An infinte word w avoids a pattern p with the involution t if there is no
substitution for the variables in p and no involution t such that the resulting
word is a factor of w. We investigate the avoidance of patterns with respect to
the size of the alphabet. For example, it is shown that the pattern a t(a) a
can be avoided over three letters but not two letters, whereas it is well known
that a a a is avoidable over two letters.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Bischoff, Bastian

Nowotka, Dirk

Electronic Proceedings in Theoretical Computer Science

English

arXiv

An infinte word w avoids a pattern p with the involution t if there is no substitution for the variables in p and no involution t such that the resulting word is a factor of w. We investigate the avoidance of patterns with respect to the size of the alphabet. For example, it is shown that the pattern a t(a) a can be avoided over three letters but not two letters, whereas it is well known that a a a is avoidable over two letters

Bastian Bischoff

Dirk Nowotka

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P. Ambrož, Š. Holub and Z. Masáková (Eds.):8th International Conference WORDS 2011EPTCS 63, 2011, pp. 65–70, doi:10.4204/EPTCS.63.10c© B. Bischoff & D. NowotkaThis work is licensed under theCreative Commons Attribution License.Pattern Avoidability with InvolutionBastian Bischoff Dirk Nowotka∗Institute for Formal Methods in Computer ScienceUniversität Stuttgart, Germanybastian.bischoff@googlemail.com nowotka@fmi.uni-stuttgart.deAn infinte wordw avoids a patternp with the involutionθ if there is no substitution for the variablesin p and no involutionθ such that the resulting word is a factor ofw. We investigate the avoidance ofpatterns with respect to the size of the alphabet. For example, it is shown that the patternα θ (α)αcan be avoided over three letters but not two letters, whereas it is well known thatα α α is avoidableover two letters.1 IntroductionThe avoidability of patterns in infinite words is an old area of interest with a first systematic study goingback to Thue [5, 6]. This field includes rediscoveries and stuies by many authors over the last onehundred years; see for example [2] and [1] for surveys. In this article, we are concerned with a variationof the theme by considering avoidable patterns with involuti n. An involution θ is a mapping suchthat θ2 is the identity. We consider morphic, whereθ(uv) = θ(u)θ(v), and antimorphic involutions,whereθ(uv) = θ(v)θ(u). The subject of this article draws quite some motivation from applications inbiology where the Watson-Crick complement corresponds to an antimorphic involution in our case. Ourconsiderations are more general, however, by considering any alphabet size and also morphic involutions.During the review phase of this article, James Currie [3] presented a solution for all those patternsunder involution in{α ,θ(α)}∗ that we do not consider here, which leads to a characterization of theavoidance index for all unary patterns under involution.2 PreliminariesOur notation is guided by what is commonly found in the literau e, see for example the first chapterof [4] as a reference. LetΣ be a finite alphabet oflettersandΣ∗ denote allfiniteandΣω denote all (right-)infinite words overΣ. Let ε denote the empty word. Letters are usually denoted bya, , or c, and wordsoverΣ are usually denoted byu, v, or w in this paper. Thei-th letter of a wordw is denoted byw[i], thatis, w= w[1]w[2] · · ·w[n] if w is finite, and the lengthn of w is denoted by|w| as usual.BesidesΣ we need another finite setE of symbols. The elements ofE are calledvariablesand weusually denote them byα , β , or γ . Words inE∗ are calledpatterns. For exampleαβα ∈ E∗ is a patternconsisting of the variablesα andβ in E. We assign to every pattern ap ttern languageover the alphabetΣ. This language contains every word, that can be generated bysubstituting all variables in the patternby non-empty words inΣ∗. For example the pattern language of the patternαα over Σ = {a,b} is{aa,bb,aaaa,abab,baba,bbbb, . . . }.We say that a wordw avoidsa pattern, if no factor ofw exists, that is in the pattern language. On theother hand, if a factor ofw is an element of the pattern language, we sayw containsthe pattern. If for a∗This work has been supported by the DFG Heisenberg grant 582013.66 Pattern Avoidability with Involutiongiven patterne and an alphabetΣ with k elements a wordw∈ Σω exists that avoidse, then we say thateis k-avoidable. Otherwise we calle k-unavoidable. We callk∈ N theavoidance indexV (e) of a patterne∈ E∗, if e is k-avoidable andk is minimal. If no suchk exists, we defineV (e) = ∞.Let f : {a,b}∗ → {a,b}∗ with a 7→ ab andb 7→ ba. The fixpoint t = limk→∞ f k(a) exists and iscalledThue–Morse word. The following result is a classical one.Theorem 1([5, 6]). The Thue–Morse word avoids the patternsααα andαβαβα .3 Patterns with InvolutionFor introducing patterns with involution, we extend the setof pattern variablesE by addingθ(α) for allvariablesα ∈ E and some involutionθ . For the rest of the article, we will stick to this definition of E.Given a morphic or antimorphic involution, we build the corresponding pattern language by replacingthe variables by non-empty words and, for variables of the form θ(α), by applying the involution afterthe substitution.For example, letθ be the morphic involution witha 7→ b andb 7→ a over Σ = {a,b}, and let thepattern beα θ(α). We get the pattern language{ab,ba,aabb,abba,baab,bbaa, . . . }. Every word in{a,b}ω \ (aω ∪bω) contains the patternα θ(α) for the morphic involutionθ with a 7→ b andb 7→ a.Observation 2. Let θ be a morphic or antimorphic involution and not the identity or reversal mapping.Then every pattern, that contains variables of theα andθ(α), is avoidable.Indeed, sinceθ is not the identity or reversal mapping, a lettera∈ Σ with θ(a) 6= a exists. Thereforew= aω avoids every pattern that includes variablesα andθ(α).Because of this observation we do not have to examine, if patterns are avoidable or unavoidable fora given involution. So we now change the point of view. For a given patterne∈ E∗, we either look atall morphic or all antimorphic involutionsΣ∗ → Σ∗ at the same time. So, we examine, for example, if aninfinite wordw∈ Σω exists, that avoids a patterne for all morphic involutions.Definition 3. Let e∈ E∗ be a pattern, possibly with variables of the formθ(α). We call k∈ N themorphic (antimorphic)θ -avoidance indexV θm (e) (V θa (e)) of e∈ E∗, if an infinite word w∈ Σω over Σwith |Σ|= k exists, that avoids the pattern e for all morphic (antimorphic) involutionsΣ∗ → Σ∗ and k isminimal. If this doesn’t hold for any k∈ N, we defineV θm (e) = ∞ (V θa (e) = ∞).We establish the first facts about avoidance of patternα θ(α)α .Lemma 4. Let Σ be a binary alphabet. Then there is no word w∈ Σω , that avoids the patternα θ(α)αfor all morphic involutionsθ : Σ∗ → Σ∗. That is,V θm (α θ(α)α)> 2.Proof. Let Σ = {a,b}. We try to construct a wordw ∈ Σω , that avoidse= α θ(α)α for all morphicinvolutions and bring this to a contradiction. For example,this word must not containaaa, bbb, abaorbabas a factor. Without loss of generalityw begins witha.Case 1: Assumed the wordw begins withab. Then this prefix must be followed byb, abb<p w. Thenext letter must be ana, the fifth must be ana too. So we haveabbaa<p w. If the following letter isana, aaa is a factor ofw. So the next letter must be the letterb. But for the morphic involutionθ witha 7→ b andb 7→ a the wordabθ(ab)ab is a factor ofw.Case 2: The argument for the caseaa≤p w is analogous to case 1.The proof of the following lemma is analogous to the previousne.Lemma 5. Let Σ be a binary alphabet. There is no word w∈ Σω , that avoids the patternα θ(α)α forall antimorphic involutionsθ : Σ∗ → Σ∗. That is,V θa (α θ(α)α) > 2.B. Bischoff & D. Nowotka 674 Main ResultIn this section, we establish theθ -avoidance indices for the patternα θ(α)α in the morphic and anti-morphic case. We start with the morphic case.Theorem 6. It holds thatV θm (αθ(α)α) = 3.Proof. Let Σ an alphabet with three elements,Σ = {a,b,c}. Let v be the infinitely long Thue–Morseword over the lettersa′ andb′. Furthermore letw ∈ Σω be the word, that is the outcome of replacingeverya′ in v by aacbandb′ by accb. We will show, thatw avoids the patternαθ(α)α for all morphicinvolutions. For better readability, we definex= aacbandy= accb.We assume it exists a morphic involutionθ and a substitution forα , such thatαθ(α)α is a factor ofw. Proof by contradiction. First, we examine the possibilities of replacing the variableα by wordsu∈ Σ+of length|u| < 7. The worduθ(u)u has a maximal length of 18. Therefore there must exist a morphicinvolution so thatuθ(u)u is a factor of a wordw′ ∈ {x,y}6. Because of Theorem 1, the wordsxxx, yyy,xyxyxandyxyxycan not be a factor ofw′. A computer program can easily check these finite possibilitieswith the result, that no wordsu andw′ exist, which fulfill the conditions. Now we assumeα gets replacedby a wordu∈ Σ+ with |u| ≥ 7. Then, the wordu containsaacbor accb. Without loss of generality,ucontainsaacb. Therefore,θ(u) contains the factorθ(aac) = θ(a)θ(a)θ(c). In additionθ(u) and forthis reasonθ(a)θ(a)θ(c) is a factor ofw. There are only two possibilities for two succeeding identicalletters inw. Either these letters are two lettersc followed by the letterb, or two lettersa are followed bythe letterc. This implies, thatuθ(u)u can only be a factor ofw, if θ is the identity mapping. Furthermorethis implies|u| = 4 · k for a k ∈ N. This is visualized in Fig. 1, wherewi,wi′ ,wi′′ ∈ {x,y} holds for all0≤ i ≤ k. If the word(w0)[2](w0)[3](w0)[4] or(w0)[1](w0)[2](w0)[3](w0)[4] = w0 is a prefix of the firstu inFig. 1, then the following equations apply:w0 = w0′ = w0′′w1 = w1′ = w1′′.........wk−1 = wk−1′ = wk−1′′The wordw0w1 . . .wk−1w0′w1′ . . .wk−1′ w0′′w1′′ . . .wk−1′′ = (w0w1 . . .wk−1)3 is a factor ofw. Becauseof wi ∈ {x,y} for all 0 ≤ i ≤ k− 1, this is a contradiction to Lemma 1. On the other hand, if only(w0)[3](w0)[4] or (w0)[4] is a prefix ofu, thenw0 6= w0′ is possible. But in this case(wk′′)[1](wk′′)[2] or(wk′′)[1](wk′′)[2](wk′′)[3] is a suffix of the thirdu. This impliesw1 = w1′ = w1′′w2 = w2′ = w2′′.........wk = wk′ = wk′′andw1w2 . . .wkw1′w2′ . . .wk′ w1′′w2′′ . . .wk′′ =(w1w2 . . .wk)3 is a factor ofw. Again, this is a contradictionto Lemma 1. The theorem follows with Lemma 4.The result of Theorem 6 transfers also to the antimorphic case.Theorem 7. It holds thatV θa (αθ(α)α) = 3.68 Pattern Avoidability with Involutionw= . . .w0 w1. . .wk = w0′ w1′. . .wk′ = w0′′ w1′′. . .wk′′. . .u u uFigure 1: Part ofw to illustrate the factoruuuw= . . .w1 w2w′2 w′′2w3 w4. . .. . .uFigure 2: Part ofw and the factoru of wProof. This proof follows the proof of the previous theorem. LetΣ be an alphabet with three elements,Σ = {a,b,c}. Further, letv be the Thue-Morse word over the lettersa′ andb′. Let w∈ Σω be the word,that we get by replacinga′ in v by aabbcand b′ by aaccb. We will show, thatw avoids the patternα θ(α)α for all antimorphic involutions. For better readability, we definex= aabbcandy= aaccb.We assume that there exists an antimorphic involution and a substitution ofα by a wordu∈ Σ+ insuch a way, thatuθ(u)u is a factor ofw. First we suppose that|u|< 9 holds. The worduθ(u)u then hasa maximal length of 24 anduθ(u)u is factor of a wordw′ ∈ {x,y}6. The wordxxx, yyy, xyxyx, andyxyxymust not be a factor ofw′ because of Lemma 1. A computer program can check these finite possibilitieswith the result, that no wordsu andw′ exist that fulfill these conditions for an antimorphic involution θ .So ,|u| ≥ 9 must hold andu contains at least one wordx or y completely. We now look at the firstu ofthe factoruθ(u)u of w. Let w1w′2 ≤s u with w1,w2 ∈ {x,y}, w2 = w′2w′′2 and |w′2| < 5. We get Fig. 2wherew3,w4 ∈ {x,y}. Without loss of generality, letw1 = x= aabbc. Thenθ(u) and thereforew2w3w4contains the wordθ(aabbc) = θ(c)θ(b)θ(b)θ (a)θ (a) with length 5 as a factor. Hence we look at thefollowing words:xx= aabbcaabbcxy= aabbcaaccbyx= aaccbaabbcyy= aaccbaaccb.Only xxcontainsθ(c)θ(b)θ(b)θ (a)θ (a) for the antimorphic involutionθ with a 7→b, b 7→ a, andc 7→ c.Because ofw1 = x, the equationw2w3 = xx is a contradiction to Lemma 1. The casew2w3w4 = yxxremains. Now there are five possibilities for the position ofu, see Fig. 3. It is easy to check, that in allfive casesθ(u) ≤p w′′2w3w4 respectivelyw′′2w3w4 ≤p θ(u) doesn’t hold. So our assumption, that thereexists an antimorphic involutionθ and a wordu ∈ Σ+ with uθ(u)u is a factor ofw, was wrong. Thetheorem follows with Lemma 5.B. Bischoff & D. Nowotka 69w= . . .w1a a b b cw2a a c c bw3a a b b cw4a a b b c. . .Case 1:. . .Case 2:. . .Case 3:. . .Case 4:. . .Case 5:. . .uFigure 3: Illustration of possible positions of the factoru of w5 Complementary PatternsIn this section, patterns similar toα θ(α)α are considered.For the next lemma we need a further definition. Lete∈E∗ be a pattern consisting of variables of theform α andθ(α) ande′ be the pattern that we get, when all variablesα andθ(α) in eare switched. Wecall e′ ∈ E theθ -complementarypattern ofe. For example theθ -complementary pattern ofα α θ(α)βis θ(α)θ(α)α θ(β ). For this definition it doesn’t matter if morphic or antimorphic involutions areexamined.Lemma 8. Let e∈ E∗ be a pattern and e′ ∈ E be theθ -complementary pattern of e. ThenV θa (e) =V θa (e′) andV θm (e) = Vθm (e′).Proof. First of all we showV θm (e) = Vθm (e′). For better readability, we replace the variableα in thepatterne′ by α ′ andθ(α) by θ(α ′). We assume a wordw ∈ Σω contains the patterne for a morphicinvolution and a substitution ofα by u ∈ Σ+. Thenw contains the patterne′ for the same morphicinvolution by substitutingα ′ by θ(u). Symmetry reasons imply:It exists a morphic involutionθ so thatw contains the patterne.⇔ It exists a morphic involutionθ ′ so thatw contains the patterne′.By negation we get:The wordw∈ Σω avoids the patterne.⇔ The wordw∈ Σω avoids the patterne′.The equationV θm (e) = Vθm (e′) follows. The proof ofV θa (e) = Vθa (e′) is identical.Note the followingθ -free patterns; see [1].Observation 9. The patternsαα , ααβ , βαα , ααβα , αββα , ααββ , αβαβ , ααβαα , andααβαβare 2-unavoidable and3-avoidable.Lemma 10. Let e∈ E∗ be a pattern, that contains the variablesα and θ(α). Further, e contains noother variable of the formθ(γ). Let e′ be the pattern when all occurrences ofθ(α) in e are replaced byα . The pattern e′′ obtained when all occurrences ofθ(α) in e are replaced by a new variableβ .ThenV (e′)≤ V θm (e)≤ V (e′′) andV θa (e)≤ V (e′′).70 Pattern Avoidability with InvolutionProof. The relationV (e′) ≤ V θm (e) holds, since the morphicθ -avoidance index considers all morphicinvolutions, including the identity mapping. Now sayV (e′′) = k, i.e., a wordw∈ Σω exists, that avoidsthe patterne′′.Then this word also avoids the patterne for all morphic and antimorphic involutions.Therefore the relationsV θm (e)≤ V (e′′) andV θa (e)≤ V (e′′) hold.Lemma 11. It holds thatV θa (α α θ(α)) = V θm (α α θ(α)) = 3.Proof. According to Observation 9 the equationV (α α β ) = 3 holds. Lemma 10 impliesV θa (α α θ(α)),V θm (α α θ(α)) ≤ 3. We show by contradiction, that it holds thatV θa (α α θ(α)) 6= 2. The proof for therelationV θm (α α θ(α)) 6= 2 is analogous. Assuming a wordw∈ Σω with Σ = {a,b} exists that avoids thepatternα α θ(α) for all antimorphic involutions. Thenw contains neitheraa nor bb as a factor. Withoutloss of generalityw begins with the lettera. It follows thatw= (ab)ω . Butw= (ab)ω contains the patternα α θ(α) for α = aband the antimorphic involution defined bya 7→ b andb 7→ a. This is a contradictionto our assumption. ThereforeV θa (α α θ(α)) 6= 2 holds and analogouslyV θm (α α θ(α)) 6= 2. We getV θa (α α θ(α)) = V θm (α α θ(α)) = 3.Lemma 12. It holds thatV θa (θ(α)α α) = V θm (θ(α)α α) = 3.Proof. The proof is analogous to the proof of Lemma 11.Corollary 13.1. V θm (θ(α)α θ(α)) = V θa (θ(α)α θ(α)) = 3 by Theorem 6 and 7.2. V θm (θ(α)θ(α)α) = V θa (θ(α)θ(α)α) = 3 by Lemma 11.3. V θm (α θ(α)θ(α)) = V θa (α θ(α)θ(α)) = 3 by Lemma 12.AcknowledgementWe would like to thank the anonymous referees for their careful r views and constructive comments.References[1] J. Cassaigne (2002):Unavoidable Patterns, chapter 3, pp. 111–134. In [4].[2] J. Currie (2005): Pattern avoidance: themes and variations. Theoret. Comput. Sci.339(1), pp. 7–18,doi:10.1016/j.tcs.2005.01.004.[3] J. Currie (2011): Pattern avoidance with involution. CoRR abs/1105.2849. Available athttp://arxiv.org/abs/1105.2849.[4] M. Lothaire (2002):Algebraic Combinatorics on Words. Cambridge University Press, Cambridge, UK.[5] A. Thue (1906):Über unendliche Zeichenreihen. Norske Vid. Skrifter I. Mat.-Nat. Kl., Christiania7, pp. 1–22.[6] A. Thue (1912): Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. SkrifterI. Mat.-Nat. Kl., Christiania1, pp. 1–67.

Pattern Avoidability with Involution

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Pattern Avoidability with Involution

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