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

2. A very general problem: Given a basis B of languages, and a set O of operations, characterize the family O(B) of languages expressible from the basis B by using the operations in O.

3. The Family REG of Regular Languages:

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

5. Shuffle Product The shuffle of two words u and v is the set u ш v = {u 1 v 1 …u n v n |n≥0, u 1 …u n =u, v 1 …v n =v} ab ш ba = {abba, baab, abab, baba} The shuffle of two languages L and K is the language L ш K = U uєL,vєK u ш v

6. Expressive Power of the Shuffle Very little is known about classes of languages closed under shuffle, and their study appears to be a difficult problem. Such a study, apart its theoretical interest, is also motivated by applications to the modeling of process algebras and to program verification

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

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

9. The Problem Problem 1: Give a (decidable) characterization if the family INT Proposition INT is not a variety (in the sense of Eilenberg) Remark: REG and SF are varieties

10. Periodicity A language L   * is aperiodic, or non-counting, if there exists an integer n  0 such that, for all x,y,z   *, one has xy n z  L  xy n+1 z  L. Theorem (M.P. Schutzenberger) A regular language L is aperiodic if and only if it is star-free

11. Periodicity The strict inclusion SF  INT implies that the shuffle of two star-free languages in general is not star-free: «the shuffle creates periodicities» Problem 2: Determine conditions under which the shuffle of two star-free languages is star-free.

12. Bounded Shuffle Let k be a positive integer. The k-shuffle of two languages L 1 and L 2 is defined as follows: L 1 ш k L 2 = = {u 1 v 1 …u m v m |m≤k, u 1 …u m  L 1, v 1 …v m  L 2 }. Any k-shuffle is called bounded shuffle Theorem (Castiglione, R.) SF is closed under bounded shuffle Corollary. The shuffle of a star-free language and a finite language is a star-free language

13. Partial Commutations 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 is closed by   if L = [L] . The closed subsets of  * are called trace languages.

14. Partial Commutations Let L 1 and L 2 be two languages over the alphabet . Let  1 and  2 be 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 (  2 resp.) corresponding to the L 1 (L 2 resp.) under the morphism  1 (  2 resp.). Let  =  1   2 and consider the partial commutation    1   2 and let  :  *   * be the morphism induced by  1 and  2 (delete colours). The  -product of L 1 and L 2 is L 1 ш  L 2 =  ( [L’ 1 L’ 2 ]  )

15. Partial Commutations bacbcaabca  L 1, babcacbab  L 2 bacbcaabca babcacbab  = {(a,a), (a,b), (b,a), (b,b), (c,a) (c,b), (c,c)} bbaabcbcaabcacacbab  L 1 ш  L 2 The  -product generalizes at the same time concatenation and shuffle: If  = , then L 1 ш  L 2 = L 1 L 2 If  =  1   2, then L 1 ш  L 2 = L 1 ш L 2

16. Partial Commutations Given the partial commutation    1   2, we define the partial commutation  ’    defined as follows: (a,b)   ’  (a,b)   Theorem (Guaiana, R., Salemi) Let L 1, L 2 be languages over , closed under  ’. If L 1, L 2  SF, then L 1 ш  L 2  SF. Corollary. The shuffle of two commutative star- free languages is star-free

17. Partial Commutations If the internal commutation  ’ (i.e.the commutation allowed inside each of the languages L 1, L 2 ) is the «same» as the external commutation  (i.e. the commutations between the letters in L 1 and the letters in L 2 ), then the  -product preserves the star-freeness.

18. Unambiguous Star-Free Languages A language L is the marked product of the languages L 0, L 1, …, L n if L = L 0 a 1 L 1 a 2 L 2 … a n L n, for some letters a 1, a 2, …, a n of . A marked product L = L 0 a 1 L 1 a 2 L 2 … a n L n is unambiguous if every word of L admits a unique decomposition u = u 0 a 1 u 1 …a n u n, with u 0  L 0, …, u n  L n. The product {a,c}*a{  }b{b,c}* is unambiguous

19. Unambiguous Star-Free Languages SF is the smallest Boolean algebra of languages which is closed under marked product The family USF of Unambiguous Star-Free languages is the smallest Boolean algebra of languages of  * containing the languages of the form A*,for A  , which is closed under unambiguous marked product.

20. Unambiguous Star-Free Languages FO : class of languages corresponding to formulas of first order logic. FO k : class of languages corresponding to formulas of first order logic with k variables. Theorem (McNaughton) SF = FO Theorem (Immerman, Kozen) FO 3 = FO Theorem (Therien, Wilke) FO 2 = USF

21. Unambiguous Star-Free Languaes USF  SF  INT  REG Theorem (Castiglione, R.) If L 1, L 2  USF then L 1 ш L 2  SF REG INT SF USF

22. Cyclic Submonoids The languages in the class USF correspond to regular expressions in which the star operation is restricted to subsets of the alphabet. The simplest languages not in USF are the languages of the form L = u*, where u is a word of length  2. Such languages are the cyclic submonoids of  *. We here study the shuffle of cyclic submonoids.

23. Cyclic submonoids Theorem (Berstel, Boasson, Carton, Pin, R.) If a word u contains at least two different letters, then u*  INT. A word u   * is primitive if the condition u=v n, for some word v and integer n, implies u=v and n=1. Theorem (McNaughton, Papert) The language u* is star-free if and only if u is a primitive word.

24. Cyclic submonoids 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 words u,v such that 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 u n v m is primitive for all n,m  2. Theorem (Shyr, Yu) If u and v are distinct primitive words, then there is at most one non-primitive word in the language u + v +.

26. Combinatorics on Words Problem 3 is related to the search for the powers (non-primitive words) that appear in the language u + ш v +. Denote by Q the set of primitive words. For u,v,w  Q, let p(u,v,w) be the integer k such that (u*ш v*)  w* = (w k )* If (u*ш v*)  w* = {  }, then p(u,v,w) = 0. formulas of first order logic

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