Languages and Codes

1. Languages and Codes Chapter 10 第 10章

2. Operations of Words and Languages • A groupoid is a non-empty closed set for a given binary operation. • When the operation is associative, groupoids are called semigroups. • Properties of operations will affect the properties of those languages that operations are applied to. 醫學影像處理實驗室

3. Catenation & Primitive Words • One of the main results about primitivity is that every non-empty word can be uniquely expressed as a power of a primitive word. • If we define the ◦-primitive words analogous to the primitive words, then every ◦-primitive word is primitive whenever ◦ is the catenation of words and ins-primitive whenever ◦ is the insertion of words. 醫學影像處理實驗室

4. Binary Word-Operations • A binary word-operation with right identity (shortly bw-operation) is defined as a mapping ◦:A*A*2A* with ◦(u,1)={u}. • Define ◦(L1,L2)=uL1,vL2◦(u,v) and ◦(L1,)= =◦(,L2). • Often identify singleton sets with their elements. 醫學影像處理實驗室

5. Iterated Bw-Operations • The iterated bw-operation ◦i is defined by ◦0(L1,L2)=L1 and ◦i(L1,L2)=◦(◦i–1(L1,L2),L2) whenever i1 for languages L1 and L2. • The i-th◦-power of a non-empty language L is defined as L◦(0)={1} and L◦(i)=◦i–1(L,L) for i1. 醫學影像處理實驗室

6. ◦-Primitive Words • A non-empty word w is called ◦-primitive if wu◦(i) for some word u and i1 yields i=1 and w=u. • Example. a2ba2ba2(a2ba)(2) and a2ba2ba2(aba2)(2).  A word could be obtained from many ins-primitive words by self-inserted. 醫學影像處理實驗室

7. +-Closures of Languages • The +-closure of a non-empty language L with respect to a bw-operation ◦, denoted by L◦(+), is defined as L◦(+)=k1L◦(k). • A language L is ◦-closed if u,vL imply ◦(u,v)L. 醫學影像處理實驗室

8. E-Closed, c-Closed andi-Closed • For LA*, consider (1) uvL  wL  uwvL, (2) uwvL  wL  uv1  uvL, (3) uvL  uwvL  wL. • A language satisfying (1), (2) or (3) is called e-closed, c-closed or i-closed, respectively. 醫學影像處理實驗室

9. Examples • Example. The following two languages are examples of e-closed and c-closed languages: (a) The Dyck language LDyck. LDyck does not satisfy condition (3). (b) The balance language Lab. Lab satisfies condition (3). 醫學影像處理實驗室

10. Insertion Closures • Let [L]=L(+)=n1L(n). • Then [L] is the smallest e-closed language containing L. [L] is called the e-closed language generated by L. • L1L2  [L1][L2], L1,L2A+. 醫學影像處理實驗室

11. Ins-Primitive Words • Let Qins denote the set of all ins-primitive words. • a2baQins, [a2ba] is e-closed but neither c-closed nor i-closed for a3b(a2ba)[a2ba]  a2ba[a2ba] but a3b[a2ba], and a2b(a3b)a[a2ba]  a2ba[a2ba] but a3b[a2ba]. 醫學影像處理實驗室

12. Properties • Prop. If uD(1), then wj[u] for j1 implies w[u]. • Prop. Let uA+. Then (1) [u] is regular;  (2) ua+ for some aA;  (3) [u] is i-closed. 醫學影像處理實驗室

13. Shuffle-Closures • For LA*, consider (1) u,vL  uvL, (2) uL  uvL  vL. • A language satisfying (1) is called shuffle-closedor sh-closed. A language satisfying (1) and (2) is called strongly shuffle-closedor ssh-closed. 醫學影像處理實驗室

14. Examples • Let A={a,b}, k a fixed positive integer and Lk(a) the language consisting of all words containing at least k letters a. Then Lk(a) is sh-closed, but not ssh-closed. • The balance language Lab is ssh-closed. • L(+) is the smallest sh-closed language containing L. 醫學影像處理實驗室

15. Properties • Rem. L1L2  L1(+)L2(+), L1,L2A*. • Rem. (L(+))(+)=L(+), LA*. • Prop. ssh-closed language is a commutative language and a free submonoid of A*. 醫學影像處理實驗室

16. Shuffle-Free Languages • A non-empty language LA+ is called -free if (L(+),L)L=. • For LA*, if  a -free language BL\{1} s.t. B(+)=(L\{1})(+), then B is called a -base of L. 醫學影像處理實驗室

17. Construction • For a non-empty LA+, define K1=L Ii={w|wKi and lg(w)lg(y) for all yKi}, i1, Ki=L\(1ji–1Ij)(+) for i2, (L)=i1Ii. • Fact. (L)L. 醫學影像處理實驗室

18. Properties • Prop. L((L))(+) and (L) is -free. • Prop. If LA+ is sh-closed, then L=((L))(+). • Fact. For LA*, (L) is a -base of L. • Prop. The -base of a language LA* is unique. • Prop. If a language L{1} is ssh-closed, then the -base of L is a commutative hypercode. 醫學影像處理實驗室

19. Generalization • Consider a bw-operation ◦. • A non-empty language LA+ is called ◦-free if ◦(L◦(+),L)L=. • For LA*, if  a ◦-free language BL\{1} s.t. B◦(+)=(L\{1})◦(+), then B is called a ◦-base of L. 醫學影像處理實驗室

20. Construction • For a non-empty LA+, define K1=L Ii={w|wKi and lg(w)lg(y) for all yKi}, i1, Ki=L\(1ji–1Ij)◦(+) for i2, ◦(L)=i1Ii. • Fact. ◦(L)L. 醫學影像處理實驗室

21. Properties 1 • A bw-operation ◦ is called plus-closed if for any non-empty language L, L◦(+) is ◦-closed. • A bw-operation ◦ is called length-increasing if for any u,vA+ and w◦(u,v), lg(w)>max{lg(u),lg(v)}. • Prop. Let ◦ be plus-closed and length-increasing. Then L(◦(L))◦(+) and ◦(L) is ◦-free. 醫學影像處理實驗室

22. Properties 2 • Prop. Let ◦ be plus-closed and length-increasing. If LA+ is ◦-closed, then L=(◦(L))◦(+). • Fact. Let ◦ be plus-closed and length-increasing. For any non-empty language LA* with L{1}, ◦(L) is a ◦-base of L. 醫學影像處理實驗室

23. Properties 3 • Prop. Let ◦ be plus-closed and length-increasing. Then the ◦-base of a language LA* is unique. 醫學影像處理實驗室

24. General Properties • A bw-operation ◦ is called left-inclusive if for u,v,wA*, ◦(◦(u,v),w)◦(u,◦(v,w)). • Lem. If ◦ is left-inclusive then for any non-empty language L, L◦(+)is ◦-closed. • Lem. If ◦ is plus-closed then for any uA*, ◦m(◦n(u,u),◦p(u,u))u◦(+), for all m,n,p0. 醫學影像處理實驗室

25. ◦-Primitivity 1 • Prop. Let ◦ be plus-closed and length-increasing. Then for wA+, ◦-primitive word uand an integer n1  wu◦(n). • A bw-operation ◦ is called propagating if for any u,vA* and w◦(u,v), na(w)=na(u)+na(v) any aA. 醫學影像處理實驗室

26. ◦-Primitivity 2 • Prop. Let ◦ be plus-closed and propagating. Then for wA+, ◦-primitive word uand a uniqueintegern1  wu◦(n). • Lem. Let ◦ be plus-closed and propagating and let |A|2. If wA+, aA, wa+ then  an integer m1 s.t. all the words v1◦(w,wm–1a), v2◦(awm–1,w), v3wma, and v4awm are ◦-primitive. 醫學影像處理實驗室

27. ◦-Primitivity 3 • Let Q◦(A) denote the set of all ◦-primitive words over A. • A language LA* is called right (resp. left) ◦-dense if for each wA+, uA* s.t. ◦(w,u)L (resp. ◦(u,w)L). • Prop. Let ◦ be plus-closed and propagating and let |A|2. Then Q◦(A) is right and left ◦-dense. 醫學影像處理實驗室

28. Home Work • How to define the ◦-density of languages? • A bw-operation ◦ must be equipped with what properties to make the set Q◦(A) ◦-dense. • How to define the ◦-disjunctivity of languages? 醫學影像處理實驗室