Unavoidable pattern

In mathematics and theoretical computer science, a pattern(term) is a sequence of characters over some alphabet $\Delta$. A word $w$ avoids pattern $p$ if $p$ does not match $w$. If $w$ avoids $p$, $w$ is also called $p$-free.

A word $w$ matches pattern $p$ if there exists a factor(subword) of $w$ which is also an instance of $p$.

A word $x\in \Sigma ^{*}$ is an instance of pattern $p\in \Delta ^{*}$, if there exists a non-erasing morphism(substitution) $f:\Delta ^{*}\rightarrow \Sigma ^{*}$ such that $f(p)=x$. Non-erasing means $f(a)\neq \varepsilon ,\forall a\in \Delta$.

A pattern $p$ is unavoidable on a finite alphabet $\Sigma$ if there exists $n\in \mathrm {N}$, word $x$ matches $p$ for any $x\in \Sigma ^{*},|x|\geq n$. Otherwise, $p$ is avoidable on $\Sigma$, which implies there exist infinitely many words over alphabet $\Sigma$ that avoid $p$. This is equivalent to saying that $p$ is avoidable on $\Sigma$ if there exists an infinite word $w\in \Sigma ^{\omega }$ that avoids $p$.

A pattern $p$ is k-avoidable if $p$ is avoidable on an alphabet $\Sigma$ of size $k$. Otherwise, $p$ is k-unavoidable, which means $p$ is unavoidable on every alphabet of size $k$.

A pattern $p$ is an unavoidable pattern(blocking term) if $p$ is unavoidable on any finite alphabet.

Let A be an alphabet of letters and E a disjoint alphabet of pattern symbols or "variables". Elements of E⁺ are patterns. For a pattern p, the pattern language is that subset of A^∗ containing all words h(p) where h is a non-erasing semigroup morphism from the free monoid E^∗ to A^∗. A word w in A^∗ matches or meets p if it contains some word in the pattern language as a factor, otherwise w avoids p.

There is a word W(k) over an alphabet of size 4k which avoids every avoidable pattern with less than 2k variables.

Zimin words

Given alphabet $\Delta =\{x_{1},x_{2},...\}$, Zimin words are defined recursively $Z_{n+1}=Z_{n}x_{n+1}Z_{n}$ for $n\in \mathrm {Z} ^{+}$ and base case $Z_{1}=x_{1}$.

A word $w$ is an unavoidable pattern if and only if it's a factor of a Zimin word.

It is unavoidable that any string, containing two unique characters, that is five or more characters long will contain a pattern of the form ABA (the second Zimin word). Using three unique characters any string containing 29 or more characters will contain a pattern of the form ABACABA

Pattern reduction

Given a pattern $p$ over some alphabet $\Delta$, we say $x\in \Delta$ is free for $p$ if there exist subsets $A,B$ of $\Delta$ such that the following hold:

1. $uv$ is a factor of $p$ and $u\in A$ ↔  $uv$ is a factor of $p$ and $v\in B$
2. $x\in A\backslash B\cup B\backslash A$

e.g. let $p=abcbab$, then $b$ is free for $p$ since there exist $A=ac,B=b$ satisfied conditions above.

A pattern $p\in \Delta ^{*}$ can reduce to pattern $q$ if there exists a character $x\in \Delta$ such that $x$ is free for $p$, and $q$ can be obtained by removing all occurrence of $x$ from $p$.

e.g. let $p=abcbab$, then $p$ is reducible to $q=aca$ since $b$ is free for $p$.

Given patterns $p,q,r$, if $p$ can reduce to $q$ and $q$ can reduce to $r$, then $p$ can reduce to $r$.

A pattern $p$ is unavoidable if and only if $p$ can reduce to a word of length 1.

Examples

• The Thue–Morse sequence avoids the patterns xxx and xyxyx.
• The patterns x and xyx are unavoidable on any alphabet.
• The power pattern xx is 3-avoidable; words avoiding this pattern are square-free.
• The power patterns xⁿ for n ≥ 3 are 2-avoidable: the Thue–Morse sequence is an example for n=3.
• Sesquipowers are unavoidable.

Avoidability index

The avoidability index of a pattern p is the smallest k such that p is k-avoidable, or ∞ if p is unavoidable. For binary patterns (two variables x and y) we have:

• The Zimin words x, xyx, xyxzxyx, xyxzxyxwxyxzxyx, etc. are unavoidable, as well as any word that can be written as a subword of a Zimin word via a homomorphism. All other words are avoidable.
• Many patterns have avoidability index 2.
• xx,xxy,xyy,xxyx,xxyy,xyxx,xyxy,xyyx,xxyxx,xxyxy,xyxyy, as well as many doubled words, have avoidability index 3, though this list is not complete.
• abwbaxbcyaczca has avoidability index 4, as well as other locked words. (Baker, McNulty, Taylor 1989)
• abvbawbcxacycdazdcd has avoidability index 5. (Clark 2004)
• no words with index greater than 5 have been found.

Square-free words

Main article: Square-free word

A square-free word is one avoiding the pattern xx. An example is the word over the alphabet {0,±1} obtained by taking the first difference of the Thue–Morse sequence.

See also

• ABACABA pattern

References

• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. Vol. 27. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4480-9. Zbl 1161.68043.
• Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183.
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.