homotopical methods in polygraphic rewriting yves...
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Homotopical methods in polygraphic rewriting
Yves Guiraud and Philippe Malbos
Categorical Computer Science, Grenoble, 26/11/2009
References.
• Higher-dimensional categories with finite derivation type, Theory and Applications of Categories, 2009.
• Identities among relations for higher-dimensional rewriting systems, arXiv:0910.4538.
Part I. Two-dimensional Homotopy and String Rewriting
String Rewriting
String Rewriting System : X a set , R⊆ X∗×X∗
ulv →R urvu
oo
l
ZZ
r
¥¥ voo (r, l) ∈ R u,v ∈ X∗
→∗R : reflexive symetrique closure of →R
Terminating :
w0 →R w1 →R · · ·→R wn →R · · ·
Confluentw
∗R
||zzzz
zzzz
∗R
""DD
DDDD
DD
w1
∗R
!!CC
CCCC
CCw2
∗R
}}{{{{
{{{{
w ′
String Rewriting and word problem
Word problem
w,w ′ ∈ X∗, is w = w ′ in X∗/ ↔∗R
↔∗R : derivation.
Normal form algorithm : (X,R) : finite + convergent (terminating + confluent)
w
ÄÄÄÄÄ
""EEE
E w ′
zzvvvv
!!BB
B
ÄÄÄÄÄ ##FFF
Fyyssss
##FFF
Fzzvvv
v&&MMMMM
ÂÂ???
zzvvvv
ÂÂ???
{{xxxx
%%KKKK{{xxx
x$$HHH
Hxxqqqqq
||zzzz
zzzz
##FFF
Fyyssss
##FFF
Fzzvvv
v&&MMMMM
""EEE
EÄÄÄÄÄ !!
BBB
zzvvvv
w?
w ′
Fact. Monoids having a finite convergent presentation are decidable.
First Squier theorem
Rewriting is not universal to decidethe word problem in finite type monoids.
Theorem. (Squier ’87) There are finite type decidable monoids which do not have a finite convergent
presentation.
Proof :
• A monoid M having a finite convergent presentation (X,R) is of homological type FP3.
kerJ−→ ZM[R]J−→ ZM[X]−→ ZM−→ Z
i.e. module of homological 3-syzygies is generated by critical branchings.
• There are finite type decidable monoids which are not of type FP3.
Second Squier Theorem
Theorem. Squier ’87 (’94) The homological finiteness condition FP3 is not sufficient for a finite typedecidable monoid to admit a presentation by a finite convergent rewriting system.
Proof : • (X,R) a string rewriting system.
• S(X,R) Squier 2-dimensional combinatorial complex.
0-cells : words on X, 1-cells : derivations ↔∗R, 2-cells : Peiffer elements
ulwl ′v88
uAwl ′vxxqqqqqqqq ff
ulwA ′v&&MMMMMMMM
urwl ′vff
urwA ′v &&MMMMMMMM ulwr ′v88
uAwr ′vxxqqqqqqqq
urwr ′v
• (X,R) has finite derivation type (FDT) if
X and R are finite and S(X,R) has a finite set of homotopy trivializer.
• Property FDT is Tietze invariant for finite rewriting systems
• A monoid having a finite convergent rewriting system has FDT.
• There are finite type decidable monoids which do not have FDT and which are FP3.
Part II. Two-dimensional Homotopy for higher-dimensional rewritingsystems
Mac Lane’s coherence theorem
monoidal category is made of:
• a category C,
• functors ⊗ : C×C → C and I : ∗→ C,
• three natural isomorphisms
αx,y,z : (x⊗y)⊗z → x⊗ (y⊗z) λx : I⊗x → x ρx : x⊗ I → x
such that the following diagrams commute:
(x⊗(y⊗z))⊗tα
// x⊗((y⊗z)⊗t)α
''NNNNN
((x⊗y)⊗z)⊗t
α 77pppppα
// (x⊗y)⊗(z⊗t)α
//
c©x⊗(y⊗(z⊗t))
x⊗(I⊗y)λ##
FFFF
(x⊗I)⊗y
α 99ssssρ
// x⊗y
c©
Mac Lane’s coherence theorem. "In a monoidal category (C,⊗, I,α,λ,ρ), all the diagrams built from C,
⊗, I, α, λ and ρ are commutative."
Program:
– General setting: homotopy bases of track n-categories.
– Proof method: rewriting techniques for presentations of n-categories by polygraphs.
– Algebraic interpretation: identities among relations.
n-categories
An n-category C is made of:
• 0-cells
• 1-cells: xu
//y with one composition
u?0 v = xu
//yv
//z
• 2-cells: x
uÃÃ
v
>>yf ¦¼ with two compositions
f?0 g = x
uÃÃ
u ′>>y
v!!
v ′== zf ¦¼
g¦¼ and f?1 g = x
u
ºº
v //
w
GGy
f ¦¼Â  ÂÂ
g¦¼ÂÂÂÂÂÂ
Exchange relation:
(f?1 g)?0 (h?1 k) = (f?0 h)?1 (g?0 k)
when · ¹¹//HH·
f ¦¼g
¦¼
¹¹//HH·
h ¦¼
k ¦¼
• 3-cells with three compositions ?0, ?1 and ?2, etc.
Track n-categories, cellular extensions and polygraphs
A track n-category is an n-category whose n-cells are invertible (for ?n−1).
A cellular extension of C is a set Γ of (n+1)-cells •f
ÃÃ
g
>>•γ¦¼ with f and g parallel n-cells in C.
C[Γ ] C(Γ) C/Γ
•
f
!!
g
==•γ
¦¼
•
f
!!
g
==•γ
µ&γ−
Rf
≈ • f
g// •
An n-polygraph is a family Σ = (Σ0, . . . ,Σn) where each Σk+1 is a cellular extension of Σ0[Σ1] · · · [Σk].
Free n-category Free track n-category Presented (n−1)-category
Σ∗ = Σ∗n−1[Σn] Σ> = Σ∗n−1(Σn) Σ = Σ∗n−1/Σn
Graphical notations for polygraphs
We draw:
• Generating 2-cells as "circuit components":
. . . . . .
• 2-cells as "circuits":
. . . .
• Generating 3-cells as "rewriting rules":
. V . .V . . V
.
• 3-cells as "rewriting paths":
.V
.V
.V
.
Example : the 2-category of monoids
Let Σ be the 3-polygraph with one 0-cell, one 1-cell, two 2-cells . and . and three 3-cells:
..
_%9. .
._%9 .
..
_%9 .
Proposition. The 2-category Σ is the theory of monoids.
i.e., there is an equivalence:
Monoids (X,×,1) in a 2-category C ↔ 2-functors M : Σ → C
M( .) = X M(. ) = × M( .) = 1
M(. ) = c© M( .) = c© M( .) = c©
Example : the track 3-category of monoidal categories
Let Γ be the cellular extension of Σ∗ with two 4-cells:
.
._%9
. .E»,EEE
EEEE
EEEE
E
.
. y2Fyyyy yyyyyyyy
.SÂ3SSSSSSSSSSSSSS
SSSSSSSSSSSSSS
SSSSSSSSSSSSSS .
..
k+?kkkkkkkkkkkkkk
kkkkkkkkkkkkkk
kkkkkkkkkkkkkk
.
ÄÂ
.
.
9µ&99
9999
99
9999
9999
9999
9999
.
. £6J£££££££
£££££££
£££££££
.
_%9 .
.
ÄÂ
Proposition. The track 3-category Σ>/Γ is the theory of monoidal categories, i.e., there is an equivalence:
Monoidal categories (C,⊗, I,α,λ,ρ) ↔ 3-functors M : Σ>/Γ → Cat
3-category Cat :– one 0-cell, categories as 1-cells, functors as 2-cells, natural transformations as 3-cells
– ?0 is ×, ?1 is the composition of functors, ?2 the vertical composition of natural transformations
The equivalence is given by:
M( .) = C M(. ) = ⊗ M( .) = I
M(. ) = α M( .) = λ M( .) = ρ
M(. ) = c© M(. ) = c©
Homotopy bases and finite derivation type
A homotopy basis of an n-category C is a cellular extension Γ such that:
For every n-cells ·f
ÀÀ
g
AA · in C, there exists an (n+1)-cell ·f
ÀÀ
g
AA ·¦¼ in C(Γ), i.e., f = g in C/Γ .
An n-polygraph Σ has finite derivation type (FDT) if it is finite and if Σ> admits a finite homotopy basis.
Theorem. Let Σ and Υ be finite and Tietze-equivalent n-polygraphs, i.e., Σ ' Υ. Then:
Σ has FDT iff Υ has FDT.
Mac Lane’s theorem revisited. Let Σ be the 3-polygraph(∗, ., . , ., . , ., .
).
Then the cellular extension{
. , .}
of Σ∗ is a homotopy basis of Σ>.
Part III. Computation of homotopy bases
Rewriting properties of an n-polygraph Σ: termination and confluence
A reduction of Σ is a non-identity n-cell uf
//v of Σ∗.
A normal form is an (n−1)-cell u of Σ∗ such that no reduction uf
//v exists.
The polygraph Σ terminates when it has no infinite sequence of reductions u1f1
// u2f2
// u3f3
// (· · ·)Termination ⇒ Existence of normal forms
A branching of Σ is a diagramu
f
ÄÄÄÄÄÄ
ÄÄ g
ÂÂ??
???
wv
in Σ∗.
– It is local when f and g contain exactly one generating n-cell of Σn.
– It is confluent when there exists a diagramv
f ′ ÂÂ??
????
u ′
w
g ′ÄÄÄÄÄÄ
Ä
The polygraph Σ is (locally) confluent when every (local) branching is confluent.
Confluence ⇒ Unicity of normal forms
Rewriting properties of an n-polygraph Σ: convergence
The polygraph Σ is convergent if it terminates and it is confluent.
Theorem [Newman’s lemma]. Termination + local confluence ⇒ Convergence.
A branching is critical when it is "a minimal overlapping" of n-cells, such as:
..
|t© ||||
||||
||||
||||
||||
|||| .
B¹*BB
BBBB
B
BBBB
BBB
BBBB
BBB
. .
..
£v ££££
£££
££££
£££
££££
£££
.
9µ&99
9999
99
9999
9999
9999
9999
..
Theorem. Termination + confluence of critical branchings ⇒ Convergence.
The homotopy basis of generating confluences
A generating confluence of an n-polygraph Σ is an (n+1)-cell
uf}}{{{
{ g
""EEE
E
v
f ′ ÃÃBB
B %9 w
g ′}}{{{{
u ′
with (f,g) critical.
Theorem. Let Σ be a convergent n-polygraph. Let Γ be a cellular extension of Σ∗ made of one generating
confluence for each critical branching of Σ. Then Γ is a homotopy basis of Σ>.
Corollary. If Σ is a finite convergent n-polygraph with a finite number of critical branchings, then it has FDT.
Generating confluences : pear necklaces
Theorem, Squier ’94. If a monoid admits a presentation by a finite convergent word rewriting system, then
it has FDT.
Theorem There exists a 2-category that lacks FDT, even though it admits a presentation by a finite
convergent 3-polygraph.
• A 3-polygraph presenting the 2-category of pear necklaces.
one 0-cell, one 1-cell, three 2-cells :
four 3-cells :αV
β
Vγ
VδV
• Σ is finite and convergent but does not have FDT.
Generating confluences : pear necklaces
• Four regular critical branching
γ
QÁ2
δ
m,@γδ
δQÁ2
γ
m,@δγ
β _%9
γ
@)@@
@@@@
@@
@@@@
@@@@
@@@@
@@@@
αy2Fyyyyyyyy
yyyyyyyy
yyyyyyyy
γ_%9
αγ
α _%9
δ
@)@@
@@@@
@@
@@@@
@@@@
@@@@
@@@@
βy2Fyyyyyyyy
yyyyyyyy
yyyyyyyy
δ_%9
βδ
• One right-indexed critical branching
k k ∈{
, , ,n}
n
Generating confluences : pear necklaces
β _%9
Peiffγ
)Á
Peiff
δ
Ä5IÄÄÄÄÄÄÄÄ
ÄÄÄÄÄÄÄÄ
ÄÄÄÄÄÄÄÄ
β _%9
δ~~~ ~~~~~~
~5I~~~~~~~~~~~~
γ _%9
δ
DDDD
DDD
DDDD
DDD
DDDD
DDD
Dº+DD
DDDD
DD
DDDD
DDDD
DDDD
DDDDα
ÂEY
β¦¼
γhhhhhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhhhhh
h*>hhhhhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhhhhhαγ
δVVVVVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVVVVV
VÃ4VVVVVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVVVVVβδ
Peiff
α _%9
γ?)??
????
??
????
????
????
????
δ _%9
γ@@
@@@
@ @@@
@)@@
@@@@
@@ @@@@Peiff
γzzzzzzz
zzzzzzz
zzzzzzz
z3Gzzzzzzzz
zzzzzzzz
zzzzzzzz
α_%9
δ
@T
Peiff
Generating confluences : pear necklaces
α _%9
β D»,DDDDDDD
DDDDDDD
DDDDDDD
δVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVV
VVVVVVVVVVVVVVV
VÃ4VVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVV
VVVVVVVVVVVVVVVVV
δ _%9
Peiff
α_%9
βδ
δ_%9
α
¦8L¦¦¦¦¦¦¦¦¦
¦¦¦¦¦¦¦¦¦
¦¦¦¦¦¦¦¦¦
β _%9 γ _%9
β
9µ&99
9999
999
9999
9999
9
9999
9999
9
αz3Gzzzzzzz
zzzzzzz
zzzzzzz
β_%9
γhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhh
hhhhhhhhhhhhhhh
h*>hhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhh
hhhhhhhhhhhhhhhhhαγ
γ_%9
Peiff... n
α
SÂ3
β
k+? ... n+1
.
αβ(
n)
Generating confluences : pear necklaces
• An infinite homotopy base :
γ
QÁ2
δ
m,@γδ
δQÁ2
γ
m,@δγ
β _%9
γ
@)@@
@@@@
@@
@@@@
@@@@
@@@@
@@@@
αy2Fyyyyyyyy
yyyyyyyy
yyyyyyyy
γ_%9
αγ
α _%9
δ
@)@@
@@@@
@@
@@@@
@@@@
@@@@
@@@@
βy2Fyyyyyyyy
yyyyyyyy
yyyyyyyy
δ_%9
βδ
... n
α
SÂ3
β
k+? ... n+1
.
αβ(
n)
• The 3-polygraph is finite and convergent but does not have finite derivation type
Generating confluences : Mac Lane’s coherence theorem
• Let Σ be the finite 3-polygraph(∗, ., . , ., . , ., .
).
Lemma. Σ terminates and is locally confluent, with the following five generating confluences:
.
._%9
..
B¹*BB
BBBB
B
BBBB
BBB
BBBB
BBB
.
. |4H|||||||
|||||||
|||||||
._%9
. ._%9
.ÄÂ
.
.
.
9µ&99
9999
99
9999
9999
9999
9999
.
. £6J£££££££
£££££££
£££££££
._%9 .
.ÄÂ
.
.
6%
.
©9M.
ÄÂ
.
.
<¶'<<
<<<<
<<<
<<<<
<<<<
<
<<<<
<<<<
<
.
. ~5I~~~~~~~
~~~~~~~
~~~~~~~
._%9 .
ÄÂ
..
<¶'<<
<<<<
<<
<<<<
<<<<
<<<<
<<<<
.
. ~5I~~~~~~~
~~~~~~~
~~~~~~~
._%9 .
ÄÂ
Theorem. The cellular extension{
. , .}
is a homotopy basis of Σ>.
Corollary (Mac Lane’s coherence theorem). "In a monoidal category (C,⊗, I,α,λ,ρ), all the diagrams
built from C, ⊗, I, α, λ and ρ are commutative."
Part IV. Identities among relations
Defining identities among relations
The contexts of an n-category C are the partial maps C : Cn → Cn generated by:
x 7→ f ?i x and x 7→ x ?i f
The category of contexts of C is the category CC with:
– Objects: n-cells of C.
– Morphisms from f to g: contexts C of C such that C[f] = g.
The natural system of identities among relations of an n-polygraph Σ is the functor Π(Σ) : CΣ → Abdefined as follows:
• If u is an (n−1)-cell of Σ, then Π(Σ)u is the quotient of
Z{bfc
∣∣ v fbb in Σ>, v = u}
by (with ? denoting ?n−1):
– bf?gc = bfc+ bgc for every vf""
gbb with v = u.
– bf?gc = bg? fc for every vf &&
wg
ee with v = w = u.
• If C is a context of Σ from u to v, then Cbfc= bB[f]c, with B = C.
Generating identities among relations
A generating set of Π(Σ) is a part X⊆ Π(Σ) such that, for every bfc:
bfc =
k∑
i=1
±Ci[xi], with xi ∈ X, Ci ∈ CΣ.
Proposition. Let Σ and Υ be finite Tietze-equivalent n-polygraphs. Then
Π(Σ) is finitely generated iff Π(Υ) is finitely generated.
Proposition. Let Γ be a homotopy basis of Σ> and Γ ={
γ = f?g−∣∣γ : f → g in Γ
}.
Then⌊Γ⌋
is a generating set for Π(Σ).
3.2. Generating identities among relations
Theorem. If a n-polygraph Σ has FDT, then Π(Σ) is finitely generated.
Proposition. If Σ is a convergent n-polygraph, then Π(Σ) is generated by the generating confluences of Σ.
Example. Let Σ be the 2-polygraph(∗, ., .
).
It is a finite convergent presentation of the monoid {1,a} with aa = a.
It has one generating confluence:
..
_%9.
Hence the following element generates Π(Σ):
⌊.
⌋=
⌊. ?1
(.
)−⌋
=
.
=⌊
.⌋
=⌊
. .⌋