Decreasing Diagrams: Two Labels Suffice
Jan Willem Klop1,2
Jörg Endrullis1
Roy Overbeek1
Presented by Bertram Felgenhauer3
2
1
Vrije Universiteit Amsterdam, the Netherlands
Centrum Wiskunde & Informatica, the Netherlands
3
University of Innsbruck, Austria
{j.w.klop, j.endrullis, r.overbeek}@vu.nl
September 9, 2016
5th International Workshop on Confluence (IWC 2016)
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Overview
1
Preliminaries
Abstract reduction systems and confluence (CR)
Decreasing Church-Rosser (DCR) and decreasing diagrams
Cofinality property (CP)
Dependencies between CR, DCR and CP
2
Two labels suffice
Departing question: DCR hierarchy
Proof sketch for CP ⇒ DCR2
Dependencies between properties (updated)
3
Further results: commutation
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Abstract reduction systems and confluence (CR)
ARS A = (A, →) with → ⊆ A × A
A is countable if A is
(A, →) is confluent (CR) if
(a b ∧ a c) ⇒ ∃d ∈ A (b d ∧ c d)
S
indexed ARS A = (A, {→α }α∈I ) letting → = α∈I →α
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Decreasing Church-Rosser (DCR) and decreasing diagrams
Definition 1 (Decreasing Church–Rosser [4])
A = (A, →) is decreasing Church–Rosser (DCR) if it equals
B = (A, {→α }α∈I ) indexed by a well-founded partial order (I , <) such
that every peak c ←β a →α b can be joined decreasingly.
α
a
b
<α
β or ≡
β
<α ∪ <β
c
<β
α
or ≡
<α
∪ <β
d
Theorem 2 (Decreasing Diagrams – De Bruijn [1] & Van Oostrom [4])
DCR ⇒ CR
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Cofinality property (CP)
Definition 3 (Cofinal Reduction)
Let A = (A, →) be an ARS. A finite or infinite reduction sequence
b0 → b1 → b2 → · · · is cofinal in A if a ∈ A implies a bi for some i.
Definition 4 (Cofinality Property)
An ARS A = (A, →) has the cofinality property (CP) if for every a ∈ A,
there exists a reduction sequence a ≡ b0 → b1 → b2 → · · · that is cofinal
in A|{b | ab} .
a
b1
...
bi
...
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Dependencies between CR, DCR and CP
For countable ARSs, the relevant properties coincide:
CP
DCR
CR
For uncountable systems, the situation is as follows:
X
CP
DCR
X
CR
?
Example 5 (Counter-example to DCR ⇒ CP)
Let A be the set of finite subsets X of the line R. Consider the reduction
rule X → X ∪ {x} for x 6∈ X . The uncountable ARS (A, →) is DCR, but
not CP.
Whether CR ⇒ DCR for uncountable systems is a long-standing open
problem in abstract rewriting.
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Departing question: DCR hierarchy
Definition 6 (DCRα )
For ordinals α, let DCRα denote the class of ARSs that can be shown to
satisfy DCR using the label set {β | β < α}. (With < the usual order on
ordinals.)
Do we have strict inclusions DCRα ⊂ DCRβ for all α < β?
If not, what does the hierarchy look like?
Our main result:
Theorem 7 (Two Labels Suffice – Klop, Endrullis & Overbeek [2])
CP ⇒ DCR2
Thus one easily obtains DCR = DCR2 for the countable case.
Our proof is an adaptation of Van Oostrom’s proof for
CP ⇒ DCR [3, Proposition 14.2.30, p. 766].
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CP ⇒ DCR2 : proof sketch (1/4)
∗
Lemma 8 (CP ⇒ CP ↔ – Mano, 1993)
Let A = (A, →) be a confluent ARS and a ∈ A. If a rewrite sequence is
cofinal in A|{b | ab} , then it is also cofinal in A|{b | a↔∗ b} .
Thus CP implies that there exists a main road in every weakly connected
component (w.r.t. ↔∗ ).
n4
n1
n5
n3
m3
m2
m1
main road
n7
n0
n6
m0
n2
m4
m5
···
We focus on a single component ARS A = (A, →) satisfying CP, and let
M denote a fixed acyclic main road in A.
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CP ⇒ DCR2 : proof sketch (2/4)
Our labelling function presupposes the following notions.
d(a) is the distance of a to the main road M
> is a linear order on A
Definition 9 (Minimizing Step)
A step a → b is minimizing if
(i) d(a) = d(b) + 1 and
(ii) b 0 ≥ b for every step a → b 0 with d(b 0 ) = d(b).
Remark: > exists by the Well-Ordering Theorem, or by construction for
countable systems
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CP ⇒ DCR2 : proof sketch (3/4)
We now label steps a → b with 0 or 1 as follows:
a →0 b ⇐⇒ a → b is on M or minimizing
a →1 b ⇐⇒ a → b is not on M and not minimizing
1
0
n4
1
n1
n5
0
1
1
0
0
0
1
0
0
m1
0
0
main road
n3
1
minimizing
n7
n0
n6
m0
n2
1
m3
0
m2
non-minimizing
0
0
m4
0
m5
0
···
1
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CP ⇒ DCR2 : proof sketch (4/4)
We show DCR. There are three cases for the peaks:
a
0
a
b
≡
0
c
≡
d
1
a
b
0
0
c
0
1
b
1
d
0
c
0
d
→0 is deterministic
there exist 0-labelled paths from any point to M
any two points on M can be joined by 0-reductions
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Dependencies between CR, DCR, CP and DCR2
For countable ARSs, we now have:
DCR2
CP
DCR
CR
And for uncountable systems:
CP
X
?
DCR2
?
DCR
?
CR
The implications DCR ⇒ DCR2 and CR ⇒ DCR2 are new open
problems for the uncountable case.
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Further results: commutation (1/2)
Relation → commutes with
in an ARS (A, →,
) if:
a
c
b
d
DCR can be used to prove commutation (although it is incomplete):
α
a
b
<α
β or ≡
β
<α ∪ <β
c
<β
α
or ≡
<α
∪ <β
d
Question: do we have DCR = DCR2 for commutation?
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Further results: commutation (2/2)
Theorem 10 (Lower DCR hierarchy for commutation)
For commutation, DCRα ⊂ DCRβ for all ordinals α < β ≤ ω
a2
a1
a5
a4
c1
a7
a3
a6
d1
c2
plan: extend system to require additional label
assume c2 ∗ d1 ←∗ c1 with two steps ≥ n on one of the reductions
peaks from a1 , a4 and a7 each contain a step ≥ n + 1
hence a1
∗
c1 and a1 →∗ c2 with three steps ≥ n + 1
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Further results: commutation (2/2)
Theorem 10 (Lower DCR hierarchy for commutation)
For commutation, DCRα ⊂ DCRβ for all ordinals α < β ≤ ω
a2
a1
a5
a4
c1
a7
a3
d2 d1
a6
b5
b7
c2
b2
b4
b6
b1
b3
plan: extend system to require additional label
assume c2 ∗ d1 ←∗ c1 with two steps ≥ n on one of the reductions
peaks from a1 , a4 and a7 each contain a step ≥ n + 1
hence a1
∗
c1 and a1 →∗ c2 with three steps ≥ n + 1
or: symmetric for c1
∗
d2 ←∗ c2 , b1 to b7
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References
[1] N.G. de Bruijn.
A Note on Weak Diamond Properties.
Memorandum 78–08, Eindhoven University of Technology, 1978.
[2] J. Endrullis, J.W. Klop, and R. Overbeek.
Decreasing Diagrams: Two Labels Suffice.
In Proc. 5th International Workshop on Confluence (IWC 2016),
2016.
[3] Terese.
Term Rewriting Systems, volume 55 of Cambridge Tracts in
Theoretical Computer Science.
Cambridge University Press, 2003.
[4] V. van Oostrom.
Confluence by Decreasing Diagrams.
Theoretical Computer Science, 126(2):259–280, 1994.
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