Pattern Avoidability with Involution Bastian Bischoff Dirk Nowotka WORDS 2011

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

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 A 2 (actually occurs in every binary word longer than four)

avoidability index: Generalization: functional dependencies between variables Example where is any mapping. We consider involutions here. Avoidability Pattern xxy unavoidable over A 2 (actually occurs in every binary word longer than four) avoidable over A 3 witness: 012021012102012 . . . (square-free Thue-word)

Generalization: functional dependencies between variables Example where is any mapping. We consider involutions here. Avoidability Pattern xxy unavoidable over A 2 (actually occurs in every binary word longer than four) avoidable over A 3 witness: 012021012102012 . . . (square-free Thue-word) avoidability index: V ( xxy ) = 3

Example where is any mapping. We consider involutions here. Avoidability Pattern xxy unavoidable over A 2 (actually occurs in every binary word longer than four) avoidable over A 3 witness: 012021012102012 . . . (square-free Thue-word) avoidability index: V ( xxy ) = 3 Generalization: functional dependencies between variables

We consider involutions here. Avoidability Pattern xxy unavoidable over A 2 (actually occurs in every binary word longer than four) avoidable over A 3 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.

Avoidability Pattern xxy unavoidable over A 2 (actually occurs in every binary word longer than four) avoidable over A 3 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.

patterns instance of 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 )

instance of 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 } ) ∗

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 )

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 011001 x ¯ xx for morphic with where ϑ ϑ (0) = 1 , ϑ (1) = 0 h ( x ) = 01 .

pattern is morphically (antimorphically) -avoidable, if there exists w such that for all morphic (antimorphic) involutions occurs in w no instance of morphic avoidance index of : minimal such that is morphically -avoidable if unavoidable analogously, antimorphic avoidance index of Avoidability with Involution . Observation . . x there exists an infinite word w such For all patterns p with both x and ¯ that no instance of p for an involution ϑ ̸ = id occurs in w . .

morphic avoidance index of : minimal such that is morphically -avoidable if unavoidable analogously, antimorphic avoidance index of Avoidability with Involution . Observation . . x there exists an infinite word w such For all patterns p with both x and ¯ that no instance of p for an involution ϑ ̸ = id occurs in w . . pattern p is morphically (antimorphically) k -avoidable, if there exists w ∈ A ω k such that for all morphic (antimorphic) involutions ϑ no instance of p occurs in w

if unavoidable analogously, antimorphic avoidance index of Avoidability with Involution . Observation . . x there exists an infinite word w such For all patterns p with both x and ¯ that no instance of p for an involution ϑ ̸ = id occurs in w . . pattern p is morphically (antimorphically) k -avoidable, if there exists w ∈ A ω k such that for all morphic (antimorphic) involutions ϑ no instance of p occurs in w morphic avoidance index V m ( p ) of p : minimal k such that p is morphically k -avoidable

analogously, antimorphic avoidance index of Avoidability with Involution . Observation . . x there exists an infinite word w such For all patterns p with both x and ¯ that no instance of p for an involution ϑ ̸ = id occurs in w . . pattern p is morphically (antimorphically) k -avoidable, if there exists w ∈ A ω k such that for all morphic (antimorphic) involutions ϑ no instance of p occurs in w morphic avoidance index V m ( p ) of p : minimal k such that p is morphically k -avoidable V m ( p ) = ∞ if p unavoidable

Avoidability with Involution . Observation . . x there exists an infinite word w such For all patterns p with both x and ¯ that no instance of p for an involution ϑ ̸ = id occurs in w . . pattern p is morphically (antimorphically) k -avoidable, if there exists w ∈ A ω k such that for all morphic (antimorphic) involutions ϑ no instance of p occurs in w morphic avoidance index V m ( p ) of p : minimal k such that p is morphically k -avoidable V m ( p ) = ∞ if p unavoidable analogously, antimorphic avoidance index V a ( p ) of p

with witness w Suppose or , do not occur in w , , , has to occur in w then but for with and Some Facts . Lemma . . V m ( x ¯ xx ) > 2 and V a ( x ¯ xx ) > 2 . .

do not occur in w , , , has to occur in w then but for with and Some Facts . Lemma . . V m ( x ¯ xx ) > 2 and V a ( x ¯ xx ) > 2 . . xx ) = 2 with witness w ∈ A ω Suppose V m ( x ¯ xx ) = 2 or V a ( x ¯ 2 ,

has to occur in w then but for with and Some Facts . Lemma . . V m ( x ¯ xx ) > 2 and V a ( x ¯ xx ) > 2 . . xx ) = 2 with witness w ∈ A ω Suppose V m ( x ¯ xx ) = 2 or V a ( x ¯ 2 , 000 , 111 , 010 , 101 do not occur in w

but for with and Some Facts . Lemma . . V m ( x ¯ xx ) > 2 and V a ( x ¯ xx ) > 2 . . xx ) = 2 with witness w ∈ A ω Suppose V m ( x ¯ xx ) = 2 or V a ( x ¯ 2 , 000 , 111 , 010 , 101 do not occur in w then 001100 has to occur in w

Some Facts . Lemma . . V m ( x ¯ xx ) > 2 and V a ( x ¯ xx ) > 2 . . xx ) = 2 with witness w ∈ A ω Suppose V m ( x ¯ xx ) = 2 or V a ( x ¯ 2 , 000 , 111 , 010 , 101 do not occur in w then 001100 has to occur in w but 00 ϑ (00)00 = 001100 for ϑ with ϑ (0) = 1 and ϑ (1) = 0

Consider morphic case. Thue-Morse word with subst. w v v w occurs in w suppose an instance of let (smaller instance individually checked) occurs in w w.l.o.g. occurs in , and hence, . then or . ; moreover implies cube in v if or prefix of , then implies cube in v if or suffix of , then contradiction (antimorphic case similar) Result . Theorem . . V m ( x ¯ xx ) = 3 and V a ( x ¯ xx ) = 3 . .

Thue-Morse word with subst. w v v w occurs in w suppose an instance of let (smaller instance individually checked) occurs in w w.l.o.g. occurs in , and hence, . then . or ; moreover implies cube in v if or prefix of , then implies cube in v if or suffix of , then contradiction (antimorphic case similar) Result . Theorem . . V m ( x ¯ xx ) = 3 and V a ( x ¯ xx ) = 3 . . Consider morphic case.

occurs in w suppose an instance of let (smaller instance individually checked) occurs in w w.l.o.g. occurs in , and hence, . then or . ; moreover implies cube in v if or prefix of , then implies cube in v if or suffix of , then contradiction (antimorphic case similar) Result . Theorem . . V m ( x ¯ xx ) = 3 and V a ( x ¯ xx ) = 3 . . Consider morphic case. Thue-Morse word v v = 0 1 1 0 1 0 0 1 1 0 . . .

occurs in w suppose an instance of let (smaller instance individually checked) occurs in w w.l.o.g. occurs in , and hence, . then . or ; moreover implies cube in v if or prefix of , then implies cube in v if or suffix of , then contradiction (antimorphic case similar) Result . Theorem . . V m ( x ¯ xx ) = 3 and V a ( 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 . . .

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