Pattern Avoidability with Involution Bastian Bischoff Dirk Nowotka - - PowerPoint PPT Presentation

Pattern Avoidability with Involution

Bastian Bischoff Dirk Nowotka WORDS 2011

Avoidability

Pattern

xxy

unavoidable over (actually occurs in every binary word longer than four) avoidable over witness: (square-free Thue-word) avoidability index: Generalization: functional dependencies between variables Example where is any mapping. We consider involutions here.

Avoidability

Pattern

xxy

unavoidable over A2 (actually occurs in every binary word longer than four) avoidable over witness: (square-free Thue-word) avoidability index: Generalization: functional dependencies between variables Example where is any mapping. We consider involutions here.

Avoidability

Pattern

xxy

unavoidable over A2 (actually occurs in every binary word longer than four) avoidable over A3 witness:

012021012102012 . . .

(square-free Thue-word) avoidability index: Generalization: functional dependencies between variables Example where is any mapping. We consider involutions here.

Avoidability

Pattern

xxy

unavoidable over A2 (actually occurs in every binary word longer than four) avoidable over A3 witness:

012021012102012 . . .

(square-free Thue-word) avoidability index: V(xxy) = 3 Generalization: functional dependencies between variables Example where is any mapping. We consider involutions here.

Avoidability

Pattern

xxy

unavoidable over A2 (actually occurs in every binary word longer than four) avoidable over A3 witness:

012021012102012 . . .

(square-free Thue-word) avoidability index: V(xxy) = 3 Generalization: functional dependencies between variables Example where is any mapping. We consider involutions here.

Avoidability

Pattern

xxy

unavoidable over A2 (actually occurs in every binary word longer than four) avoidable over A3 witness:

012021012102012 . . .

(square-free Thue-word) avoidability index: V(xxy) = 3 Generalization: functional dependencies between variables Example

x f(x) y

where f is any mapping. We consider involutions here.

Avoidability

Pattern

xxy

unavoidable over A2 (actually occurs in every binary word longer than four) avoidable over A3 witness:

012021012102012 . . .

(square-free Thue-word) avoidability index: V(xxy) = 3 Generalization: functional dependencies between variables Example

x f(x) y

where f is any mapping. We consider involutions here.

Notation

involution ϑ ◦ ϑ = id

morphic ϑ(uv) = ϑ(u)ϑ(v) antimorphic ϑ(uv) = ϑ(v)ϑ(u)

patterns instance

• f pattern

for involution , if

there exists morphism such that (for all variables ) and

Example is an instance of for morphic with where

Notation

involution ϑ ◦ ϑ = id

morphic ϑ(uv) = ϑ(u)ϑ(v) antimorphic ϑ(uv) = ϑ(v)ϑ(u)

patterns P = (X ∪ {¯

x | x ∈ X})∗

instance

• f pattern

for involution , if

there exists morphism such that (for all variables ) and

Example is an instance of for morphic with where

Notation

involution ϑ ◦ ϑ = id

morphic ϑ(uv) = ϑ(u)ϑ(v) antimorphic ϑ(uv) = ϑ(v)ϑ(u)

patterns P = (X ∪ {¯

x | x ∈ X})∗

instance w of pattern p for involution ϑ, if

there exists morphism h such that h(¯

x) = ϑ(h(x)) (for all variables x) and w = h(p)

Example is an instance of for morphic with where

Notation

involution ϑ ◦ ϑ = id

morphic ϑ(uv) = ϑ(u)ϑ(v) antimorphic ϑ(uv) = ϑ(v)ϑ(u)

patterns P = (X ∪ {¯

x | x ∈ X})∗

instance w of pattern p for involution ϑ, if

there exists morphism h such that h(¯

x) = ϑ(h(x)) (for all variables x) and w = h(p)

Example

011001

is an instance of

x¯ xx

for morphic

ϑ

with

ϑ(0) = 1, ϑ(1) = 0

where

h(x) = 01.

Avoidability with Involution

.

Observation

. . . For all patterns p with both x and ¯

x there exists an infinite word w such

that no instance of p for an involution ϑ ̸= id occurs in w. pattern is morphically (antimorphically)

• avoidable, if

there exists w such that for all morphic (antimorphic) involutions no instance of

• ccurs in w

morphic avoidance index

• f

: minimal such that is morphically

• avoidable

if unavoidable analogously, antimorphic avoidance index

• f

Avoidability with Involution

.

Observation

. . . For all patterns p with both x and ¯

x there exists an infinite word w such

that no instance of p for an involution ϑ ̸= id occurs in w. pattern p is morphically (antimorphically) k-avoidable, if

there exists w ∈ Aω

k such that

for all morphic (antimorphic) involutions ϑ no instance of p occurs in w

morphic avoidance index

• f

: minimal such that is morphically

• avoidable

if unavoidable analogously, antimorphic avoidance index

• f

Avoidability with Involution

.

Observation

. . . For all patterns p with both x and ¯

x there exists an infinite word w such

that no instance of p for an involution ϑ ̸= id occurs in w. pattern p is morphically (antimorphically) k-avoidable, if

there exists w ∈ Aω

k such that

for all morphic (antimorphic) involutions ϑ no instance of p occurs in w

morphic avoidance index Vm(p) of p: minimal k such that p is morphically k-avoidable if unavoidable analogously, antimorphic avoidance index

• f

Avoidability with Involution

.

Observation

. . . For all patterns p with both x and ¯

x there exists an infinite word w such

that no instance of p for an involution ϑ ̸= id occurs in w. pattern p is morphically (antimorphically) k-avoidable, if

there exists w ∈ Aω

k such that

for all morphic (antimorphic) involutions ϑ no instance of p occurs in w

morphic avoidance index Vm(p) of p: minimal k such that p is morphically k-avoidable

Vm(p) = ∞ if p unavoidable

analogously, antimorphic avoidance index

• f

Avoidability with Involution

.

Observation

. . . For all patterns p with both x and ¯

x there exists an infinite word w such

that no instance of p for an involution ϑ ̸= id occurs in w. pattern p is morphically (antimorphically) k-avoidable, if

there exists w ∈ Aω

k such that

for all morphic (antimorphic) involutions ϑ no instance of p occurs in w

morphic avoidance index Vm(p) of p: minimal k such that p is morphically k-avoidable

Vm(p) = ∞ if p unavoidable

analogously, antimorphic avoidance index Va(p) of p

Some Facts

.

Lemma

. . .

Vm(x¯ xx) > 2

and

Va(x¯ xx) > 2.

Suppose

• r

with witness w , , , , do not occur in w then has to occur in w but for with and

Some Facts

.

Lemma

. . .

Vm(x¯ xx) > 2

and

Va(x¯ xx) > 2.

Suppose Vm(x¯

xx) = 2 or Va(x¯ xx) = 2 with witness w ∈ Aω

2 ,

, , , do not occur in w then has to occur in w but for with and

Some Facts

.

Lemma

. . .

Vm(x¯ xx) > 2

and

Va(x¯ xx) > 2.

Suppose Vm(x¯

xx) = 2 or Va(x¯ xx) = 2 with witness w ∈ Aω

2 ,

000, 111, 010, 101 do not occur in w

then has to occur in w but for with and

Some Facts

.

Lemma

. . .

Vm(x¯ xx) > 2

and

Va(x¯ xx) > 2.

Suppose Vm(x¯

xx) = 2 or Va(x¯ xx) = 2 with witness w ∈ Aω

2 ,

000, 111, 010, 101 do not occur in w

then 001100 has to occur in w but for with and

Some Facts

.

Lemma

. . .

Vm(x¯ xx) > 2

and

Va(x¯ xx) > 2.

Suppose Vm(x¯

xx) = 2 or Va(x¯ xx) = 2 with witness w ∈ Aω

2 ,

000, 111, 010, 101 do not occur in w

then 001100 has to occur in w but 00ϑ(00)00 = 001100 for ϑ with ϑ(0) = 1 and ϑ(1) = 0

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w

v v w

suppose an instance of

• ccurs in w

let (smaller instance individually checked) w.l.o.g.

• ccurs in

, and hence,

• ccurs in w

then . .

• r

; moreover if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w

v v w

suppose an instance of

• ccurs in w

let (smaller instance individually checked) w.l.o.g.

• ccurs in

, and hence,

• ccurs in w

then . .

• r

; moreover if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word v

v = 0 1 1 0 1 0 0 1 1 0 . . .

suppose an instance of

• ccurs in w

let (smaller instance individually checked) w.l.o.g.

• ccurs in

, and hence,

• ccurs in w

then . .

• r

; moreover if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of

• ccurs in w

let (smaller instance individually checked) w.l.o.g.

• ccurs in

, and hence,

• ccurs in w

then . .

• r

; moreover if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of x¯

xx occurs in w

let (smaller instance individually checked) w.l.o.g.

• ccurs in

, and hence,

• ccurs in w

then . .

• r

; moreover if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of x¯

xx occurs in w

let |h(x)| ≥ 7 (smaller instance individually checked) w.l.o.g.

• ccurs in

, and hence,

• ccurs in w

then . .

• r

; moreover if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of x¯

xx occurs in w

let |h(x)| ≥ 7 (smaller instance individually checked) w.l.o.g. 0021 occurs in h(x), and hence, ϑ(002) occurs in w then . .

• r

; moreover if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of x¯

xx occurs in w

let |h(x)| ≥ 7 (smaller instance individually checked) w.l.o.g. 0021 occurs in h(x), and hence, ϑ(002) occurs in w then ϑ(002) = 002 or ϑ(002) = 221 if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of x¯

xx occurs in w

let |h(x)| ≥ 7 (smaller instance individually checked) w.l.o.g. 0021 occurs in h(x), and hence, ϑ(002) occurs in w then ϑ(002) = 002 . .

• r ϑ(002) = 221

if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of x¯

xx occurs in w

let |h(x)| ≥ 7 (smaller instance individually checked) w.l.o.g. 0021 occurs in h(x), and hence, ϑ(002) occurs in w then ϑ(002) = 002 . .

• r ϑ(002) = 221 ; moreover |h(x)| = 4n

if

• r

prefix of , then implies cube in v if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of x¯

xx occurs in w

let |h(x)| ≥ 7 (smaller instance individually checked) w.l.o.g. 0021 occurs in h(x), and hence, ϑ(002) occurs in w then ϑ(002) = 002 . .

• r ϑ(002) = 221 ; moreover |h(x)| = 4n

if 021 or 221 prefix of h(x), then 0h(x¯

xx) implies cube in v

if

• r

suffix of , then implies cube in v contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of x¯

xx occurs in w

let |h(x)| ≥ 7 (smaller instance individually checked) w.l.o.g. 0021 occurs in h(x), and hence, ϑ(002) occurs in w then ϑ(002) = 002 . .

• r ϑ(002) = 221 ; moreover |h(x)| = 4n

if 021 or 221 prefix of h(x), then 0h(x¯

xx) implies cube in v

if 00 or 02 suffix of h(x), then h(x¯

xx)21 implies cube in v

contradiction (antimorphic case similar)

Result

.

Theorem

. . .

Vm(x¯ xx) = 3

and

Va(x¯ xx) = 3.

Consider morphic case. Thue-Morse word with subst. w = v[0 → 0021, 1 → 0221]

v = 0 1 1 0 1 0 0 1 1 0 . . . w = 0021 0221 0221 0021 0221 0021 0021 0221 0221 0021 . . .

suppose an instance of x¯

xx occurs in w

let |h(x)| ≥ 7 (smaller instance individually checked) w.l.o.g. 0021 occurs in h(x), and hence, ϑ(002) occurs in w then ϑ(002) = 002 . .

• r ϑ(002) = 221 ; moreover |h(x)| = 4n

if 021 or 221 prefix of h(x), then 0h(x¯

xx) implies cube in v

if 00 or 02 suffix of h(x), then h(x¯

xx)21 implies cube in v

contradiction (antimorphic case similar)

Stripped Patterns

.

Lemma

. . . Let p ∈ (X ∪ {¯

x})∗ contain both x and ¯

• x. Then

V(p[¯ x → x]) ≤ Vm(p) ≤ V(p[¯ x → y]) Va(p) ≤ V(p[¯ x → y])

with y new in p. Example and and

Stripped Patterns

.

Lemma

. . . Let p ∈ (X ∪ {¯

x})∗ contain both x and ¯

• x. Then

V(p[¯ x → x]) ≤ Vm(p) ≤ V(p[¯ x → y]) Va(p) ≤ V(p[¯ x → y])

with y new in p. Example

V(xxx) = 2

and

Vm(x¯ xx) = 3

and

V(xyx) = ∞

Stripped Patterns

.

Lemma

. . . Let p ∈ (X ∪ {¯

x})∗ contain both x and ¯

• x. Then

V(p[¯ x → x]) ≤ Vm(p) ≤ V(p[¯ x → y]) Va(p) ≤ V(p[¯ x → y])

with y new in p. Example

V(xxx) = 2

and

Vm(x¯ xx) = 3

and

V(xyx) = ∞ V(xxx) = 2 ≤ Vm(xx¯ x) ≤ V(xxy) = 3

Other Patterns

.

Lemma

. . .

Vm(xx¯ x) = Va(xx¯ x) = 3.

and by previous lemma Suppose, with witness w , both and do not occur in w, that is, w and

• ccurs in w for

id (morphic) or

and (antimorphic); contradiction

Other Patterns

.

Lemma

. . .

Vm(xx¯ x) = Va(xx¯ x) = 3. V(xxy) = 3 and Vm(xx¯ x) ≤ 3 by previous lemma

Suppose, with witness w , both and do not occur in w, that is, w and

• ccurs in w for

id (morphic) or

and (antimorphic); contradiction

Other Patterns

.

Lemma

. . .

Vm(xx¯ x) = Va(xx¯ x) = 3. V(xxy) = 3 and Vm(xx¯ x) ≤ 3 by previous lemma

Suppose, Vm(xx¯

x) = 2 with witness w ∈ Aω

2 ,

both and do not occur in w, that is, w and

• ccurs in w for

id (morphic) or

and (antimorphic); contradiction

Other Patterns

.

Lemma

. . .

Vm(xx¯ x) = Va(xx¯ x) = 3. V(xxy) = 3 and Vm(xx¯ x) ≤ 3 by previous lemma

Suppose, Vm(xx¯

x) = 2 with witness w ∈ Aω

2 ,

both 00 and 11 do not occur in w, that is, w = (01)ω and

• ccurs in w for

id (morphic) or

and (antimorphic); contradiction

Other Patterns

.

Lemma

. . .

Vm(xx¯ x) = Va(xx¯ x) = 3. V(xxy) = 3 and Vm(xx¯ x) ≤ 3 by previous lemma

Suppose, Vm(xx¯

x) = 2 with witness w ∈ Aω

2 ,

both 00 and 11 do not occur in w, that is, w = (01)ω and 0101ϑ(01) occurs in w for ϑ = id (morphic) or ϑ(0) = 1 and

ϑ(1) = 0 (antimorphic); contradiction

Complementary Patterns

.

Observation

. . . Let p and p′ be such that p′ is p with x and ¯

x switched for all variables x.

Then

Vm(p) = Vm(p′)

and

Va(p) = Va(p′) .

In particular

Complementary Patterns

.

Observation

. . . Let p and p′ be such that p′ is p with x and ¯

x switched for all variables x.

Then

Vm(p) = Vm(p′)

and

Va(p) = Va(p′) .

In particular

Vm(¯ xx¯ x) = Va(¯ xx¯ x) = Vm(¯ x¯ xx) = Va(¯ x¯ xx) = 3

In Summary

.

Theorem

. . . Let p ∈ {x, ¯

x}3, then Vm(p) = Va(p) = { 3

if p ∈ {x, ¯

x}3 \ {xxx, ¯ x¯ x¯ x} 2

• therwise.

.

Theorem (James Currie)

. . . Let and , then . .

Corollary

. . . Let , then if if

• therwise

In Summary

.

Theorem

. . . Let p ∈ {x, ¯

x}3, then Vm(p) = Va(p) = { 3

if p ∈ {x, ¯

x}3 \ {xxx, ¯ x¯ x¯ x} 2

• therwise.

.

Theorem (James Currie)

. . . Let p ∈ {x, ¯

x}∗ and |p| ≥ 4, then Vm(p) = Va(p) = 2.

.

Corollary

. . . Let , then if if

• therwise

In Summary

.

Theorem

. . . Let p ∈ {x, ¯

x}3, then Vm(p) = Va(p) = { 3

if p ∈ {x, ¯

x}3 \ {xxx, ¯ x¯ x¯ x} 2

• therwise.

.

Theorem (James Currie)

. . . Let p ∈ {x, ¯

x}∗ and |p| ≥ 4, then Vm(p) = Va(p) = 2.

.

Corollary

. . . Let p ∈ {x, ¯

x}∗, then Vm(p) = Va(p) =      ∞

if p ∈ {x, ¯

x, x¯ x, ¯ xx} 3

if p ∈ {x, ¯

x}3 \ {xxx, ¯ x¯ x¯ x} 2

• therwise.

— End of Talk —