Unique Normal Forms in Infinitary Weakly Orthogonal Term Rewriting - - PowerPoint PPT Presentation

▶

Nov 28, 2022 194 likes •1.03k views

Counterexample in Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Unique Normal Forms in Infinitary Weakly Orthogonal Term Rewriting Jrg Endrullis Clemens

SLIDE 1

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Unique Normal Forms in Infinitary Weakly Orthogonal Term Rewriting

Jörg Endrullis⋆ Clemens Grabmayer△ Dimitri Hendriks⋆ Jan Willem Klop⋆ Vincent van Oostrom△

△) Universiteit Utrecht ⋆) Vrije Universiteit Amsterdam

RTA 2010, Edinburgh, UK July 11–13, 2010

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 2

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Weakly orthogonal vs. orthogonal

Weakly orthogonal (first-/higher-order) rewrite systems:

◮ definition: ‘harmless’ weakening of orthogonality ◮ for finitary TRSs: most ‘nice’ properties of orthogonal systems

are preserved

◮ but: new concepts, and non-trivial adaptations are needed

In this paper we:

◮ investigate infinitary weakly orthogonal rewrite systems ◮ show that uniqueness of infinitary normal forms fails in contrast

to orthogonal systems

◮ explain how this failure can be repaired

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 3

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Weakly orthogonal vs. orthogonal

Weakly orthogonal (first-/higher-order) rewrite systems:

◮ definition: ‘harmless’ weakening of orthogonality ◮ for finitary TRSs: most ‘nice’ properties of orthogonal systems

are preserved

◮ but: new concepts, and non-trivial adaptations are needed

In this paper we:

◮ investigate infinitary weakly orthogonal rewrite systems ◮ show that uniqueness of infinitary normal forms fails in contrast

to orthogonal systems

◮ explain how this failure can be repaired

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 4

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Overview

◮ Definitions: weakly orthogonal, UN∞ ◮ Counterexample to UN∞ for weakly orthogonal TRSs ◮ Counterexample to UN∞ for λ∞βη ◮ Restoring infinitary confluence ◮ Diamond and triangle properties for developments

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 5

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Weakly orthogonal

Weakly orthogonal (first-/higher-order) systems:

◮ left-linear ◮ all critical pairs are trivial.

Examples.

◮ Successor/Predecessor TRS:

P(S(x)) → x S(P(x)) → x with critical pairs: S(x) ← S(P(S(x))) → S(x) P(x) ← P(S(P(x))) → P(x)

◮ Parallel-Or TRS (‘almost orthogonal’):

por(true, x) → true por(x, true) → true por(false, false) → false

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 6

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Weakly orthogonal

Weakly orthogonal (first-/higher-order) systems:

◮ left-linear ◮ all critical pairs are trivial.

Examples.

◮ Successor/Predecessor TRS:

P(S(x)) → x S(P(x)) → x with critical pairs: S(x) ← S(P(S(x))) → S(x) P(x) ← P(S(P(x))) → P(x)

◮ Parallel-Or TRS (‘almost orthogonal’):

por(true, x) → true por(x, true) → true por(false, false) → false

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 7

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Weakly orthogonal

Weakly orthogonal (first-/higher-order) systems:

◮ left-linear ◮ all critical pairs are trivial.

Examples.

◮ Successor/Predecessor TRS:

P(S(x)) → x S(P(x)) → x with critical pairs: S(x) ← S(P(S(x))) → S(x) P(x) ← P(S(P(x))) → P(x)

◮ Parallel-Or TRS (‘almost orthogonal’):

por(true, x) → true por(x, true) → true por(false, false) → false

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 8

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Weakly orthogonal

Weakly orthogonal (first-/higher-order) systems:

◮ left-linear ◮ all critical pairs are trivial.

Examples.

◮ Successor/Predecessor TRS:

P(S(x)) → x S(P(x)) → x with critical pairs: S(x) ← S(P(S(x))) → S(x) P(x) ← P(S(P(x))) → P(x)

◮ Parallel-Or TRS (‘almost orthogonal’):

por(true, x) → true por(x, true) → true por(false, false) → false

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 9

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Weakly orthogonal

Weakly orthogonal (first-/higher-order) systems:

◮ left-linear ◮ all critical pairs are trivial.

Examples.

◮ Successor/Predecessor TRS:

P(S(x)) → x S(P(x)) → x with critical pairs: S(x) ← S(P(S(x))) → S(x) P(x) ← P(S(P(x))) → P(x)

◮ Parallel-Or TRS (‘almost orthogonal’):

por(true, x) → true por(x, true) → true por(false, false) → false

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 10

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Weakly orthogonal

Weakly orthogonal (first-/higher-order) systems:

◮ left-linear ◮ all critical pairs are trivial.

Examples.

◮ Successor/Predecessor TRS:

P(S(x)) → x S(P(x)) → x with critical pairs: S(x) ← S(P(S(x))) → S(x) P(x) ← P(S(P(x))) → P(x)

◮ Parallel-Or TRS (‘almost orthogonal’):

por(true, x) → true por(x, true) → true por(false, false) → false

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 11

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Weakly orthogonal

Weakly orthogonal (first-/higher-order) systems:

◮ left-linear ◮ all critical pairs are trivial.

Examples.

◮ Successor/Predecessor TRS:

P(S(x)) → x S(P(x)) → x with critical pairs: S(x) ← S(P(S(x))) → S(x) P(x) ← P(S(P(x))) → P(x)

◮ Parallel-Or TRS (‘almost orthogonal’):

por(true, x) → true por(x, true) → true por(false, false) → false

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 12

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

CR∞ en UN∞ (definitions). Situation in OTRSs

◮ CR∞:

t1 և և t ։ ։ t2 = ⇒ ∃s. t1 ։ ։ s և և t2

◮ UN∞:

t1 և և t ։ ։ t2 ∧ t1, t2 normal forms = ⇒ t1 = t2

◮ SN∞:

all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known):

◮ SN∞ =

⇒ CR∞, and CR∞ = ⇒ UN∞.

◮ CR∞ fails (Kennaway).

◮ But for non-collapsing TRSs: CR∞ holds.

◮ UN∞ holds (Kennaway/Klop).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 13

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

CR∞ en UN∞ (definitions). Situation in OTRSs

◮ CR∞:

t1 և և t ։ ։ t2 = ⇒ ∃s. t1 ։ ։ s և և t2

◮ UN∞:

t1 և և t ։ ։ t2 ∧ t1, t2 normal forms = ⇒ t1 = t2

◮ SN∞:

all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known):

◮ SN∞ =

⇒ CR∞, and CR∞ = ⇒ UN∞.

◮ CR∞ fails (Kennaway).

◮ But for non-collapsing TRSs: CR∞ holds.

◮ UN∞ holds (Kennaway/Klop).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 14

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

CR∞ en UN∞ (definitions). Situation in OTRSs

◮ CR∞:

t1 և և t ։ ։ t2 = ⇒ ∃s. t1 ։ ։ s և և t2

◮ UN∞:

t1 և և t ։ ։ t2 ∧ t1, t2 normal forms = ⇒ t1 = t2

◮ SN∞:

all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known):

◮ SN∞ =

⇒ CR∞, and CR∞ = ⇒ UN∞.

◮ CR∞ fails (Kennaway).

◮ But for non-collapsing TRSs: CR∞ holds.

◮ UN∞ holds (Kennaway/Klop).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 15

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

CR∞ en UN∞ (definitions). Situation in OTRSs

◮ CR∞:

t1 և և t ։ ։ t2 = ⇒ ∃s. t1 ։ ։ s և և t2

◮ UN∞:

t1 և և t ։ ։ t2 ∧ t1, t2 normal forms = ⇒ t1 = t2

◮ SN∞:

all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known):

◮ SN∞ =

⇒ CR∞, and CR∞ = ⇒ UN∞.

◮ CR∞ fails (Kennaway).

◮ But for non-collapsing TRSs: CR∞ holds.

◮ UN∞ holds (Kennaway/Klop).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 16

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

CR∞ en UN∞ (definitions). Situation in OTRSs

◮ CR∞:

t1 և և t ։ ։ t2 = ⇒ ∃s. t1 ։ ։ s և և t2

◮ UN∞:

t1 և և t ։ ։ t2 ∧ t1, t2 normal forms = ⇒ t1 = t2

◮ SN∞:

all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known):

◮ SN∞ =

⇒ CR∞, and CR∞ = ⇒ UN∞.

◮ CR∞ fails (Kennaway).

◮ But for non-collapsing TRSs: CR∞ holds.

◮ UN∞ holds (Kennaway/Klop).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 17

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

CR∞ en UN∞ (definitions). Situation in OTRSs

◮ CR∞:

t1 և և t ։ ։ t2 = ⇒ ∃s. t1 ։ ։ s և և t2

◮ UN∞:

t1 և և t ։ ։ t2 ∧ t1, t2 normal forms = ⇒ t1 = t2

◮ SN∞:

all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known):

◮ SN∞ =

⇒ CR∞, and CR∞ = ⇒ UN∞.

◮ CR∞ fails (Kennaway).

◮ But for non-collapsing TRSs: CR∞ holds.

◮ UN∞ holds (Kennaway/Klop).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 18

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

CR∞ en UN∞ (definitions). Situation in OTRSs

◮ CR∞:

t1 և և t ։ ։ t2 = ⇒ ∃s. t1 ։ ։ s և և t2

◮ UN∞:

t1 և և t ։ ։ t2 ∧ t1, t2 normal forms = ⇒ t1 = t2

◮ SN∞:

all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known):

◮ SN∞ =

⇒ CR∞, and CR∞ = ⇒ UN∞.

◮ CR∞ fails (Kennaway).

◮ But for non-collapsing TRSs: CR∞ holds.

◮ UN∞ holds (Kennaway/Klop).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 19

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

31 A(x) x B(x) x C A(B(C))

A(B(C)) A(C) B(C) A(A(B(C))) B(A(B(C))) A(A(C)) B(B(C)) A(A(A(B(C)))) B(B(A(B(C)))) A(A(A(C))) B(B(B(C))) A B

......

......

ABC ABABC ABABABC ABABABABAB... A B

(b)

Failure of infinitary confluence

not CR

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 20

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Overview

1. Counterexample to UN∞ for weakly orthogonal iTRSs
2. Counterexample to UN∞ in λ∞βη
3. Restoring infinitary confluence
4. Diamond and triangle properties for developments
5. Summary

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 21

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: P(S(x)) → x S(P(x)) → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly:

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 22

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly:

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 23

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly:

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 24

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly:

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 25

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly:

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 26

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly:

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 27

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly:

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 28

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly:

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 29

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly:

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 30

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly: ψ = P SS PPP SSSS PPPPP SSSSSS . . .

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 31

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly: ψ → P S PP SSSS PPPPP SSSSSS . . .

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 32

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly: ψ ։ P P SSSS PPPPP SSSSSS . . .

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 33

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly: ψ ։ P P SSS PPPP SSSSSS . . .

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 34

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly: ψ ։ P P SS PPP SSSSSS . . .

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 35

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly: ψ ։ P P S PP SSSSSS . . .

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 36

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly: ψ ։ P P P SSSSSS . . .

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 37

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

In the Successor/Predecessor TRS: PS → x SP → x with the normal forms Sω = SSS . . . and Pω = PPP . . . we consider: ψ = P1 S2 P3 S4 P5 S6 . . . = P SS PPP SSSS PPPPP SSSSSS . . . We find: ψ = P SS PPP SSSS PPPPP SSSSSS . . . → S PPP SSSS PPPPP SSSSSS . . . → S PP SSS PPPPP SSSSSS . . . → SP SS PPPPP SSSSSS . . . → S S PPPPP SSSSSS . . . ։ ։ S S S S S S . . . = Sω And similarly: ψ ։ ։ P P P P P . . . = Pω

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 38

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

P SP S S P P P P P S S P S S P P P S S P P P S

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 39

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails weakly-ortho iTRS

sum(w, n) +∞ −∞

Graph for the oscillating PS-word ψ = P1 S2 P3 . . . .

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 40

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Overview

1. Counterexample to UN∞ for weakly orthogonal iTRSs
2. Counterexample to UN∞ in λ∞βη
3. Restoring infinitary confluence
4. Diamond and triangle properties for developments
5. Summary

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 41

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

λ∞βη

Terms of λ∞βη: the (potentially) infinite λ-terms in Ter∞(λ) The rewrite rules of λ∞βη are: (λx.M)N

→ M[x:=N] λx.Mx

→ M (x not free in M) λ∞βη is weakly orthogonal, since the critical pairs are trivial: Mx

← (λx.Mx)x

→ Mx (x not free in M) λx.M[y:=x]

← λx.(λy.M)x

→ λy.M (x not free in λy.M)

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 42

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

P SP S S P P P P P S S P S S P P P S S P P P S

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 43

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

SP P S S P S S P P P S S S S P P P P P P P P S

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 44

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η @ @ λ λ λ λ @ @ @ @ λ λ λ λ @ @ @ @ @ @

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 45

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η @ @ @ @ @ @ @ @ @ @ @ @ λ λ λ λ λ λ λ λ

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 46

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η @ @ @ @ @ @ @ @ @ @ @ @ λ λ λ λ λ λ λ λ

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 47

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η @ @ @ @ @ @ @ @ @ @ @ @ λ λ λ λ λ λ λ λ

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 48

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η @ @ @ @ @ @ @ @ @ @ @ @ λ λ λ λ λ λ λ λ

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 49

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η @ @ @ @ @ @ @ @ @ @ @ @ λ λ λ λ λ λ λ λ

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 50

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η @ @ @ x0 @ x0 @ x0 @ @ @ @ @ x0 @ x0 @ x0 λ λ λ λ λ λ λ λ

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 51

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η @ @ @ x0 @ x0 @ x0 x1 λ @ x1 x1 λ @ x1 @ @ x1 λ x1 λ @ x0 @ x0 @ x0 λ λ λ λ

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 52

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η x2 λ @ x2 @ @ x0 @ x0 @ x0 x1 λ @ x1 x1 λ @ x1 @ x2 x2 λ @ x1 λ x1 λ @ x0 @ x0 @ x0 λ λ

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 53

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

β η x2 λ @ x2 λx3 x3 @ @ x0 @ x0 @ x0 x1 λ @ x1 x1 λ @ x1 @ x2 x2 λ @ x1 λ x1 λ @ x0 @ x0 @ x0 λ

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 54

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Counterexample: UN∞ fails in λ∞βη

β η x2 λ @ x2 λx3 x3 @ @ x0 @ x0 @ x0 x1 λ @ x1 x1 λ @ x1 @ x2 x2 λ @ x4 λx4 x1 λ x1 λ @ x0 @ x0 @ x0

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 55

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Translating the S-P-example to λ∞βη

_ : {P, S}ω → Ter∞(λ) defined by:

◮ w = w0; ◮ for all w ∈ {P, S}ω, and i ∈ Z:

Pwi = wi−1 xi Swi = λxi+1.wi+1 Lemma PSw (λxi.wi) xi w wi _i PS _i β SPw λxi+1.wi xi+1 w wi _i SP _i η

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 56

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Contrast with λ∞β

We saw for λ∞βη:

◮ UN∞ fails ◮ Consequently: CR∞ fails

However for λ∞β it holds:

◮ CR∞ fails ◮ But: UN∞ holds!

Due to this, λ∞β is important for the model theory of λ-calculus: for several models equality is captured by λ∞β-convertibility:

◮ Böhm Trees ◮ Lévy–Longo Trees ◮ Berarducci Trees Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 57

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Contrast with λ∞β

We saw for λ∞βη:

◮ UN∞ fails ◮ Consequently: CR∞ fails

However for λ∞β it holds:

◮ CR∞ fails ◮ But: UN∞ holds!

Due to this, λ∞β is important for the model theory of λ-calculus: for several models equality is captured by λ∞β-convertibility:

◮ Böhm Trees ◮ Lévy–Longo Trees ◮ Berarducci Trees Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 58

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Contrast with λ∞β

We saw for λ∞βη:

◮ UN∞ fails ◮ Consequently: CR∞ fails

However for λ∞β it holds:

◮ CR∞ fails ◮ But: UN∞ holds!

Due to this, λ∞β is important for the model theory of λ-calculus: for several models equality is captured by λ∞β-convertibility:

◮ Böhm Trees ◮ Lévy–Longo Trees ◮ Berarducci Trees Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 59

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Overview

1. Counterexample to UN∞ for weakly orthogonal iTRSs
2. Counterexample to UN∞ in λ∞βη
3. Restoring infinitary confluence
4. Diamond and triangle properties for developments
5. Summary

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 60

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Restoring infinitary confluence (preview)

Theorem Weakly orthogonal TRSs without collapsing rules are inf. confluent. Proof. s s1 t1 s2 t2 ≥ d ≥ d > d > d

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 61

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Restoring infinitary confluence (preview)

Theorem Weakly orthogonal TRSs without collapsing rules are inf. confluent. Proof. s s1 t1 s2 t2 ≥ d ≥ d > d > d

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 62

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Restoring infinitary confluence (preview)

Theorem Weakly orthogonal TRSs without collapsing rules are inf. confluent. Proof. s s1 t1 s2 t2 s′ ≥ d ≥ d > d > d ≥ d ≥ d finitary diagram

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 63

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Restoring infinitary confluence (preview)

Theorem Weakly orthogonal TRSs without collapsing rules are inf. confluent. Proof. s s1 t1 s2 t2 s′ t′

≥ d ≥ d > d > d ≥ d ≥ d > d ≥ d finitary diagram PML∞

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 64

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Restoring infinitary confluence (preview)

Theorem Weakly orthogonal TRSs without collapsing rules are inf. confluent. Proof. s s1 t1 s2 t2 s′ t′

t′

≥ d ≥ d > d > d ≥ d ≥ d > d > d ≥ d ≥ d finitary diagram PML∞ PML∞

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 65

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Restoring infinitary confluence (preview)

Theorem Weakly orthogonal TRSs without collapsing rules are inf. confluent. Proof. s s1 t1 s2 t2 s′ t′

t′

u ≥ d ≥ d > d > d ≥ d ≥ d > d > d ≥ d ≥ d finitary diagram PML∞ PML∞ repeat construction with d + 1

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 66

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Orthogonalization (of parallel steps)

Proposition For parallel steps φ : s − → t1 and ψ : s − → t2 in a w-o TRS there exists orthogonal steps φ′ and ψ′ such that φ′ : s − → t1 and ψ′ : s − → t2 (the pair φ′, ψ′ is an orthogonalization of φ and ψ). Proof. In case of overlaps, we replace the outer redex with the inner one (by weak orthogonality overlapping redexes have the same effect).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 67

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Orthogonalization (of parallel steps)

Proposition For parallel steps φ : s − → t1 and ψ : s − → t2 in a w-o TRS there exists orthogonal steps φ′ and ψ′ such that φ′ : s − → t1 and ψ′ : s − → t2 (the pair φ′, ψ′ is an orthogonalization of φ and ψ). Proof. In case of overlaps, we replace the outer redex with the inner one (by weak orthogonality overlapping redexes have the same effect).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 68

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Orthogonalization (of parallel steps)

Proposition For parallel steps φ : s − → t1 and ψ : s − → t2 in a w-o TRS there exists orthogonal steps φ′ and ψ′ such that φ′ : s − → t1 and ψ′ : s − → t2 (the pair φ′, ψ′ is an orthogonalization of φ and ψ). Proof. In case of overlaps, we replace the outer redex with the inner one (by weak orthogonality overlapping redexes have the same effect).

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 69

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Infinitary Parallel Moves Lemma PML∞

Using additionally a:

◮ refined compression lemma (preservation of min. depth of steps)

we show: Lemma Let R be a non-collapsing weakly orthogonal TRS. Then: s t1 t2 u ≥ dκ ≥ dξ ≥ min(dκ, dξ + 1) ≥ min(dξ, dκ + 1)

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 70

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Restoring infinitary confluence

Theorem Weakly orthogonal TRSs without collapsing rules are inf. confluent. Proof. s s1 t1 s2 t2 s′ t′

t′

u ≥ d ≥ d > d > d ≥ d ≥ d > d > d ≥ d ≥ d finitary diagram PML∞ PML∞ repeat construction with d + 1

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 71

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Overview

1. Counterexample to UN∞ for weakly orthogonal iTRSs
2. Counterexample to UN∞ in λ∞βη
3. Restoring infinitary confluence
4. Diamond and triangle properties for developments
5. Summary

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 72

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Diamond and triangle properties for developments

Definition A binary relation → on A has:

◮ the diamond property if: ← · →

⊆ → · ← ;

◮ the triangle property if:

∀a ∈ A. ∃a• ∈ A. a → a• ∧ (∀b ∈ A. a → b ⇒ b → a•) . a b a•

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 73

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Diamond and triangle properties for developments

Definition A binary relation → on A has:

◮ the diamond property if: ← · →

⊆ → · ← ;

◮ the triangle property if:

∀a ∈ A. ∃a• ∈ A. a → a• ∧ (∀b ∈ A. a → b ⇒ b → a•) . a b a•

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 74

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Diamond and triangle properties for developments

Definition A binary relation → on A has:

◮ the diamond property if: ← · →

⊆ → · ← ;

◮ the triangle property if:

∀a ∈ A. ∃a• ∈ A. a → a• ∧ (∀b ∈ A. a → b ⇒ b → a•) . a b a•

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 75

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Diamond and triangle properties for developments

Definition A binary relation → on A has:

◮ the diamond property if: ← · →

⊆ → · ← ;

◮ the triangle property if:

∀a ∈ A. ∃a• ∈ A. a → a• ∧ (∀b ∈ A. a → b ⇒ b → a•) . a b a•

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 76

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Diamond and triangle properties for developments

Definition A binary relation → on A has:

◮ the diamond property if: ← · →

⊆ → · ← ;

◮ the triangle property if:

∀a ∈ A. ∃a• ∈ A. a → a• ∧ (∀b ∈ A. a → b ⇒ b → a•) . a a• b

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 77

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Diamond and triangle properties for developments

Definition A binary relation → on A has:

◮ the diamond property if: ← · →

⊆ → · ← ;

◮ the triangle property if:

∀a ∈ A. ∃a• ∈ A. a → a• ∧ (∀b ∈ A. a → b ⇒ b → a•) . a b a•

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 78

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Diamond and triangle properties for developments

Theorem For every weakly orthogonal TRS without collapsing rules, for infinitary developments there hold:

the diamond property;

the triangle property. Our proof proceeds by:

◮ refining an earlier cluster analysis (I-clusters and Y-clusters) from

the finite case;

◮ a top-down orthogonalization algorithm.

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 79

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Diamond and triangle properties for developments

Theorem For every weakly orthogonal TRS without collapsing rules, for infinitary developments there hold:

the diamond property;

the triangle property. Our proof proceeds by:

◮ refining an earlier cluster analysis (I-clusters and Y-clusters) from

the finite case;

◮ a top-down orthogonalization algorithm.

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 80

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Overview

1. Counterexample to UN∞ for weakly orthogonal iTRSs
2. Counterexample to UN∞ in λ∞βη
3. Restoring infinitary confluence
4. Diamond and triangle properties for developments
5. Summary

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 81

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Summary

◮ Counterexample to UN∞/CR∞ for weakly orthogonal TRSs ◮ By translation: counterexample to UN∞/CR∞ for λ∞βη ◮ Restoring CR∞ (hence UN∞) for non-collapsing w-o TRSs ◮ Diamond and triangle properties for developments in

non-collapsing w-o TRSs

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs

SLIDE 82

Introduction Counterexample in w-o iTRSs Counterexample in λ∞βη Restoring infinitary confluence Diamond and triangle properties Summary

Summary

yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes no no no no no no no no no no no no yes no ? ? yes yes yes yes yes no no no yes no yes yes yes yes yes PML CR UN NF PML∞ CR∞ UN∞ NF∞ WOCRS λβη fe-OCRS λβ 1c-WOTRS nc-WOTRS WOTRS OTRS finitary infinitary higher-order first-order

Endrullis, Grabmayer, Hendriks, Klop, van Oostrom UN∞ in weakly-orthogonal iTRSs