.pdf · musings around...

MotivationLinear Inverse Monoids

The Danos-Regnier CategoryCoherence Completions

Conclusion

Musings Around the Geometry of Interaction,and Coherence

Jean Goubault-Larrecq

Futurs

GeoCal’06 — Feb 22, 2006

Jean Goubault-Larrecq GoI and Coherence



Conclusion

Outline

MotivationLinear Inverse MonoidsThe Danos-Regnier Category

Inductive GroupoidsIntroducing DR(M)The Weakly Cantorian CaseMiserable Failures (Ignoring Great Successes)

Coherence CompletionsThe Modified Hu-Joyal ConstructionLifting Monoidal Structure, Creating the RestRelation to Syntax?Towards Fibrations?

Conclusion




Conclusion

DPDA Equivalence

◮ A famous hard problem, first described in 1966, solved byG. Senizergues (1997);

◮ Shown primitive recursive by C. Stirling (2002);◮ Basically: given two Deterministic PushDown Automata,

do they generate the same language?




Conclusion

DPDA Equivalence



do they generate the same language?◮ Note: Determinism is essential: NPDA equivalence is

undecidable.




Conclusion

DPDA Equivalence



do they generate the same language?◮ Note: Determinism is essential: NPDA equivalence is

undecidable.◮ Some constructions are constructions in coherence

spaces.◮ Can we make a bridge with linear logic? Can we extend

this?




Conclusion

DPDA and Coherence

◮ A pushdown automaton is a prefix-rewrite systemconsisting of rules:

pX a−→qα

p is current control state, q is new control state,X is top stack symbol, α is new sequence of stack symbols,

a is letter read or ǫ.




Conclusion

DPDA and Coherence

◮ A pushdown automaton is a prefix-rewrite systemconsisting of rules:

pX a−→qα

p is current control state, q is new control state,X is top stack symbol, α is new sequence of stack symbols,

a is letter read or ǫ.◮ Build a strict deterministic grammar (Harrison and Havel,

1973). From a coherence relation ⌢⌣ on letters, define

w ⌢⌣

∗ w ′ iff w = w ′,

or w = w0aw1, w ′ = w0bw ′1, a 6= b, a ⌢

⌣ b

w0

w1

w’1

a

b

This is Reddy’s (2002) free object space...Jean Goubault-Larrecq GoI and Coherence



Conclusion

DPDA and Coherence, cont’d

◮ If you build formal sums∑

i wi of coherent words (cliques),




Conclusion



i wi of coherent words (cliques),◮ and look at the action of each letter a:

a

(

∑

i

wi

)

=∑

j

w ′j

where w ′j ranges over all words such that wi

a−→w ′

j forsome i ,




Conclusion




a

(

∑

i

wi

)

=∑

j

w ′j


a−→w ′

j forsome i ,

◮ then∑

j w ′j is well-defined (a clique again),




Conclusion




a

(

∑

i

wi

)

=∑

j

w ′j


a−→w ′

j forsome i ,

◮ then∑

j w ′j is well-defined (a clique again),

◮ and a is linear:

wi⌢⌣

∗ wk ,wi

a−→w ′

j

wka−→w ′

ℓ

}

⇒ w ′j

⌢⌣

∗ w ′ℓ

and if w ′j = w ′

ℓ then wi = wk

◮ Essential to model determinism.Jean Goubault-Larrecq GoI and Coherence



Conclusion

Coherence of Paths, and Trees

Let Σ be a first-order signature. For each n-ary symbol f(n ≥ 1), define letters f/i .

Let f/i ⌢⌣ g/j iff f = g (whatever i , j).

◮ Every tree (first-order term) defines a coherence spaceover paths, with coherence ⌢

⌣∗.




Conclusion





⌣∗.

f

a c

f

a c

f

g f

a b b




Conclusion





⌣∗.

f

a c

f

a b b

c

f

g

f

a

f/1 g/1 f/1 a




Conclusion





⌣∗.

f

a c

f

a b

c

f/1 g/1 f/1 af

g

bf

a

f/1 g/3 b




Conclusion





⌣∗.

◮ Conversely, every clique of paths embeds into some tree(possibly infinite).




Conclusion




⌣∗.


f/2 g/1f

g




Conclusion




⌣∗.


f/2 g/1

f/2 g/2 f/2

f

g

f




Conclusion




⌣∗.


g

f/2 g/1

f/2 g/2 f/2

f/2 g/2 f/1 g/3

f

g

f




Conclusion




⌣∗.


g

f

f/2 g/1

f/2 g/2 f/2

f/2 g/2 f/1 g/3

f/2 g/3 f/1f

g

f Motto: Cliques =chunks of infinitetrees.




Conclusion

Paths in the λ-Calculus

◮ Can we describe λ-terms, or rather Bohm(-like) trees, bythe sets of (finite) paths through them?




Conclusion



◮ Yes, at least for linear λµY -terms (MLL+fixpoint),




Conclusion



◮ Yes, at least for linear λµY -terms (MLL+fixpoint),◮ Recipe: take the weights from the dynamic algebra, forget

about the underlying λ-term.




Conclusion



◮ Yes, at least for linear λµY -terms (MLL+fixpoint),◮ Recipe: take the weights from the dynamic algebra, forget

about the underlying λ-term.◮ This will give us a nice *-autonomous category DR(M), for

any linear inverse monoid M.




Conclusion

An example: λx , y , z · zxy

λ

λ

λ

@

@

x

y

z




Conclusion


λ

λ

λ

@

@

x

y

z

p




Conclusion


λ

λ

λ

@

@

x

y

z

pp




Conclusion


λ

λ

λ

@

@

x

y

z

ppp




Conclusion


λ

λ

λ

@

@

x

y

z

pppp*




Conclusion


λ

λ

λ

@

x

y

z

pppp*

@

p*




Conclusion


λ

λ

λ

@

p*q*

x

y

z

pppp*

@




Conclusion


λ

λ

λ

@

p*q*p*

x

y

z

pppp*

@




Conclusion


λ

λ

λ

@

p*q*p*p*

x

y

z

pppp*

@




Conclusion


λ

λ

λ

@

p*q*p*p*p*p*p*

x

y

z

pppp*

@

+ ppqpp




Conclusion


λ

λ

λ

@

p*q*p*p*p*p*p*q*p*x

y

z

pppp*

@

+ ppqpp+ ppqpq




Conclusion


λ

λ

λ

@

p*q*p*p*p*p*p*q*p*

pppp*+ ppqpp+ ppqpq

q*+ ppqqx

y

z

@




Conclusion


λ

λ

@

λ

p*q*p*p*p*p*p*q*p*

pppp*+ ppqpp+ ppqpq

q*+ ppqqq*p*q* p*p*+ pq

x

y

z

@




Conclusion


λ

λ

@

λ

p*q*p*p*p*p*p*q*p*

pppp*+ ppqpp+ ppqpq

q*+ ppqqq*p*q* p*p*

q*q*p*p*

x

y

z

@

+ q+ pq




Conclusion

Static Denotations vs. Dynamic Behaviors

◮ GoI is usually described as some kind of dynamic process(stress on computation);




Conclusion



◮ I don’t care about computation here.




Conclusion



◮ I don’t care about computation here.◮ My interest is into denotational (static) models,




Conclusion



◮ I don’t care about computation here.◮ My interest is into denotational (static) models,◮ and ultimately deciding equality of Bohm(-like) trees for a

large class of programs,




Conclusion



◮ I don’t care about computation here.◮ My interest is into denotational (static) models,◮ and ultimately deciding equality of Bohm(-like) trees for a

large class of programs,◮ including deterministic higher-order program schemes (still

an open problem).




Conclusion

Outline




Conclusion




Conclusion

Inverse Monoids

◮ In the example above, paths were words over p, p∗, q, q∗,plus some relations;




Conclusion

Inverse Monoids

◮ In the example above, paths were words over p, p∗, q, q∗,plus some relations;

◮ Think of p : n ∈ N 7→ 2n,01234567 .

.

.

01234567

p

.

.

.

p*

01234567 .

.

.

01234567

q

.

.

.

q*

q : n ∈ N 7→ 2n + 1;◮ p∗, q∗ their inverses;◮ They all live in PI(N)

(partial injections over N),the small model of GoI;




Conclusion

Inverse Monoids

◮ Think of p : n ∈ N 7→ 2n,01234567 .

.

.

01234567

p

.

.

.

p*

01234567 .

.

.

01234567

q

.

.

.

q*

q : n ∈ N 7→ 2n + 1;◮ p∗, q∗ their inverses;◮ They all live in PI(N)

(partial injections over N),the small model of GoI;

◮ In general, let M be aninverse monoid (e.g., PI(N)):

(u∗)∗ = u (uv)∗ = v∗u∗ uu∗u = u uu∗vv∗ = vv∗uu∗

◮ The idempotents are all of the form 〈u〉 = uu∗ (codomain).Domain is 〈u∗〉 = u∗u.4th equation: idempotents commute.(In PI(N), they are [identities on] sets of integers.)




Conclusion

The Natural Ordering ≤

◮ Every inverse monoid can be equipped with an naturalordering ≤: u ≤ v iff any of the following equivalentconditions hold:

(i) vu∗ = uu∗ (iii) 〈u〉 v = u (v) u∗v = u∗u

(ii) uv∗ = uu∗ (iv) v 〈u∗〉 = u (vi) v∗u = u∗u

(This is standard, and well-known.)◮ In PI(N), ≤ is graph inclusion:

01234567 .

.

.

01234567

01234567 .

.

.

01234567.

.

. ...

<




Conclusion

The Natural Coherence ⌢⌣

◮ This one seems to be new.◮ Let u ⌢

⌣ v iff

u 〈v∗〉 = v 〈u∗〉 and 〈v〉u = 〈u〉 v

◮ In PI(N), means that u and v agree on their domain and ontheir codomain:

01234567 .

.

.

01234567.

.

.

01234567 .

.

.

01234567.

.

.

01234567 .

.

.

01234567.

.

.

agree on common intersection:




Conclusion

Linear Inverse Monoids

DefinitionAn inverse monoid M is linear iff:

◮ Every clique (ui)i∈I has a sup∑

i∈I ui ;◮ Sups distribute over product: v

(∑

i∈I ui)

w =∑

i∈I vuiw .




Conclusion

Linear Inverse Monoids

DefinitionAn inverse monoid M is linear iff:

◮ Every clique (ui)i∈I has a sup∑

i∈I ui ;◮ Sups distribute over product: v

(∑

i∈I ui)

w =∑

i∈I vuiw .

This is certainly the case in PI(N), where sups are unions:

01234567 .

.

.

01234567.

.

.

01234567 .

.

.

01234567.

.

.

01234567 .

.

.

01234567.

.

.

+ =




Conclusion

Bi-unambiguous Automata

b

a∗

b∗

qI

a

b

qF

b∗

is notation for b∗b +∑

n∈Na∗(bb∗)na.




Conclusion


b

a∗

b∗

qI

a

b

qF

b∗


n∈Na∗(bb∗)na.

One can write whole words over arrows: no confusion canarise, because of distributivity.




Conclusion


b

a∗

b∗

qI

a

b

qF

b∗


n∈Na∗(bb∗)na.

One can write whole words over arrows: no confusion canarise, because of distributivity.Bi-unambiguous as soon as makes sense: whichever way youtravel from qI to qF or conversely, you get the same result.




Conclusion


b

a∗

b∗

qI

a

b

qF

b∗


n∈Na∗(bb∗)na.

One can write whole words over arrows: no confusion canarise, because of distributivity.Bi-unambiguous as soon as makes sense: whichever way youtravel from qI to qF or conversely, you get the same result.Special case: Bideterminism (here, when ab∗ = a∗b = 0): atmost one path forward, at most one path backward.




Conclusion

A Few Useful Theorems

Let M be an inverse linear monoid.

TheoremDefine # (conflict) as the negation of ⌢

⌣; Then (M,≤, #) is anevent structure.




Conclusion

A Few Useful Theorems

Let M be an inverse linear monoid.

TheoremDefine # (conflict) as the negation of ⌢

⌣; Then (M,≤, #) is anevent structure.

Theorem [Preston-Wagner]Every inverse monoid M embeds into some linear inversemonoid (i.e., PI(M)), preserving product, inverse, unit, andpreserving and reflecting order.




Conclusion

Constructions of Linear Inverse Monoids

◮ Fisg(M) = the free linear inverse monoid over an inversemonoid M:

◮ Elements are down-closed cliques of M;◮ UV = {uv |u ∈ U, v ∈ V};◮ U∗ = {u∗|u ∈ U}.




Conclusion




◮ Inf-semi-lattices (product is inf), groups, semi-directproducts of groups and lattices;




Conclusion




◮ Inf-semi-lattices (product is inf), groups, semi-directproducts of groups and lattices;

◮ Bimonotonic partial injections on some partially orderedspace;

◮ Bicontinuous partial injections on some topological space;◮ Coherence-preserving-and-reflecting partial injections on

some coherence space E ;




Conclusion

Constructions of Linear Inverse Monoids, cont’d

◮ MΣ: Girard’s (1995) rudimentary clauses s ← t , s and thaving same free variables.

◮ (s ← t) · (s′ ← t ′) =

{

sσ ← t ′σ if t , s′ unify, with mgu σ0 otherwise

◮ (s ← t)∗ = t ← s




Conclusion

Constructions of Linear Inverse Monoids, cont’d

◮ MΣ: Girard’s (1995) rudimentary clauses s ← t , s and thaving same free variables.

◮ (s ← t) · (s′ ← t ′) =

{

sσ ← t ′σ if t , s′ unify, with mgu σ0 otherwise

◮ (s ← t)∗ = t ← s

◮ F0isg(MΣ) = cliques of rudimentary clauses =

bideterministic automata on terms.




Conclusion

Inductive GroupoidsIntroducing DR(M)

The Weakly Cantorian CaseMiserable Failures (Ignoring Great Successes)

Outline




Conclusion




Conclusion



The Inductive Groupoid of an Inverse Monoid

◮ A well-known construction. IG(M) has:◮ Objects: all idempotents 〈u〉 of M;

◮ Morphisms: A u B provided 〈u∗〉 = A and 〈u〉 = B.




Conclusion




◮ A well-known construction. IG(M) has:◮ Objects: all idempotents 〈u〉 of M;

◮ Morphisms: A u B provided 〈u∗〉 = A and 〈u〉 = B.◮ IG(M) is an inductive groupoid , a groupoid such that:

◮ ≤ on objects is an inf-semi-lattice (A ∧ B = AB = BA);◮ ≤ on morphisms makes IG(M) order-enriched;

◮ If

A2u2 B2

A1 u1B1

≤ then A1 ≤ A2 and B1 ≤ B2.




Conclusion




◮ IG(M) is an inductive groupoid , a groupoid such that:◮ ≤ on objects is an inf-semi-lattice (A ∧ B = AB = BA);◮ ≤ on morphisms makes IG(M) order-enriched;

◮ If

A2u2 B2

A1 u1B1

≤ then A1 ≤ A2 and B1 ≤ B2.

◮ If

A2u2 B2

A1

≤ then

A2u2 B2

A1

≤

(u2|A1)B1

≤≤

for a unique B1 and a unique (u2|A1).




Conclusion



The Inverse Monoid of an Inductive Groupoid

◮ Elements are morphisms (whatever source and target);

◮ Product of A2u2 B2 by A1

u1 B1 is

A1u1 B1

•

≤

(B1∧A2|u1) B1 ∧ A2≤

≤

(u2|B1∧A2)

A2

u2

• ≤ B2

◮ Inverse is. . . inverse.




Conclusion



Outline




Conclusion




Conclusion



The Danos-Regnier Category of M

DR(M) is defined similarly. . . but morphisms are morecomplex.

◮ Objects: idempotents of M again;

◮ Morphisms from A to B: diagrams

•A

a

β

•A

◦B ◦B

a∗

γ

where:

◮ (co)domains of a, β, γ are smaller than A, resp. B;◮ β and γ are self-inverses: β∗ = β, γ∗ = γ;◮ Bideterminism: aβ = 0, γa = 0.




Conclusion



The Danos-Regnier Category of M

◮ Morphisms from A to B: diagrams

•A

a

β

•A

◦B ◦B

a∗

γ

where:

◮ (co)domains of a, β, γ are smaller than A, resp. B;◮ β and γ are self-inverses: β∗ = β, γ∗ = γ;◮ Bideterminism: aβ = 0, γa = 0.

◮ Intuition:◮ γ is the sum of paths from output (B) to output (see the

λx , y , z.zxy example);◮ a is the sum of paths from input (A) to output (B);◮ β is the sum of paths from input to input.




Conclusion



Identity, Composition

◮ Identity on A =•A

A

•A

◦A ◦A

A(Zero arrows omitted.)




Conclusion





A

•A

◦A ◦A


◮ Composing:

•B

a′

β′

•B

◦C ◦C

a′∗

γ′

◦

•A

a

β

•A

◦B ◦B

a∗

γ

=

•A

a

β

•A

B

a′

β′

B

a∗

γ

◦C ◦C

a′∗

γ′




Conclusion





A

•A

◦A ◦A


◮ Composing:

•B

a′

β′

•B

◦C ◦C

a′∗

γ′

◦

•A

a

β

•A

◦B ◦B

a∗

γ

=

•A

a

β

•A

B

a′

β′

B

a∗

γ

◦C ◦C

a′∗

γ′

Note non-trivial loops: this is the feedback equation.




Conclusion



This is a Category

Proof

•A

a

β

•A

B

a′

β′

B

a∗

γ

C

a′′

β′′

C

a′∗

γ′

◦D ◦D

a′′∗

γ′′




Conclusion



Isos

TheoremThe groupoid of DR(M) is IG(M). Ie., the isos in DR(M) are•A

a

•A

◦B ◦B

a∗ with 〈a∗〉 = A and 〈a〉 = B (totality).




Conclusion



Epis, Monos

Theorem

Epis in DR(M) are

•A

a

β

•A

◦B ◦B

a∗ with 〈a〉 = B (right totality).

Monos in DR(M) are

•A

a

•A

◦B ◦B

a∗

γ

with 〈a∗〉 = A (left totality)




Conclusion



Epis, Monos

Theorem

Epis in DR(M) are

•A

a

β

•A

◦B ◦B

a∗ with 〈a〉 = B (right totality).

Monos in DR(M) are

•A

a

•A

◦B ◦B

a∗

γ

with 〈a∗〉 = A (left totality)

TheoremDR(M) has an epi-mono factorization. Epis, monos are split.Iso = Epi ∩Mono.




Conclusion



CPO Enrichment

TheoremDR(M) is enriched overCPO.




Conclusion



CPO Enrichment


Curiously, not over COH oreven STAB.




Conclusion



CPO Enrichment


In particular, every f : A →A has a least fixpoint µf =∑

n∈Nf n ◦ ⊥ (where I = 0,

⊥ = all-zero morphism):

•I •I

A

a

β

A

γ

...a

...

a∗

A

a

β

A

γ

a∗

◦A ◦A

a∗

γ




Conclusion



Duality

TheoremThere is a dualizing functor, inducing an isomorphism ofcategories, ⊥ : DR(M)→ DR(M)op:

◮ Objects: A⊥ = A;◮ Morphisms:

•A

a

β

•A

◦B ◦B

a∗

γ

⊥

=

•B

a∗

γ

•B

◦A ◦A

a

β




Conclusion



Outline




Conclusion




Conclusion



Remember Our Example?

λ

λ

@

λ

p*q*p*p*p*p*p*q*p*

pppp*+ ppqpp+ ppqpq

q*+ ppqqq*p*q* p*p*

q*q*p*p*

x

y

z

@

+ q+ pq




Conclusion



(Weakly) Cantorian Linear Inverse Monoids

This can be encoded provided M is weakly Cantorian, i.e.,contains two elements p, q such that:

p∗q = 0 〈p∗〉 = 〈q∗〉 = 1

(Cantorian if additionally, 〈p〉+ 〈q〉 = 1.)




Conclusion



(Weakly) Cantorian Linear Inverse Monoids

This can be encoded provided M is weakly Cantorian, i.e.,contains two elements p, q such that:

p∗q = 0 〈p∗〉 = 〈q∗〉 = 1

(Cantorian if additionally, 〈p〉+ 〈q〉 = 1.)This provides a symmetric monoidal structure:

•A1

a1

β1

•A1

◦B1 ◦B1

a1∗

γ1

⊗

•A2

a2

β2

•A2

◦B2 ◦B2

a2∗

γ2

=

•A1⊗A2

a1⊗a2

β1⊗β2

•A1⊗A2

◦B1⊗B2 ◦B1⊗B2

a∗

1⊗a∗

2

γ1⊗γ2

where u ⊗ v abbreviates pup∗ + qvq∗.Jean Goubault-Larrecq GoI and Coherence



Conclusion



DR(M) is Compact-Closed

◮ In other words, it is a model of (classical) MLL.◮ We already have ⊗ and ( )⊥.◮ Let us define:

◮ A−◦ B = qAq∗ + pBp∗ (= B ⊗ A);◮ Linear application appA,B : (A−◦ B)⊗ A→ B;◮ Linear abstraction λC

A,B,so that λC

A,B(f ) : C → (A−◦ B) for each f : C ⊗ A→ B;◮ The CA :∼∼∼ ∼∼∼ A−◦ A operator, where ∼∼∼ A = A−◦ 0;◮ Of course, all usual equations should be satisfied.




Conclusion



Linear Application

appA,B =

Aq∗

Bp∗

q

•(A−◦B)⊗A

p∗

q∗

(A−◦B)⊗A•

qA

p

◦B B◦pB




Conclusion



Linear Abstraction

λCA,B

•C⊗A

a

β

C⊗A•

◦B B◦

a∗

γ

=

•C

pC

•C

qA

C⊗A

a

β

C⊗A

Cp∗

q∗ q

B

pB

B

a∗

γ

◦A−◦B A−◦B◦

Bp∗

Aq∗




Conclusion



Linear Continuations

Let ∼∼∼ A be A−◦ 0 (linear negation, ∼= A⊥ = A).

CA =

•∼∼∼∼∼∼A

Aq∗2

•∼∼∼∼∼∼A

◦A ◦A

q2A




Conclusion



Compact Closure

TheoremDR(M) is compact-closed.




Conclusion



Compact Closure

TheoremDR(M) is compact-closed.

Proof

Funny calculations, repeatedly replacing crossings

p p∗

q∗q

by double connections .




Conclusion



My Idea of Fun?

•DpD

D•

qA

D⊗Ap∗

q∗

D⊗ADp∗

q∗ q

D

a′

β′A

q

D

p

C

pq

AC

a′∗

γ′

C⊗A

a

β C⊗Ap∗

q∗

BpB

Ba∗

γ

◦A−◦B A−◦B◦Bp∗

Aq∗




Conclusion



Traces

Every compact-closed category has a canonical trace. InDR(M),

TrXA,B

•A⊗X

a

β

A⊗X•

◦B⊗X B⊗X◦

a∗

γ

=

•A

pA

A•

A⊗X

a

β

A⊗XAp∗

qXq∗

B⊗XBp∗

qXq∗

B⊗X

a∗

γ

◦B B◦

pB




Conclusion



Relation to GoI Situations

◮ Start from PIG(M), whose morphisms are A u B where〈u∗〉 ≤ A and 〈u〉 ≤ B (instead of equality as in IG(M)).

◮ Apply the G construction. Composition by symmetricfeedback:

−

C − C

B + B

+

g ◦−

B − B

A + A

+

f =

−

B − B

A + A

+

f

−

C − C

B + B

+

g




Conclusion



Relation to GoI Situations, cont’d

Each−

B − B

A + A

+

f in PIG(M) is, equivalently:

A+ f ++

B+ A+ f +−

A−

B− f−+

B+ B− f−−

A−




Conclusion



Relation to GoI Situations, cont’d

Each−

B − B

A + A

+

f in PIG(M) is, equivalently:

A+ f ++

B+ A+ f +−

A−

B− f−+

B+ B− f−−

A−

Write this

•A+

f ++

f +−

•A−

◦B+

◦B−

f−−

f−+

, by imitation. . .




Conclusion



Relation to GoI Situations, cont’d (2)

TheoremDR(M) is a subcategory of G(PIG(M)).

Proof (essentially)

−

B − B

A + A

+

f =

•A+

f ++

f +−

•A−

◦B+

◦B−

f−−

f−+




Conclusion



Relation to GoI Situations, cont’d (2)

TheoremDR(M) is a subcategory of G(PIG(M)).

Proof (essentially)

−

B − B

A + A

+

f =

•A+

f ++

f +−

•A−

◦B+

◦B−

f−−

f−+

Only difference: in DR(M), you require some symmetry:f ++∗ = f−−, f +−∗ = f +−, f−+∗ = f−+.Intuitively, if you can go from A to B by u, you can come backfrom B to A by u∗.




Conclusion



Outline




Conclusion




Conclusion



Great Successes

◮ New insights into implementations of the λ-calculus;◮ Wadsworth-Levy labels and sharing graphs;◮ Optimal λ-reduction.




Conclusion



Great Successes

◮ New insights into implementations of the λ-calculus;◮ Wadsworth-Levy labels and sharing graphs;◮ Optimal λ-reduction.

This being said, let me gently recall that I am not interested incomputation here, but in equality of Bohm(-like) trees.I would therefore like to have an implementation of fullλ-calculus, at least, and why not a categorical model of linearlogic?




Conclusion



Additives

Equipping DR(M) with additives means finding enough limits:◮ Terminal, initial objects;◮ Products, coproducts.




Conclusion



No Limits (1): Terminal, Initial Objects

TheoremThe following statements are equivalent:

1. DR(M) has a terminal object;

2. 0 is terminal in DR(M);

3. DR(M) has an initial object;

4. 0 is initial in DR(M);

5. M = {0}.

ProofCalculation. Uniqueness of morphisms is essential!




Conclusion



No Limits (2): Products, Coproducts

TheoremLet A and B be any two objects of DR(M). Then the followingconditions are equivalent:

1. A× B exists;

2. A + B exists;

3. M = {0}.

ProofEven more boring calculation.




Conclusion





1. A× B exists;

2. A + B exists;

3. M = {0}.


Consequence: DR(M) hates additives.




Conclusion





1. A× B exists;

2. A + B exists;

3. M = {0}.


Consequence: DR(M) hates additives. (But do we care?)




Conclusion



Exponentials, Anyone?

Try to turn DR(M) into a model of MELL. Several possibilities(essentially equivalent, see Mellies 2002):

◮ as a new-Lafont category , i.e.:◮ a ∗-autonomous category C (= DR(M)?),◮ a full sub-monoidal category M of the category of

cocommutative comonoids over C,◮ U : M→ C has a right adjoint F .




Conclusion







◮ as an LNL category (Benton, 1995), i.e.:◮ a ∗-autonomous category C (= DR(M)?),◮ a category M with finite products,◮ a symmetric monoidal adjunction U ⊣ F between some

U : M→ C and some F : C→M.




Conclusion







◮ as an LNL category (Benton, 1995), i.e.:◮ a ∗-autonomous category C (= DR(M)?),◮ a category M with finite products,◮ a symmetric monoidal adjunction U ⊣ F between some

U : M→ C and some F : C→M.◮ as a linear category (Bierman, 1995) [definition omitted].

In any case, studying cocommutative comonoids over DR(M)must be the key.




Conclusion



Comonoids in DR(M) (M Weakly Cantorian)

A comonoid (A, dA, eA) verifies 1. left unit, 2. right unit, 3.associativity.

TheoremIf (A, dA, eA) verifies 1 and 2, there is a partition Ap, Aq of A(i.e., A = Ap + Aq, ApAq = 0) such that

dA =

•ApAp+qAq

A•

◦A⊗A A⊗A◦

App∗+Aqq∗

qβ0p∗+pβ0q∗

eA =

•Aβ0+β∗

0

A•

◦0 0◦

where β0 is iso between Ap and Aq.Then dA is associative, so (A, dA, eA) is a comonoid in DR(M).

Proof Jean Goubault-Larrecq GoI and Coherence



Conclusion



(Almost) No Comonoid is Cocommutative

TheoremThe only cocommutative comonoid of DR(M) is

A = 0 d0 =

•0 0•

◦0⊗0 0⊗0◦

e0 =

•0 0•

◦0 0◦

ProofCocommutativity implies Ap = Aq. But Ap and Aq do notintersect, so Ap = Aq = 0. So A = Ap + Aq = 0.




Conclusion



(Almost) No Comonoid is Cocommutative

TheoremThe only cocommutative comonoid of DR(M) is

A = 0 d0 =

•0 0•

◦0⊗0 0⊗0◦

e0 =

•0 0•

◦0 0◦

ProofCocommutativity implies Ap = Aq. But Ap and Aq do notintersect, so Ap = Aq = 0. So A = Ap + Aq = 0.

Note that this is the only cocommutative comonoid of DR(M)which is guaranteed to exist in any symmetric monoidalcategory. . .




Conclusion



No Exponentials

Corollary

◮ DR(M) is the C (linear) component of a new-Lafontcategory if and only if M = {0};

◮ DR(M) is a linear category if and only if M = {0};◮ DR(M) is the C (linear) component of an LNL category if

and only if M = {0}.




Conclusion



No Exponentials

Corollary




Consequence: DR(M) hates exponentials, too.




Conclusion



No Exponentials

Corollary




Consequence: DR(M) hates exponentials, too. (We do care.)




Conclusion



DR(M) is a Weak Linear Category

◮ Every linear inverse semigroup endomorphism on Mdefines a traced sym. mon. endofunctor ! on DR(M);(And there are not that many others.)

◮ The retractions θ : A ⊳ B : θ′ of A into B in PIG(M) are themorphisms θ such that 〈θ∗〉 = A, 〈θ〉 ≤ B, and θ′ = θ∗.

◮ The natural monoidal retractions are of the formuF (A) : F (A) ⊳ G(A) : F (A)u∗ for some fixed u ∈ M suchthat 〈u∗〉 = F (1).

◮ . . . Essentially allows one to get back the Danos-Regnierinterpretation over PI(N), or a Girard-like interpretationover rudimentary clauses.

◮ Note that δ, ǫ fail all the comonad equations. Forcomputation, this is enough, though.




Conclusion

The Modified Hu-Joyal ConstructionLifting Monoidal Structure, Creating the RestRelation to Syntax?Towards Fibrations?

Outline




Conclusion




Conclusion


Modifying the Hu-Joyal Construction

There is a way out!Modify a construction by Hu and Joyal (1997):

◮ The coherence completion of C is ∗-autonomous when C is.◮ Any other structure (comonad, product) lifts from C to its

coherence completion.




Conclusion


Modifying the Hu-Joyal Construction

There is a way out!Modify a construction by Hu and Joyal (1997):

◮ The coherence completion of C is ∗-autonomous when C is.◮ Any other structure (comonad, product) lifts from C to its

coherence completion.◮ New: if you take the identity comonad on C, and the

multiclique exponential on coherence spaces, then thisconstruction creates exponentials and additives in thecoherence completion.




Conclusion


Coherence Completion of C: Objects

Pairs (X , (Ai)i∈X ) where each Ai is an object of C.

A2A1

A3A4

A5

A6

|X| 12

35

64




Conclusion


Coherence Completion of C: Morphisms

Pairs of a linear map between the bases, and morphisms in C

for each connected elements:

A2A1

A3A4

A5

A6

|X| 12

35

64 B2

B3B4

B5

B6B1

B7

|Y| 12

35

64

7




Conclusion


Coherence Completion of C: Morphisms

Pairs of a linear map between the bases, and morphisms in C

for each connected elements:

A2A1

A3A4

A5

A6

|X| 12

35

64

f17

f11

B2

B3B4

B5

B6B1

B7

|Y| 12

35

64

7

f33 f

53

f64




Conclusion


This is a Category!

The following never happens:

A1

B2

B3

C4

f12

f13

g34

g24

1

2 3

4




Conclusion


Outline




Conclusion




Conclusion


Monoidal Structure

Inherited from C (= DR(M)) and COH:◮ (X , (Ai)i∈|X | ⊗ (Y , (Bj)j∈|Y |

) = (X ⊗ Y , (Xi ⊗ Yj)(i,j)∈|X⊗Y |);

◮ Remember that in coherence spaces, |X ⊗ Y | = |X | × |Y |,with pairwise coherence.




Conclusion


Monoidal Structure

Inherited from C (= DR(M)) and COH:◮ (X , (Ai)i∈|X | ⊗ (Y , (Bj)j∈|Y |

) = (X ⊗ Y , (Xi ⊗ Yj)(i,j)∈|X⊗Y |);

◮ Remember that in coherence spaces, |X ⊗ Y | = |X | × |Y |,with pairwise coherence.

◮ (X , (Ai)i∈|X |−◦(Y , (Bj)j∈|Y |) = (X−◦Y , (Xi −◦ Yj)(i,j)∈|X−◦Y |

);

◮ Remember that in coherence spaces, |X −◦ Y | = |X | × |Y |,with (x , y) ⌢

⌣ (x ′, y ′) iff x ⌢⌣ x ′ implies (y ⌢

⌣ y ′ and y = y ′

implies x = x ′) (linear maps);




Conclusion


Coproducts, Products

Coproducts inherited purely from COH:

A2A1

A3A4

A5

A6

B2

B3B4

B5

B6B1

B7

12

35

64|Y| 1

2

35

64

7|X|




Conclusion


Coproducts, Products

Coproducts inherited purely from COH:

A2A1

A3A4

A5

A6

B2

B3B4

B5

B6B1

B7

12

35

64|Y| 1

2

35

64

7|X|

Products: similar, except all dots from X are coherent with alldots from Y .




Conclusion


The ! Comonad

Inherited from COH, using the symmetric monoidal structurefrom C = DR(M):

◮ !(X , (Ai)i∈I) = (!X , (⊗

i∈e Ai)e∈|!X |);

◮ Requires some fine, boring, formal points (e.g., equippingthe X part with a total ordering to make

⊗

i∈e Ai

well-defined. . . )◮ |!X | is set of multicliques of X—Would not work with just

cliques.




Conclusion


The ! Comonad in Pictures

The shriek (well, part of it) of

A2A1

A3

|X| 12

3is

A2A1

A3

0

A1 A2A1 A3

A1 A1

A1 A2 A2

{1,2,2}{2}

{3}

I

{1} {1,2}{2,3}

{1,1}

|!X|




Conclusion


And Fixpoints!

◮ Again inherited purely from COH:◮ (Recursion game.) For any (multi)clique e of !X −◦ X ,

define recae ⊆ |X | as the smallest subset such thatwhenever (e′, x) ∈ e and e′ ⊆ recae, then x ∈ recae.

◮ For every coherence space A = (|A|, ⌢

⌣A),

YA = {(0, (e, x)) | x ∈ recae, e multiclique of !A−◦ A}

is a fixpoint operator :!(!A−◦ A)−◦ A.◮ Application to the coherence completion: exercice!




Conclusion


And Fixpoints!

◮ Again inherited purely from COH:◮ (Recursion game.) For any (multi)clique e of !X −◦ X ,

define recae ⊆ |X | as the smallest subset such thatwhenever (e′, x) ∈ e and e′ ⊆ recae, then x ∈ recae.

◮ For every coherence space A = (|A|, ⌢

⌣A),

YA = {(0, (e, x)) | x ∈ recae, e multiclique of !A−◦ A}

is a fixpoint operator :!(!A−◦ A)−◦ A.◮ Application to the coherence completion: exercice!

◮ This is slightly strange: the fact that DR(M) itself has(linear) fixpoints does not play any role in the construction.




Conclusion


Epitome

TheoremThe coherence completion of any ∗-autonomous category(including DR(M) when M is weakly Cantorian) is a model offull linear logic (a linear category) with fixpoints.




Conclusion


Outline




Conclusion




Conclusion


Proof Nets for Full Linear Logic: LLPA

Inspired from Abramsky’s (1993) Linear CHAM:

Termss, t , . . . = x name

| nil stop| nil⊥ abort| λx · y λ-abstraction| xy application| LMN empty tuple| LS · x 8 T · yM ext. choice, pairing| 2x first projection| 3x second projection| ∂x dissolve| S · x box

SolutionsS ::= m1, . . . , mn

Matchesm ::= N ⇋ s

Name setsN ::= {x1, . . . , xm




Conclusion


Well-Formedness, Types

◮ Unusual notion of free/bound name:◮ x is free in t (S, m) iff it occurs once in it;◮ x is bound in t (S, m) iff it occurs twice in it;◮ No name can occur more than twice.




Conclusion



◮ Unusual notion of free/bound name:◮ x is free in t (S, m) iff it occurs once in it;◮ x is bound in t (S, m) iff it occurs twice in it;◮ No name can occur more than twice.◮ Caveat: “once” and “twice” have a different meaning in the

presence of additive boxes LS · x 8 T · yM, where S · x andT · y are constrained to have exactly the same free names.




Conclusion



◮ Unusual notion of free/bound name:◮ x is free in t (S, m) iff it occurs once in it;◮ x is bound in t (S, m) iff it occurs twice in it;◮ No name can occur more than twice.◮ Caveat: “once” and “twice” have a different meaning in the

presence of additive boxes LS · x 8 T · yM, where S · x andT · y are constrained to have exactly the same free names.

◮ Judgments Γ ⊢ S ⊣ ∆, where each name occurs twice.◮ Use a minimal set of connectives:

F , G, H, . . . ::= A atomic type| ⊥⊥⊥ false (multiplicative)| ⊤⊤⊤ true (additive)| F −◦G linear implication| FNG with (Cartesian product)| !F of course




Conclusion


Typing Rules (1): Identity Group

(Ax)x:A y:A

S1

Γ

y:F∆

1

1 Γ2∆2

y:FS2

(Cut)

Γ2

Γ1

∆2

S1 ∆1

S2




Conclusion


Typing Rules (2): Multiplicatives, −◦ Right

SΓ

x:F

∆

y:G

(⊢ −◦)

λ Gz:F

SΓ ∆




Conclusion


Typing Rules (3): Multiplicatives, −◦ Left

S1

Γ∆

1

1x :G1

Γ2∆2

S2

x :F2

(−◦ ⊢)

∆2

Gz:F

Γ1

Γ2

S1 ∆1

@

S2




Conclusion


Typing Rules (4): Multiplicative Unit⊥⊥⊥

S ∆Γ

(⊢ ⊥⊥⊥)

S ∆Γ

nil z:

(⊥⊥⊥ ⊢)

nilz:




Conclusion


Typing Rules (5): Exponential Promotion

!Γ S x:F

(⊢!)

S z: Fx!Γ !




Conclusion


Typing Rules (6): Exponential Dereliction

SΓ

∆x:F

(! ⊢)

!z: FS ∆

Γ




Conclusion


Typing Rules (7): Exponential Fan

SΓ

∆...!x : F2

!x : Fn

!x : F1

(Fan)

!z: FS ∆

...

Γ




Conclusion


Typing Rules (8): Additives

◮ Omitted. . . hard to draw!




Conclusion


The Free Linear Category

TheoremLLPA with the right reduction rules defines the free linearcategory over the given set of atomic formulae.

ProofFree is obvious. What is harder is that the reduction rulesterminate and obey all the right laws. Currently under way.




Conclusion


Fixpoints

One can also add typed fixpoints. No need for a new operator.

Γ, x1 :!F , . . . , xn :!F ⊢ S ⊣ y :!F , ∆(FanRec)

Γ ⊢ S, {z, x1, . . . , xn}⇋ y ⊣ z :!F , ∆

Then the following is Y :

ηF[ x2 , y2 ]

[ x3 , y3 ]

λ

ηF[ x1 , y1 ]

@

ηF!

Of course, termination fails in the presence of (FanRec).Jean Goubault-Larrecq GoI and Coherence



Conclusion


Outline




Conclusion




Conclusion


Towards Fibrations

◮ Coherence completions are categories of trivial fibrationsover webs of coherence spaces.




Conclusion


Towards Fibrations


◮ ConjectureLLPA solutions define (non-trivial) fibrations over webs ofcoherence spaces.

Necessarily non-trivial: the construction of X −◦ Y (!X ,too?) is too big for syntax.




Conclusion


Towards Fibrations


◮ ConjectureLLPA solutions define (non-trivial) fibrations over webs ofcoherence spaces.

Necessarily non-trivial: the construction of X −◦ Y (!X ,too?) is too big for syntax.

◮ Idea:◮ total space of the fibration=solution;◮ get fibers by erasing boxes and selecting one input per

exponential fan;◮ base space will then be space of fibers.




Conclusion


Reconstructing Solutions From Fibrations

◮ Think of each fiber Ai as a chunk of a linear λ-term drawnon tracing paper .

◮ Then superpose each layer; coherence in the base spaceshould ensure this makes sense.




Conclusion





!F

!F




Conclusion





!F!F

!F




Conclusion





!F!F!F

!F




Conclusion





!F

!F!F!F!F...




Conclusion

Outline




Conclusion




Conclusion

Conclusion

In a nutshell: Coherence!◮ In the base space: coherence spaces;◮ In each fiber: coherence in linear inverse monoids.


musings around the geometry of interaction, and coherencegoubault/geocal2006.pdf · musings around...

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