Page 1
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.
Page 2
Part I. Two-dimensional Homotopy and String Rewriting
Page 3
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 ′
Page 4
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.
Page 5
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.
Page 6
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.
Page 7
Part II. Two-dimensional Homotopy for higher-dimensional rewritingsystems
Page 8
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.
Page 9
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.
Page 10
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
Page 11
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
.
Page 12
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©
Page 13
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©
Page 14
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>.
Page 15
Part III. Computation of homotopy bases
Page 16
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
Page 17
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.
Page 18
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.
Page 19
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.
Page 20
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
Page 21
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
Page 22
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)
Page 23
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
Page 24
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."
Page 25
Part IV. Identities among relations
Page 26
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.
Page 27
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).
Page 28
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
(.
)−⌋
=
.
=⌊
.⌋
=⌊
. .⌋