cyclic ordering through partial...
TRANSCRIPT
![Page 1: Cyclic Ordering through Partial Orderspageperso.lif.univ-mrs.fr/~luigi.santocanale/TRECOLOCOCO/rencontr… · cuts: Maximal cliques of co Γ = (X,∢) is round-oriented (ROCO) iff](https://reader035.vdocument.in/reader035/viewer/2022071220/6059eec278b9b83dd00b4167/html5/thumbnails/1.jpg)
Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Cyclic Ordering through Partial Orders
Stefan Haar
INRIA / LSV (ENS Cachan + CNRS France)
Nov 17, 2010
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
![Page 2: Cyclic Ordering through Partial Orderspageperso.lif.univ-mrs.fr/~luigi.santocanale/TRECOLOCOCO/rencontr… · cuts: Maximal cliques of co Γ = (X,∢) is round-oriented (ROCO) iff](https://reader035.vdocument.in/reader035/viewer/2022071220/6059eec278b9b83dd00b4167/html5/thumbnails/2.jpg)
Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Contents
1 Cyclic Orders and Orientability
2 Windings
3 Separation and Saturation
4 Characterization of Orientability
5 Conclusion
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
![Page 3: Cyclic Ordering through Partial Orderspageperso.lif.univ-mrs.fr/~luigi.santocanale/TRECOLOCOCO/rencontr… · cuts: Maximal cliques of co Γ = (X,∢) is round-oriented (ROCO) iff](https://reader035.vdocument.in/reader035/viewer/2022071220/6059eec278b9b83dd00b4167/html5/thumbnails/3.jpg)
Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Contents
1 Cyclic Orders and Orientability
2 Windings
3 Separation and Saturation
4 Characterization of Orientability
5 Conclusion
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
![Page 4: Cyclic Ordering through Partial Orderspageperso.lif.univ-mrs.fr/~luigi.santocanale/TRECOLOCOCO/rencontr… · cuts: Maximal cliques of co Γ = (X,∢) is round-oriented (ROCO) iff](https://reader035.vdocument.in/reader035/viewer/2022071220/6059eec278b9b83dd00b4167/html5/thumbnails/4.jpg)
Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
In girum imus nocte et consumimur igni
Cyclic Orders
Let ∢ be a ternary relation over set X . Then Γ = (X ,∢) is a cyclicorder (CyO) iff it satisfies, for a, b, c , d ∈ X :
1 inversion asymmetry: ∢(a, b, c) ⇒ ¬∢(b, a, c)
2 rotational symmetry: ∢(a, b, c) ⇒ ∢(c , a, b);
3 ternary transitivity: [∢(a, b, c) ∧ ∢(a, c , d)] ⇒ ∢(a, b, d).
A CyO Γtot = (X ,∢tot) is called total or a TCO if for all a, b, c ∈X ,
(x 6= y 6= z 6= x) ⇒ ∢(a, b, c) ∨ ∢(b, a, c).
If a total CyO Γtot on X extends Γ, call Γ orientable, and Γtot anorientation of Γ.
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Cyclic Transitivity
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Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Short History
Huntington (1916,1924,1938) introduces cyclic (total) orderover words of variable length
Partial Cyos as ternary relations , orientability problem:
Power({Alles, Nesetril, Novotny, Poljak, Chaida})(1982,1984,1985,1986,1991,1994,...),Galil and Megiddo (1977)Genrich (1971): links to synchronization graphs (conflict freePetri nets)Petri: Concurrency Theory; Rozenberg et al: Square structuresStehr (1998) axiomatizes partial cyclic order over words. . .
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
(Some) known results
Orientability is NP-complete (Megiddo 1976, Galil andMegiddo 1977)
Compactness: a Cyo is orientable iff all its finite sub-CyOs are(Alles, Nesetril, Poljak 1991)
Genrich 1971, Stehr 1998: orientable finite CyOs correspondto 1-safe live Synchronization Graphs
HERE: explore new link with partial orders : WINDING
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Non-Orientable CyOs
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Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Preliminaries : Global vs round-orientedness
li , {(x , y) | ∃ z : ∢(x , y , z) ∨ ∢(x , z , y)}
co , X 2 − (idX ∪ li)
Rounds: Maximal cliques of li
cuts: Maximal cliques of co
Γ = (X ,∢) is round-oriented (ROCO) iff for any round{a, b, c}, either ∢(a, b, c) or ∢(b, a, c).
Not a ROCO:
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Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Contents
1 Cyclic Orders and Orientability
2 Windings
3 Separation and Saturation
4 Characterization of Orientability
5 Conclusion
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Shifts and Windings
Definitions
Fix poset Π = (X , <). An order automorphism is a bijectionG : X → X such that ∀ x , y ∈ X : [x < y ] ⇔ [Gx < Gy ].
G is called a shift if x < Gx for all x ∈ X
The group G of Π-automorphisms generated by G isisomorphic to (Z,+)
Equivalence : x ∼G y iff ∃ k ∈ Z : Gkx = y
Let [x ] , [x ]∼G
LEt βG : X → X , x 7→ [x ]∼Gbe the winding map.
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Winding a shift-invariant PO to a CyO
0 γ0 0ε 0η κ 0 1−1κ α α2 3αα 1γ ε1 η1 κ 1 γ2 ε2 η2 2κ
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Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Windings and their Properties
Set ∢(x , y , z) iff ∃ x ∈ x , y ∈ y , z ∈ z : x < y < z < Gx .
Γ = (X ,∢) is wound from Π via G (or βG)
Note : this leaves more concurrency than the usualconstruction of CyO from a poset: ∢(x , y , z) iff∃ x ∈ x , y ∈ y , z ∈ z : x0 < y0 < z0.
Inheritance: If Γ1 is obtained by winding, then so are all itsSub-CyOs.
We say that a winding is loop-free iff for all x ∈ X , thereexist y , z ∈ X that x < y < z < Gx . Loop-free windingsgenerate ROCOs.
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Mind the Gap !
Not all windings preserve successor relations
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Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Mind the Gap !
Let x ∢· y iff1 x li y and2 for all z ∈ X − {x , y}, x li z and z li y imply ∢(x , y , z).
x covers y from below, written x ⊻ y , iff (a) x < y , and (b)for all z ∈ X , z < y implies z < x .
y covers x from above, written y ⊼ x , iff (a) x < y , and (b)for all z ∈ X , x < z implies y < z.
In (X ,∢), x covers y , written x�y , iff (a) x li y , and (b) forall z , u ∈ X , ∢(z , u, y) implies ∢(z , u, x).
A gap in (X , <): x <·y holds, but neither x ⊼ y nor y ⊻ x .
A gap in (X ,∢): x∢· y holds, and x�y does not hold.
Suppose (X , <) is gap-free and βG winds (X , <) to (X ,∢). IfβG is loop-free, then βG maps <· surjectively to ∢·.
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Contents
1 Cyclic Orders and Orientability
2 Windings
3 Separation and Saturation
4 Characterization of Orientability
5 Conclusion
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Separation properties
Definitions
A cut c is called a separator iff c ∩ O 6= ∅ for all O ∈ O(Γ),and
a cycle separator iff c ∩ γ 6= ∅ for all γ ∈ D(Γ)
Γ is called weakly (cycle) separable iff there exists a (cycle)separator c ∈ C(Γ), and
strongly (cycle) separable iff all its cuts are (cycle) cuts.
If some superstructure cyclic order Γ′ of Γ is (strongly) cycleseparable, then Γ is called (strongly) saturable.
Every total CyO is strongly separable.
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Separation properties
Cycle separation implies round separation; the converse is not true:
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Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Contents
1 Cyclic Orders and Orientability
2 Windings
3 Separation and Saturation
4 Characterization of Orientability
5 Conclusion
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Main Result
THEOREM
Let X 6= ∅, and Γ = (X ,∢) a ROCO. Then the following are equiv-alent:
1Γ is weakly saturable;
2there exists a winding representation for Γ, i.e. a partial orderΠ = (X , <) with shift G such that X = X /G
, and φG winds Πto Γ ;
3Γ is orientable.
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Sketch of Proof
1 ⇒ 2
Saturability to winding:Given (X , <) and c, set X , X × Z, xk , (x , k).Let G : X → X be given by Gxk = xk+1 for k ∈ Z.
R0 , {(a0, b0) | a ∈ c ∧ a li b}
∪ {(ak , ak+l ) | a ∈ X , k ∈ Z, l ∈ N}
≺ , {(al+k , bm+k) | al R0 bm, k ∈ Z}
< , transitive closure of ≺
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Sketch of Proof
2 ⇒ 3
Winding to orientability:Take a weakly separable SuperPO Π♯ of Π, with separator and
c0.ck , Gkc0, and define Uk , U ♯k∩ X , where
U ♯k
,⋃
yk ∈ ck , y
k+1 ∈ ck+1
[yk , yk+1[.
The Uk are pairwise disjoint and cover X
Set Πk , (Uk , <k); G induces order isomorphismsGn,m : Un → Um from Πn to Πm.
Take total ordering Πtot0 on U0 that embeds <0 (Szpilrajn !)
Export Πtot0 to Uk via G0,k ⇒ Done
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Sketch of Proof
3 ⇒ 1
Orientability to saturability:Let Γtot = (X ,∢tot) be an orientation of Γ; fix x ∈ X .
Let Y be a set disjoint from X , and ψ : O → Y injective
Set Ax , {O ∈ O(Γ) | x 6∈ O} and Xx , X ∪ ψ(Ax )
Let ιx : X → Xx be the insertion of X into Xx .
For every cycle O ∈ Ax , set xO , ψ(O)
For every edge [si , ei ] of O, let
∢x(si , xO , ei) if ∢tot(si , x , ei ), and∢x(si , ei , xO) otherwise.
Check that Γx = (Xx ,∢x) is a superstructure of Γ that hascx , ψ(O(Γ)) as a separator.
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Winding a shift-invariant PO to a CyO
0 γ0 0ε 0η κ 0 1−1κ α α2 3αα 1γ ε1 η1 κ 1 γ2 ε2 η2 2κ
−1λ 0β δ0 β1 β2λ 10ζ θ0 λ0 1δ 1ζ θ1 δ2 ζ2 θ2 λ 2 β3
κ
λ
θ
γ
εη
ζ
δ
β
α
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Contents
1 Cyclic Orders and Orientability
2 Windings
3 Separation and Saturation
4 Characterization of Orientability
5 Conclusion
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders
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Cyclic Orders and OrientabilityWindings
Separation and SaturationCharacterization of Orientability
Conclusion
Final Remarks
Outlook
Characterization provides link to
concurrent process dynamicsPartial order properties
The difficulty of cyclic orientation is lifted once a separator isknown
Future Work:
formalize abstraction mappingsDevelop morphisms, study category of CyOs vs PosetsInvestigate functorial properties (adjunctions etc ?)Correlate (?) with topological coverings etc.Things to explore : e.g. cyclic analogs of lattices ...
Stefan Haar INRIA / LSV (ENS Cachan + CNRS France) Cyclic Ordering through Partial Orders