ON THE EXPRESSIVE POWER OF SHUFFLE PRODUCT

1. ON THE EXPRESSIVE POWER OF SHUFFLE PRODUCT Antonio Restivo Università di Palermo

2. A very general problem: Given a basisB of languages, and a set Oof operations, characterize the familyO(B)of languagesexpressible from the basisB by using the operations in O.

3. The Family REG of Regular Languages: The basis: B = { {a} | a  } U {} The operations: O = {union, concatenation, (Kleene) star} REG = O(B) REG isclosedalso under allBooleanoperations

4. The Family SF of Star-Free Languages The basis: B = { {a} | a } U {ε} The operations: O = {Booleanoperations , concatenation} SF = O(B)

5. Shuffle Product The shuffle of twowordsu and vis the set uшv = {u1v1…unvn|n≥0, u1…un=u, v1…vn=v} abшba = {abba, baab, abab, baba} The shuffle of twolanguagesL and Kis the language LшK = UuєL,vєKuшv

6. ExpressivePowerof the Shuffle Verylittleisknownaboutclasses of languagesclosed under shuffle, and theirstudyappears to be a difficultproblem. Such a study, apartitstheoreticalinterest, isalsomotivated by applications to the modeling of processalgebras and to programverification

7. The Family INT of IntermixedLanguages The basis: B = { {a} | a } U {ε} The operations: O = {Booleanoperations, concatenation, shuffle} INT = O(B)

8. Theorem(Berstel, Boasson, Carton, Pin, R.) SF  INT  REG REG INT SF

9. The Problem Problem 1: Givea (decidable) characterizationif the family INT Proposition INT isnot a variety (in the sense of Eilenberg) Remark: REG and SF are varieties

10. Periodicity A language L  * isaperiodic , or non-counting, ifthereexists an integer n  0 suchthat, for allx,y,z *, onehas xynz L  xyn+1z  L. Theorem(M.P. Schutzenberger) A regular language L isaperiodicif and onlyifitis star-free

11. Periodicity The strictinclusion SF  INT impliesthatthe shuffle of two star-free languages in general isnot star-free: «the shufflecreatesperiodicities» Problem 2: Determineconditions under which the shuffle of two star-free languagesis star-free.

12. BoundedShuffle Let k be a positive integer. The k-shuffle of twolanguages L1 and L2isdefinedasfollows: L1шk L2 = = {u1v1…umvm|m≤k, u1…umL1, v1…vmL2}. Any k-shuffleiscalledboundedshuffle Theorem(Castiglione, R.) SF isclosed under boundedshuffle Corollary. The shuffle of a star-free language and a finite languageis a star-free language

13. PartialCommutations Let be an alphabet and let      be a symmetric and reflexive relation, called(partial) commutation. Consider the congruence  of * generated by the set of pairs (ab,ba) with (a,b). If L  * is a language, [L] denotes the closure of L by . L isclosed by  if L = [L]. The closedsubsets of * are calledtrace languages.

14. PartialCommutations Let L1 and L2 be twolanguages over the alphabet. Let 1and 2be two disjoint copies of the alphabet  (colored copies), and i: i   , for i=1,2, the corresponding bijections. Let L’1 (L’2 resp.) be the subset of 1(2resp.) corresponding to the L1 (L2 resp.) under the morphism 1 (2 resp.). Let = 1 2and considerthe partialcommutation  1 2 and let: * * be the morphisminduced by 1and 2(delete colours). The -product of L1 and L2is L1ш L2 =  ( [L’1L’2])

15. PartialCommutations bacbcaabcaL1, babcacbab  L2 bacbcaabcababcacbab • = {(a,a), (a,b), (b,a), (b,b), (c,a) (c,b), (c,c)} bbaabcbcaabcacacbab L1ш L2 The -productgeneralizesat the same time concatenation and shuffle: If  = , then L1ш L2 = L1L2 If  = 1  2, then L1ш L2 = L1ш L2

16. PartialCommutations Given the partialcommutation  1  2, wedefine the partialcommutation ’    definedasfollows: (a,b)  ’  (a,b)   Theorem(Guaiana, R., Salemi) Let L1, L2 be languages over , closedunder ’. If L1, L2  SF, then L1ш L2  SF. Corollary. The shuffle of two commutative star-free languagesis star-free

17. PartialCommutations If the internalcommutation’ (i.e.thecommutationallowed inside eachof the languages L1, L2) is the «same» as the externalcommutation (i.e. the commutationsbetween the letters in L1 and the letters in L2), thenthe -productpreserves the star-freeness.

18. Unambiguous Star-Free Languages A language L is the markedproduct of the languages L0, L1, …, Lnif L = L0a1L1a2L2 … anLn, for some letters a1, a2, … , an of  . A markedproduct L = L0a1L1a2L2… anLnisunambiguousifevery word of L admits a uniquedecomposition u = u0a1u1…anun, with u0 L0, … , un Ln. The product{a,c}*a{}b{b,c}*isunambiguous

19. Unambiguous Star-Free Languages SFis the smallestBoolean algebra of languageswhichisclosed under markedproduct The family USF of Unambiguous Star-Free languagesis the smallestBoolean algebra of languages of * containing the languages of the form A*,for A , whichisclosed under unambiguousmarkedproduct.

20. Unambiguous Star-Free Languages FO: class of languagescorresponding to formulas of first orderlogic. FOk: class of languagescorresponding to formulas of first orderlogic with k variables. Theorem(McNaughton)SF = FO Theorem(Immerman, Kozen)FO3 = FO Theorem(Therien, Wilke) FO2 = USF

21. Unambiguous Star-Free Languaes REG USF  SF  INT  REG Theorem(Castiglione, R.) If L1, L2  USFthen L1ш L2  SF INT SF USF

22. CyclicSubmonoids The languages in the classUSFcorrespond to regular expressions in which the star operationisrestricted to subsets of the alphabet. The simplestlanguagesnot in USF are the languages of the form L = u*, where u is a word of length 2. Suchlanguages are the cyclicsubmonoids of *. Weherestudy the shuffle of cyclicsubmonoids.

23. Cyclicsubmonoids Theorem(Berstel, Boasson, Carton, Pin, R.) If a word u containsatleasttwodifferentletters, then u*  INT. A word u  * isprimitiveif the condition u=vn, for some word v and integer n, implies u=v and n=1. Theorem(McNaughton, Papert) The language u* is star-free if and onlyif u is a primitive word.

24. Cyclicsubmonoids u = b, v = ab b* ш (ab)* = (b + ab)*  SF u = aab, v = bba (aab)* ш (bba)*  (ab)* = ((ab)3)*   (aab)* ш (bba)*  SF Problem 3: Characterize the pairs of primitive wordsu,vsuchthat u*ш v* is a star-free language.

25. Combinatorics on Words Theorem(Lyndon, Schutzenberger) If u and v are distinct primitive words, then the word unvmis primitive for alln,m 2. Theorem(Shyr, Yu) If u and v are distinct primitive words, thenthereisatmostone non-primitive word in the languageu+v+.

26. Combinatorics on Words Problem 3 isrelated to the search for the powers (non-primitive words) thatappear in the language u+ш v+. Denote by Qthe set of primitive words. For u,v,w Q, letp(u,v,w) be the integer k suchthat (u*ш v*)  w* = (wk)* If (u*ш v*)  w* = {}, then p(u,v,w) = 0. formulasof first orderlogic

27. Combinatorics on Words For u,v Q, define the set of integers P(u,v) = { p(u,v,w) | w  Q}. For instance, ifu = a10b and v = b then P(u,v) = {0,1,2,5,10}. Problem 4: Giventwo primitive words u, v, characterize the set P(u,v) in terms of the combinatorialproperties of u and v.