We show that the language of primitive non-strong words is not context-free. We also discuss the relation of del-robuste primitive word language to other formal languages.
Words
A word is said to be a conjugate of a word x if it is a cyclic shift of x, that is, if w=uv and x=vu for a u,v ∈V∗[16]. Then the words u and v are powers of the same word if and only if the words ur and vr have a common prefix of length.
Finite Automata
For a regular languageL ⊆ V∗, there exists an integerp≥1such that for each word w∈Lmet|w| ≥p, there is a factorizationw=xyzinV∗ satisfyingy6=λ,|xy| ≤pandxynz∈Lfor all∈N. A language L ⊆ T∗ is a context-free language if there exists a context-free grammarG=hN, T, Pi and a variablev∈N such thatL=L(G, v) ={w∈T∗ |v→∗w }. Pumping lemma for context-free languages [30,31]) Let L⊆V∗ be a context-free language.
Primitive Words
We call a wordw ∈V+θ-primitive if there exists no non-empty word∈ V+ such that w is aθ-power oftand|w|> |t|. If every position in a string is covered by an occurrence of a string, then we say that t covers w.
L-Primitive Words
Symbol Substitution and Primitivity
A primitive word with length is said to be a sub-robust primitive word if and only if the word is . Clearly, the language of sub-robust primitive words is a subset of the set of primitive words, Q.
Recognizing Subst-Robust Primitive Words
In this case the pump lemma does not satisfy for i = 0, since uviwxiy = ap0bp+1ap+2bp. Case (d) As in case (c), in this case too we can find such a case that the pump lemma does not hold. Thenu is a non-subst-robust primitive word if and only if the word uu is a periodic word of length|u| contains −1 with periodicity psuch that p is a division of|u|and |u|.
The word uu = trt1at2tstrt1at2ts thus contains a factor t2tstrt1 of length |u| −1 which is equal to the primitive word(t2t1b)r+st2t1. If the word uu contains a maximal repetition of length at least |u| −1 with a period p on which divides|u|and p < |u|thenu is a non-subst-robust primitive word.
Del-Robust Primitive Words
Recognizing Del-Robust Primitive Words
Then u is a nonpartially robust primitive word if and only if worduu contains at least one nonprimitive word of length|u| −1. The word uu=trt1at2tstrt1at2ts thus contains a factor t2tstrt1of length|u| −1 which is equal to the non-primitive word(t2t1)r+s+1. If the word uu contains a maximal repetition of length at least |u| −1 with a point p where p divides |u| −1 and p < |u| −1 then u is a non-partially robust primitive word.
We then present a linear-time algorithm to test the consistency of a primitive word using the algorithm FINDMAXIMALREPETITIONS, which finds the maximum repetitions and tests primitiveness in linear time for a given word. The correctness of the algorithm follows from Corollary 3.6 which is used in step 8 of the algorithm.
Counting Del-Robust Primitive Words
In step (3), the set of pairs of periods and the corresponding lengths of maximum repetitions of uucan are also calculated in linear time [17].
Ins-Robust Primitive Words
- Ins-Robust Primitive Words and Density
- Relation of Q I with Other Formal Languages
- Counting Ins-Robust Primitive Words
- Recognizing Ins-Robust Primitive Words
Since QI is reflective, therefore every cyclic permutation of w is also a non-strong primitive word. This proves that every cyclic permutation of an ins-robust primitive word is ins-robust. We then prove that the language of primitive non-strong words is not context-free in general.
Therefore, the language of QI non-persistent primitive words is not context-free. Thus, the number of ins-robust primitive words lengthn,QI(n) in the alphabet V is the same.
Exchange-Robust Primitive Words
Structural Characterization of Exchange-Robust Primitive Words
Then there exists at least one consecutive position whose interchange makes the word non-primitive. There are both finite-length and arbitrary-length primitive words that are non-exchange robust; for example abba and (ab)nba(ab)nforn≥1. Unlike the languages of del-robust and ins-robust primitives which are closed under the cyclic permutation [44], the set QX is not closed under the cyclic permutation.
Before proving the density of a language of exchange-non-robust primitive words, we prove the following result that we need to prove the density of QX. We consider two different possibilities depending on whether w is a primitive word or a non-primitive word.
Context-freeness of Q X
If we choosex =λandz=u1bau2 then xwz∈Qandxwz∈QX. b) x contains at least one of the distinguished positions. The language of non-commuting strongwords is not context-free over the alphabet V ={a, b}. As L0 satisfies Ogden's lemma, then every time∈L0, |w| ≥N can be decomposed into w=uvxyz such that the following conditions hold: (i)vxy contains at most N marked symbols (ii)v have at least one marked symbol and (iii)uvixyiz∈L0 for alli≥ 0.
Since the family of context-free languages is closed under sequential converters and the intersection with regular languages [47], we conclude that QX is not context-free.
Conclusions
The set of L-primitive words over an alphabetV is denoted by QL(V) or simply QAnd the set of non-L-primitive words over an alphabetV is denoted by ZL. A word over an alphabet has a unique primitive root, but it can have more than one L primitive root. On the contrary, letZIs not reflective, then there exists a wordw=uv∈ZL such thatvu∈QLbut then uv∈QL, which is a contradiction.
If the part: LetQL=Qampak Q6⊆L, then there exist ∈Q which is not in L, i.e. there existssk≥2 such that uk ∈QL. Only if part: Let be a unique L-primitive word such that and for any x∈L,x=uk for some ≥1.
Other Formal Languages and L-Primitive Words
Lnif and only if there exists a uniqueL-primitive ordu∈Lsuch that and for anyx∈L,x=ukfor somek≥1. Then the language of L-primitive words is regular if any of the following conditions hold. Then ZL is regular if and only if ZL\L is a regular language and primitive root of ZL\Lis a finite language.
If the part:class of regular languages is closed under union and considers the difference of two strings, i.e. if ZL\Li is a regular language, then (ZL\L)∪L=ZL. By the pumping lemma for regular languages, there exists an integer n for w = xyz such that|y| ≥ 1 in|xy| ≤nandxyiz ∈ZL\L for alli≥0, so xyizi is not L-primitive and therefore non-primitive for alli.
Ins-Robust L-Primitive Words
Let us be a language over an alphabet V.QI⊆QLI, for any language⊆B∗, where QI is the language of ins-robust primitive words and QLI is a set of ins-robust L-primitive words.
Del-Robust L-Primitive Words
QD ⊆ QLD, for any language L⊆V∗, where QD is the language of rigid primitive words and QLD is set of rigid L-primitive words. Letw∈QD butw /∈QLD, then for some partition orw(sayw1aw2),w1w2 =uk for some∈Landk≥2. Toebaba∈QLD, since it is not the proper power of any word contained in L nor the proper power of any word of L after the removal of any symbol from this word, butabab /∈QLD since it is not even inQL is not.
From Corollary 3.5, we know that the word w is in the set QD if and only if it has the form u or its cyclic permutation for some u∈Q,u6=ain n≥2. However, since QLD is not reflective, this result may not hold for such words.
Exchange Robust L-Primitive Words
Hence there exists a partition of a wordw = w1abw2 such that w1baw2 = uk for some u ∈ Earth k > 1. It is clear that the word abba is a strong L-primitive exchange word, but not a strong primitive exchange word.
Conclusions
Ins-Robustness of θ-Primitive Words
For an involutional morphismθ of alphabetV, a θ-primitive word of length is said to be an ins-robustθ-primitive word if wordpref(w, i). Clearly, the language of ins-robust θ-primitive words is a subset of the language of θ-primitive words. Aθ-primitive word is not ins-robust if and only if w can be expressed in the form u1u2.
We can easily prove that for a morphic involutionθ:V∗ →V∗, the language ofθ-primitive words is reflective if and only ifθ is identity function. In the next lemma, we discuss the condition on the words so that the theorem holds for non-ins-robust pseudo-primitive words.
Context-freeness of Q θI
For a morphic involutionθover alphabetV, the word uu contains at least one θ-periodic word of length |u|with θ-period p such that p divides the length|u|+ 1ogp≤ |u|. But QθI is regular for alphabet V, so that|V| = 2ifθ 6=idV, where idV is the identity mapping over V. Letθbe an involution morphism.QθI is not a context-free language for alphabetV such that|V| ≥3.
Other Robustness in θ-Primitive Words
Pseudo L-Primitive Words
We define theθL-primitive root (in short, θL-root) of w, denoted by ρθL(w), as the shortest word ∈L such that aθ-power of is and there is no wordx ∈L that θL-root often en|x |< is |t|. Since t is a θ-power of s, w is therefore a θ-power of s, which contradicts that it is the θL-primitive root of w, because|s|<|t|ands∈L. Then there exists some measure of t∈Lsuch thattw=tnwithn≥2en|t|<|w|, thus also a θ-power often, which is a contradiction.
The θL-primitive root of a word need not be θ-primitive and thus need not be primitive. Similarly, we can show that the class of θ-superprimitive words is not necessarily closed under circular permutations.
Robustness of Primitive Morphism
Denote by fQ the set of all primitive wordsu ∈Q such thatf(u)∈QandfZthe set of all primitive wordsu∈Qsuch thatf(u)∈Z[15]. Denote by fD the set of all del-robust primitive words u ∈ QD such thatf(u) ∈ QD and byfD0 the set of all del-robust primitive words u ∈ QD such thatf(u) ∈/ QD, i.e. denote byfd the set of all primitive wordsu ∈Qzodatf(u) ∈QD and byfd0 the set of all primitive wordsu∈Qzodatf(u)∈/QD.
Hence the word au-primitive is a word that is not reducible to any injective morphism of V∗. A word u ∈ V+ is said to be universally del-robust primitive or simply ud-primitive if for every injective morphism f of V∗, the word f(u) is del-robust primitive.
Robustness of Abelian Primitive Words
A wordw is part-robust abelian primitive (or DA-primitive, for short) if the primitive wordw cannot be written asuu1u2. We know that the language of de-robust primitive wordsQD over an alphabetV is reflective of Theorem 3.3. Similarly, we have the property of reflectivity for the language of part-robust abelian primitive words AQD.
We can now show that the set of all del-robustA-primitive words is not context-free. Since context-free languages are closed under quotients with regular groups and inverse homomorphism, M0 is context-free if M is context-free.
Conclusions
It is also proved that the language of non-del-robust primitive words QD is not context-free. We characterized ins-robust primitive words and identified several properties and proved that the language of ins-robust primitive wordsQI is not regular. Finally, we have given a lower bound for the number of ins-robust primitive words of a given length.
Some of them that we plan to explore in the near future are:. ii) Is the language of subst-robust primitive words QS deterministically context-free?. We have characterized ins-robust pseudo-primitive words and identified several properties. and proved that the language of ins-robust primitive words QθI is not context-free.
