Decreasing Diagrams: Two Labels Suffice

Transcription

1 Decreasing Diagrams: Two Labels Suffice Jan Willem Klop 1,2 Jörg Endrullis 1 Roy Overbeek 1 Presented by Bertram Felgenhauer 3 1 Vrije Universiteit Amsterdam, the Netherlands 2 Centrum Wiskunde & Informatica, the Netherlands 3 University of Innsbruck, Austria {j.w.klop, j.endrullis, r.overbeek}@vu.nl September 9, 216 5th International Workshop on Confluence (IWC 216) 1 / 15

2 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 DCR 2 Dependencies between properties (updated) 3 Further results: commutation 2 / 15

3 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) indexed ARS A = (A, { α } α I ) letting = α I α 3 / 15

4 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. a α b <α β β or c <β α or d <α <β <α <β Theorem 2 (Decreasing Diagrams De Bruijn [1] & Van Oostrom [4]) DCR CR 4 / 15

5 Cofinality property (CP) Definition 3 (Cofinal Reduction) Let A = (A, ) be an ARS. A finite or infinite reduction sequence b b 1 b 2 is cofinal in A if a A implies a b i 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 b b 1 b 2 that is cofinal in A {b a b}. a b 1... b i... 5 / 15

6 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 CR X 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 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. 6 / 15

7 Departing question: DCR hierarchy Definition 6 (DCR α ) For ordinals α, let DCR α 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 DCR α DCR β 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 DCR 2 Thus one easily obtains DCR = DCR 2 for the countable case. Our proof is an adaptation of Van Oostrom s proof for CP DCR [3, Proposition , p. 766]. 7 / 15

8 CP DCR 2 : 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 a b}, 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. ). n 5 n 4 main road n n 6 n 7 m m 1 m 2 n 1 n 2 n 3 m 3 m4 m 5 We focus on a single component ARS A = (A, ) satisfying CP, and let M denote a fixed acyclic main road in A. 8 / 15

9 CP DCR 2 : 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 b for every step a b with d(b ) = d(b). Remark: > exists by the Well-Ordering Theorem, or by construction for countable systems 9 / 15

10 CP DCR 2 : proof sketch (3/4) We now label steps a b with or 1 as follows: a b a b is on M or minimizing a 1 b a b is not on M and not minimizing 1 n 4 1 n 5 1 n 6 n 1 m 2 m m 1 1 n n n 3 n 7 m 3 m4 m 5 1 main road minimizing non-minimizing 1 / 15

11 CP DCR 2 : proof sketch (4/4) We show DCR. There are three cases for the peaks: a b a 1 b a 1 b 1 c d c d c d is deterministic there exist -labelled paths from any point to M any two points on M can be joined by -reductions 11 / 15

12 Dependencies between CR, DCR, CP and DCR 2 For countable ARSs, we now have: CP DCR 2 DCR CR And for uncountable systems: X??? CP DCR 2 DCR CR The implications DCR DCR 2 and CR DCR 2 are new open problems for the uncountable case. 12 / 15

13 Further results: commutation (1/2) Relation commutes with in an ARS (A,, ) if: a b c d DCR can be used to prove commutation (although it is incomplete): a α b <α β β or c <β α or d <α <β <α <β Question: do we have DCR = DCR 2 for commutation? 13 / 15

14 Further results: commutation (2/2) Theorem 1 (Lower DCR hierarchy for commutation) For commutation, DCR α DCR β for all ordinals α < β ω a 2 a 5 c 1 a 1 a 4 a 7 d 1 a 3 a 6 c 2 plan: extend system to require additional label assume c 2 d 1 c 1 with two steps n on one of the reductions peaks from a 1, a 4 and a 7 each contain a step n + 1 hence a 1 c 1 and a 1 c 2 with three steps n / 15

15 Further results: commutation (2/2) Theorem 1 (Lower DCR hierarchy for commutation) For commutation, DCR α DCR β for all ordinals α < β ω a 2 a 5 c 1 b 5 b 2 a 1 a 4 a 7 d 2 d 1 b 7 b 4 b 1 a 3 a 6 c 2 b 6 b 3 plan: extend system to require additional label assume c 2 d 1 c 1 with two steps n on one of the reductions peaks from a 1, a 4 and a 7 each contain a step n + 1 hence a 1 c 1 and a 1 c 2 with three steps n + 1 or: symmetric for c 1 d 2 c 2, b 1 to b 7 14 / 15

16 References [1] N.G. de Bruijn. A Note on Weak Diamond Properties. Memorandum 78 8, Eindhoven University of Technology, [2] J. Endrullis, J.W. Klop, and R. Overbeek. Decreasing Diagrams: Two Labels Suffice. In Proc. 5th International Workshop on Confluence (IWC 216), 216. [3] Terese. Term Rewriting Systems, volume 55 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 23. [4] V. van Oostrom. Confluence by Decreasing Diagrams. Theoretical Computer Science, 126(2):259 28, / 15