Factors of words under an involution

Transcription

1 Journal o Mathematics and Inormatics Vol 1, , 5-59 ISSN: (P), (online) Published on 8 May 014 wwwresearchmathsciorg Journal o Factors o words under an involution C Annal Deva Priya Darshini 1, V Rajkumar Dare 1, Ibrahim Venkat and KG Subramanian 1 Department o Mathematics, Madras Christian College, Chennai , India annaldevapriyadarshini@gmailcom;rajkumardare@yahoocom School o Computer sciences, UniversitiSains Malaysia, Penang, Malaysia Ibrahim@csusmmy; kgsmani1948@yahoocom Received February 014; accepted 7 May 014 Abstract Generalizing the notion o Watson-Crick complementarity which is central in DNA computing, involutively bordered words have been introduced and investigated in the literature In this paper, motivated by these studies, we introduce a general notion o involutive actors o words and obtain several combinatorial properties o such words under a morphic or anti-morphic involution Propertieso the complementary notion o involutive actor-ree words are also obtained Keywords: Factor o a word, Involution AMS Mathematics Subject Classiication (010): 68R15 1 Introduction Combinatorics on words [6], which is classiied as a branch o Discrete Mathematics, is an area o study and research on general properties o words which are inite or ininite sequences o symbols A inite word or simply, a word is a inite sequence o symbols Words play a central role in many branches o study, including DNA computing [1, ] A DNA strand which is the basic unit o every living cell is a sequence o nucleotides adenine, guanine, cytosine, thymine denoted by the symbols A, G, C, T with these nucleotides binding to each other with A to T and C to G A DNA strand can thus be viewed as a word over the our letter DNA alphabet {A, T, C, G} and the relation between the nucleotides can be expressed in terms o an involution mapping, also known as the Watson Crick involution and the ends o the DNA strand can bind with each other i they are images o each other with respect to this involution The involutively bordered words were introduced by Kari et al [3] as a generalization o the classical notions o bordered and unbordered words, in the context o DNA computing In act, Kari et al [3] point out that theoretical properties o bioinormation are studied in their work by investigating models o DNA encoded inormation based on ormal language theoretic and combinatorial properties o words Also, combinatorial properties o words have been investigated in other worksrelated to DNA computing (See, or example [4, 5, 7]) Here, motivated by these studies, we introduce the concept o actor or involutive 5

2 Factors o words under an involution actor o a inite word which extends the notion o involutive border considered in [3] and obtain several combinatorial properties o such actors The paper is organized as ollows: Section introduces and investigates involutive actors o words Section 3deals with the complementary notion o involutive actor ree words Section 4 gives the concluding remarks Involutive actors o a word We irst recall certain basic notions [4] An alphabet is a inite set o symbols A inite word or simply a word w over is a inite sequence o symbols o For example, abaab is a word over = {a,b} The set o all words over is denoted by and this includes the empty word λ, which has no symbols In act is a ree monoid over under the operation o string concatenation deined by u v = a a b b, or words u = a a, v = b b, a, Σ, 1 i n, 1 n 1 m 1 j m We write u v as uv Let = 1 n 1 m i b j - { λ } A language L over is a subset o For w, alph(w) is the set o symbols o that appear in w For example, i = {a, b, c}, w = abaab, thenalph(w) = {a,b} The length o a word w denoted by w, is the number o symbols in w, counting repetitions A word w is a palindrome i it is the same read rom the let to the right or the right to the let For example, the word abba over {a, b} is a palindrome For a word w, the word u is a actor o w i there are words α, β such that w = αuβ Iα = λ then u is called a preix o w and i β = λ then u is called a suix o w The set o all actors o w is denoted by F(w) A mapping : satisying the property that ( xy) = ( x) ( y) is a morphism on I ( xy) = ( y) ( x), then is an antimorphism on is an involution on, i ( ( x )) = x, or all x We now recall notions o bordered and unbordered words For properties and other results on involutively bordered words we reer to [3] Throughout the paper, is considered an alphabet unless stated otherwise Deinition 1 Let be a morphic or an antimorphic involution on A word u is said to be - bordered i there exists a v such that u = vx = y ( or x, y In this case, we say that v is a - border o u Otherwise, the word u is called - unbordered We now deine the concept o involutive actor or actor o a inite word Deinition Let be a morphic or an antimorphic involution on A word u Σ is a - actor or an involutive actor o w i u and ( w) ie both u and its image 53

3 C A Deva Priya Darshini, VRDare, I Venkat and KG Subramanian ( are actors o w We also say that w is a word with -actors The wordw can then be written as w = xuy = x' ( y', x x', x, y, x', y' Remark: I v is a - border o a word u, then v is also a - actor o u Example 1 Let = {a, t, c, g} be an alphabet Let be a mapping on deined by ( a ) = T, ( t) = A, ( c) = g, ( g) = c (i) I is a morphic involution then the actor v = acga o the wordu = tacgaatcgtgctttis a - actor since ( = tgctis also a actor o u (ii) I is an antimorphic involution on, then the actor v =ggta o w = cagtgactaccggtac is a - actor since ( = (ggta) =tacc is also a actor o w We note that or a word u = tctccccttover, no symbol o u has its - image also in u, when is a morphic involution on deined as above We call u an involutive actor ree or -actor ree word Deinition 3 Let be a morphic or an antimorphic involution on we denote by (w), the set o all actors o w ie L L ( w) = { v Σ / v, ( F ( w)} 54 For w Σ, Example Let = { a, b} and w = abaabaaabbe a word on Let be a morphic involution on deined by ( a ) = b, ( b) = a Then the only - actors o ware a, b, ab, ba L ( w) = { a, b, ab, ba} The ollowing result is a consequence o the deinitions Theorem 1 Let be a morphic or an antimorphic involution on Then (i) i u Σ, then or every v L (, we have F( L (, (ii) or a word u Σ, we have F ( ( F ( ) = L (, (iii) iu 1, u are any two words over then L u ) L ( ), i u 1 is a actor o ( 1 u u and in particular, u 1 is a preix or suix o u Proo: In order to prove (i), let x F ( Since v L (, we have v, ( F( Now x F( as x is a actor o v and v is a actor o u Also (x) is a actor o ( and so ( x) F( Since both x, ( x) F( we have x L ( This proves F( L ( To prove (ii), let x ( F( ) Then, by deinition o F (, u = αxβ, α, β Σ and x = ( or some v Also u = α' vβ ' or some α ', β ' Σ Since the mapping is an involution, we have ( x ) = ( ( ) = v Hence u = α' ( x) β ' This implies that ( x) and so x L ( This proves that

4 Factors o words under an involution F ( ( F ( ) L ( Conversely, let x L ( Then x, ( x) so that u = αxβ; α, β Σ; u = α ' ( x) β '; α ', β ' Σ Now, ( x) implies that x = ( ( x)) ( F( ) Since x is arbitrary, ( F ( ( F ( ) L Finally, to prove (iii), let v be a actor o the word u 1 Then v, ( u1) Since u 1 is a actor o u, we have F u ) F( ) Thereore v ( u ) Hence the proo ( 1 u, Theorem Let be a morphic or an antimorphic involution on Σ Letu Σ Then ( L ( ) = L ( = L ( ( ) Proo: We irst prove ( L ( ) = L ( Let v' ( L ( ) Then v' = ( or some v L ( so that both v and v' F ( by deinition o ( L Also ( v ') = ( ( ) = v and so ( v' ) F ( Hence v' L ( This proves that ( L ( ) L ( Conversely, let v L ( This implies that both v and ( F ( Since is an involution, setting v' = ( we ind that ( v ') = v Hence v and (v ) F ( This means that v' L ( so that v = ( v' ) ( L ( ) Hence, L ( ( L ( ) Thus ( L ( ) = L ( To prove L ( = L ( ( ), let x L ( Then x, ( x), so that α 1, α, β1, β Σ ( u ) ( α1) ( x) ( β1), ( = ( α ) x ( β ( u ) ( β1) ( x) ( α1), ( = ( β ) x ( α x L ( ( u Thus L ( L ( ( ) u = α1xβ1, u = α ( x) β, or some I is a morphic involution, then = ) I is an antimorphic involution, then = ) In either case, x, ( x) ( ) Hence )) Let y L ( ( ) Then y, ( y) F ( ( ) This implies ( y), ( ( y)) ( ( )) ie y, ( y) Hence y L ( This proves L ( ( ) L ( Hence the proo Deinition 4 Let L be a language over an alphabet Let be a morphic or an antimorphic involution Then we deine L ( L) = L ( Example 3 Let L ( ab) n { 1} = n u L be a language over = {a, b} Let given by ( a ) = b, ( b) = a be a morphic involution on Then L ( ab) = { a, b}, L ( abab) = { a, b, ab, ba, aba, bab} {( ) } ( ) and so on Thus L ( L) = L ( { } {( ) } {( ) } n n n n = ab n 1 ba n 1 ab a n 0 ba b n 0 We give some properties o this language (L) L u L 55

5 C A Deva Priya Darshini, VRDare, I Venkat and KG Subramanian Theorem 3 Let be a morphic or an antimorphic involution on Let language Then (i) or every v L (L), the image word ( L ( L), 56 L Σ be a (ii) or any two languages L1, L with L1 L, we have L ( L1 ) L ( L ) Proo: (i) Let v L (L) Then v L (, or some u L, by deinition o (L) This implies v, ( F ( Let ( v ) = w Then v = (w), since is an involution This implies that w, ( w) F( and hence w L ( Thus w = ( L ( L) (ii) Let 1 L some u L1 But 1 L ( L1 ) L ( L L To prove L ), let v L L ) Then v L (, or ( 1 L means u L Thereore, v L ( or u L Since v is arbitrary, we can conclude that L L ) L ( ) ( 1 L 3 Involutive actor ree words In this section we obtain properties o the complementary notion o - actor ree or involutive actor ree words Deinition 31 Let be a morphic or an antimorphic involution on A word is - actor ree or involutive actor ree i none o its actors is a actor o w L w Σ Example 31 Consider the word w = acccccaaa over Σ = {a, t, c, g} Let be a morphic involution on deined by ( a ) = t, ( t) = a, ( c) = g, ( g) = c No actor u ow is a actor o w, as (w) is not a actor o w Theorem 31 I w Σ is a - actor ree word, then or a morphic or an anti-morphic involution on ( w) Σ is also - actor ree, Proo: Let w Σ be a - actor ree word Then or any actor u F (w) we have ( F ( w) Then w can be written as w = αuβ, or some α, β Σ But w α ' ( β ', or any α ', β ' Σ Assume that (w) has a actor u Then u, ( F ( ( w)) Then ( w ) = α 1 uβ 1, or some α, β Σ 1 1 and w = ( ( w )) = ( α1) ( ( β1), i is a morphic involution This implies ( is a actor o w, a contradiction Hence (w) is also - actor ree A similar argument holds i is an antimorphic involution Theorem 3 Let Σ= {a, b} and given by (a) = b, (b) = a be a morphic or an antimorphic involution Then or any word u Σ which is - actor ree, either ua is - actor ree or ub is - actor ree

6 Factors o words under an involution Proo: Let u Σ be a - actor ree word Then alph( cannot contain both a and b since u is -actor ree I alph ( = {a} then ua also does not contain b and so is - actor ree but i alph ( = {b} then, likewise, ub is - actor ree We now give a characterization o - actor ree words Theorem 33 A word u Σ is - actor ree i and only i or every a alph(, we have ( a) alph( Proo: Assume u is -actor ree This means that none o the actors o u is a actor Then everya in alph( is a actor o u Hence ( a) alph( Conversely, let ( a) alph(, or every a To show that u Σ is - actor ree Suppose u has a actor v Then v as well as ( are actors o u This implies that a alph ( alph ( and ( a) alph ( ( ) alph ( Thereore, or some a Σ such that a, the image ( a) which is a contradiction to our assumption Hence the proo Corollary 31 i)i w Σ is a - actor ree word, then w and (w )are also - actor ree, or a morphic or an anti-morphic involution on where w is the product o w with itsel certain number o times This ollows rom Theorem 33 and the act that alph(w ) = alph(w) ii) A word u Σ is - actor ree i and only i L ( = φ This again ollows rom Theorem 33 In act, or any actor v o u, ( cannot be a actor o u i u is - actor ree and this means (u ) is empty and the converse is similar We now examine the problem o whether the concatenation o two - actor ree words is also - actor ree Theorem 34 Let be a morphic or antimorphic involution on Σ Let 57 L u, v Σ be - actor ree Thenuv Σ is - actor ree i and only i ( alph ( ) = φ Proo: Assume that w = uv Σ is - actor ree Then by Theorem 33, or every a w), we have ( a) alph( w) In order to show that ( alph( ) = ϕ, assume the contrary ie, ( alph ( ) φ Then there exists an element a Σ such that a ( alph( ) Now, a ( alph( ) implies that ( a) But also a which implies a w) and ( a) implies ( a) w) which is a contradiction Hence, ( alph ( ) = φ Conversely, let ( alph ( ) = φ To prove that uv is - actor ree Suppose uvhasa actor Then uv = w is such that there exists a Σ with a w) as well as ( a) w) Both a, ( a) cannot happen as u is - actor ree

7 C A Deva Priya Darshini, VRDare, I Venkat and KG Subramanian Likewise, both a, ( a) cannot also happen as v is - actor ree On the other hand, i a, ( a) then ( a) ( alph( ) which is a contradiction I a, ( a) a similar contradiction arises This proves the Theorem Corollary 3 Let be a morphic or antimorphic involution on Σ Let u, v Σ be - actor reewith ( alph ( ) = φ Then u v,(u, are - actor ree Proo: By Corollary 31 and Theorem 34, we have u v,(u are - actor ree 4 Concluding remarks A preliminary version o part o this paper was presented in thesixth International Conerence on Bio-Inspired Computing: Theories and Applications, in 011 [8] The originality o the contribution in this paper lies in introducing a general notion o involutive actor or - actor o a word and in obtaining several combinatorial properties o such words The complementary notion o - actor ree words is also examined and their properties are derived We can also consider a notion o involutive actorization o a word A actorization o a word w Σ is w = u 1 u u n where the u i Σ or i= 1,, n This actorization need not be unique and in general, a word w can have more than one actorization I is a morphic or an antimorphic involution on Σ, then a actorization o w Σ, given by w = u 1 u u n where the u i Σ, or i= 1,,, n, can be called a - actorization or involutive actorization o w i and only i w = u ) ( u ) ( u ),where i j ϵ{1,,, n}or j = 1,,, m It will be o interest ( i 1 i i m to study properties o such actorizations, which is or uture work Acknowledgement: The authors thank the reerees or their useul comments which helped to improve the presentation o the paper REFERENCES 1 LM Adleman, Computing with DNA, Scientiic American, August 1998, N Jonoska, Computing with Biomolecules: Trends and Challenges, XVI Tarragona Seminar on Formal Syntax and Semantics, 7(003) 3 L Kari, K Mahalingam, Involutively bordered words, Intern J Found Comp Sci,18 (007) L Kari, K Mahalingam, Watson-crick bordered words and their syntactic monoid, Intern J Found Comp Sci,19 (008) L Kari, S Seki, On pseudoknot-bordered words and their properties, J Comp System Sci, 75 (009) MLothaire, Combinatorics on words, Addison Wesley Publishing Company, K Mahalingam and KG Subramanian, Palindromic completion o a word, Proc IEEE Fith Int Con on Bio-inspired Computing: Theories and Applications (BIC- TA), (010)

8 Factors o words under an involution 8 V Rajkumar Dare, C Annal Deva Priya Darshini, Bordered actors o a inite word, Proceedings o the 011 Sixth International Conerence on Bio-Inspired Computing: Theories and Applications (011)