Languages and Codes Chapter 8 8 Disjunctive Languages

Languages and Codes Chapter 8 第 8章 醫學影像處理實驗室

Disjunctive Languages n n n A language L is disjunctive if the principal congruence PL is the identity. Def. A language L is disjunctive if for any two distinct words u, v A*, x, y A* s. t. xuy L and xvy L or vice versa. Every regular language L has finitely many PL congruence classes. No disjunctive language is regular. 10/28/2020 醫學影像處理實驗室 2

Properties n n Prop. Let A={a} and L A*. Then L is disjunctive L is not regular. Prop. Let L A*. Then (1) (2) (3). (1) L contains a disjunctive language. (2) L is dense. (3) |L A*w. A*|= for all w A*. 10/28/2020 醫學影像處理實驗室 3

Semi-Discrete n n n A language L is semi-discrete if k 1 s. t. |L An| < k for all n 1. If k =1, a semi-discrete language is called discrete. Prop. Every semi-discrete dense language is disjunctive. 10/28/2020 醫學影像處理實驗室 4

Disjunctive Pairs n n n L 1 L 2 = . The pair (L 1, L 2) is a disjunctive pair if u v, x, y A* s. t. either xuy L 1 xvy L 2 or xvy L 1 xuy L 2. If (L 1, L 2) is a disjunctive pair, then L 1 and L 2 are disjunctive languages. Rem. Let L A*. Then L is disjunctive (L, A* L) is a disjunctive pair. 10/28/2020 醫學影像處理實驗室 5

Examples Prop. (D(1), D(1)(i) ) is a disjunctive pair for any i 2. n Prop. (Q, Q(i) ) is a disjunctive pair for any i 2. n n Prop. Let | A| 2. Then Regular Languages. (1) A+D(1) = A+A. (2) A+D(2) = A+A 2 ( a A A(A {a})+a). 10/28/2020 醫學影像處理實驗室 6

Completely Disjunctive Languages 1 n n Def. An infinite language L is called completely disjunctive (resp. completely dense) if every infinite subset of L is disjunctive (resp. dense). Def. A dense language is called quasicompletely disjunctive if every dense subset of it is disjunctive. 10/28/2020 醫學影像處理實驗室 7

Examples n n Rem. Let | A| =1. Then every infinite subset of A* is completely dense. Ex. For | A| 2, let be a total order defined on A* and A+ ={u 1< u 2 < }. Every infinite subset of L ={u 1 u 2 ui | i 1} is dense and semi-discrete. Hence every infinite subset of L is disjunctive. L is quasi-completely and completely disjunctive. 10/28/2020 醫學影像處理實驗室 8

Completely Disjunctive Languages 2 n n Prop. Let | A| =1 and let L A* be an infinite language. Then (1) (2) (3): (1) L is completely disjunctive; (2) L is regular-free; (3) L is quasi-completely disjunctive. Prop. A language is completely disjunctive it is completely dense. 10/28/2020 醫學影像處理實驗室 9

Completely Disjunctive Languages 3 Prop. Let L 1, L 2 A* and L 1 L 2 completely disjunctive. If L 1 (or L 2 ) is infinite, then L 1 (or L 2 ) is completely disjunctive. n Prop. Let L be infinite. Then the following are equivalent: (1) L is completely disjunctive; (2) L(n) is completely disjunctive n 2; (3) L A*w. A* is finite w A*. n 10/28/2020 醫學影像處理實驗室 10

Quasi-Completely Disjunctive Languages 1 n n Prop. Lpal is quasi-completely disjunctive. Prop. semi-discrete dense language is quasi -completely disjunctive. Let FCD and FQCD denote the families of all completely disjunctive and quasi-completely disjunctive languages, respectively. Prop. L 1 L 2 FQCD L 1 FQCD or L 2 FQCD. 10/28/2020 醫學影像處理實驗室 11

Quasi-Completely Disjunctive Languages 2 Prop. Let L A*. Then the following are equivalent: (1) L is dense; (2) L contains a completely disjunctive lang. ; (3) L contains a quasicompletely disjunctive language; (4) L contains a disjunctive language. n FCD FQCD , A+ Q FQCD but A+ Q FCD. n 10/28/2020 醫學影像處理實驗室 12

Right Disjunctive Languages n n For L A*, except the principal congruence PL , another congruence - right congruence is studied frequently too. The right congruence RL is defined on A* by u v (RL) (ux L vx L, x A* ). n L is right disjunctive if RL is the identity. n L is right dense if u A*, L u. A* . 10/28/2020 醫學影像處理實驗室 13

Quasi-Completely Right Disjunctive Languages 1 n n Def. A completely right disjunctive language is an infinite language of which every infinite subset is right disjunctive. Def. A right dense language is called quasicompletely right disjunctive if every right dense subset of it is right disjunctive. 10/28/2020 醫學影像處理實驗室 14

Quasi-Completely Right Disjunctive Languages 2 n Prop. Let | A| =1 and let L A* be an infinite language. Then the following are equivalent: (1) L is regular-free; (2) L is completely disjunctive; (3) L is quasi-completely disjunctive; (4) L is quasi-completely right disjunctive; (5) L is completely right disjunctive. 10/28/2020 醫學影像處理實驗室 15

Quasi-Completely Right Disjunctive Languages 3 n n Prop. Completely right disjunctive language do not exist for the case | A| 2. Prop. Let L A* be a right dense language. If L is not quasi-completely right disjunctive, then LL 1 is not quasi-completely right disjunctive for every L 1 A*. 10/28/2020 醫學影像處理實驗室 16

Quasi-Completely Right Disjunctive Languages 4 n Prop. Let L A*. Then (1) (2): (1) L is right dense; (2) L contains a quasi-completely right disjunctive language. 10/28/2020 醫學影像處理實驗室 17

Disjunctive Domains 1 Def. A language L A* is disjunctive if x y A*, x y (PL ). n Def. A language L A* is called a (right) disjunctive domain if it satisfies the following condition for every L 1 A*: if (x y (RL 1 )) x y (PL 1 ) for every x y L, then L 1 is (right) disjunctive. n 10/28/2020 醫學影像處理實驗室 18

L-Representation n n For L A+ and w A+, an L-representation of w is defined as w =x 1 y 1 x 2 y 2 xn yn xn+1 , where yj L, j =1, 2, , n, F (xi) L = , i =1, 2, , n+1. If L is a solid code, then the L-representation of each word is unique. 10/28/2020 醫學影像處理實驗室 19

(u, v)-Related Pairs 1 n Def. Let u, v A+. For two words w 1, w 2 A+, (w 1, w 2) is said to be a (u, v)-related pair if there are some {u, v}-representations of w 1 and w 2 s. t. w 1= x 1 y 1 x 2 y 2 xn yn xn+1 , w 2= x 1 z 1 x 2 z 2 xn zn xn+1 , where yi, zi {u, v} and u, v F(xj), 1 i n, 1 j n+1. 10/28/2020 醫學影像處理實驗室 20

(u, v)-Related Pairs 2 n n Rem. If u v (PL ) for some L A* and (w 1, w 2) is a (u, v)-related pair, then w 1 w 2 (PL ). Ex. Let z 1= a(a 2)ab(b 2) and z 2= a(b 2)ab(b 2). Then (z 1, z 2) is a (a 2, b 2)-related pair. If z 3= aba 2 b 3, then (z 1, z 3) is a (a 2, ba)-related pair, since z 1= a(a 2)ab 3 and z 3= a(ba)ab 3. 10/28/2020 醫學影像處理實驗室 21

(u, v)-Related Pairs 3 n n n For u 1, u 2, v 1, v 2 A+, ui vi , i =1, 2, by (u 2, v 2) (u 1, v 1), we mean that (u 2, v 2) is a (u 1, v 1)related pair. 方便證明及使用 Prop. is a partial order and not compatible with multiplication. Rem. For any u v A+, (xuy, xvy) (u, v) for any x, y A*. 10/28/2020 醫學影像處理實驗室 22

(u, v)-Related Pairs 4 n n n Let Psolid denote the set of those pairs (u, v) which satisfy (1) {u, v} a. A+b An, n 3; (2) {u, v} is a solid code; (3) {u, v} {a+b, ab+}= . Lem. For any u v A+, let n max(lg(u), lg(v)). {a 3 nbuab 3 n, a 3 nbvab 3 n} is a solid code. Lem. For any u v A+, (u, v)-related pair (z 1, z 2) s. t. (z 1, z 2) Psolid. 10/28/2020 醫學影像處理實驗室 23

A Special Language 1 n n n Consider (u, v) Psolid. Let n = lg(u) = lg(v). For any x A+ with u, v F(x), we let = bmabnxanbam, where m is the position of x in A* ordered by the standard total order. Define H ={ | x A+, u, v F(x)}. Clearly, for any w H, u, v F(w). And, u F(w) v F(w) xwy H for any x, y A*. 10/28/2020 醫學影像處理實驗室 24

A Special Language 2 n For 0 r s, define D = H E. (Not disjunctive. ) 10/28/2020 醫學影像處理實驗室 25

Properties of the Set D Lem. Let (u, v) Psolid w 1, w 2 A* s. t. (w 1, w 2) is not a (u, v)-related pair. Then F(w 1) {u, v} F(w 2) {u, v} w 1 w 2 (PD). n Lem. Let (u, v) Psolid. Then (1) uv vu (PD). (2) If w 1 w 2 A* is s. t. (w 1, w 2) is not a (u, v)related pair, then w 1 w 2(PD). n 10/28/2020 醫學影像處理實驗室 26

Disjunctive Domains 2 n n n Th. Let {a, b} A. Then (1) (2). (1) L A* is a disjunctive domain; (2) (u, v) Psolid , (u, v)-related pair (z 1, z 2) s. t. {z 1, z 2} L. Cor. u v A+, x, y A* s. t. xuy L xvy L L is a disjunctive domain. Cor. Q is a disjunctive domain. 10/28/2020 醫學影像處理實驗室 27

Properties of Disjunctive Domains n n Prop. Let A={a}. Then L A* is a disjunctive domain L is infinite. Prop. disjunctive domain is dense. Prop. No discrete language is a disjunctive domain. Prop. If L is a disjunctive domain, then LL 1 and L 1 L are disjunctive domains L 1 . 10/28/2020 醫學影像處理實驗室 28

Right Disjunctive Domains Th. L A* is a right disjunctive domain u v A*, x A* s. t. {u, v}x L. n Prop. right disjunctive domain is right dense. n Prop. No discrete language is a right disjunctive domain. n n Cor. Q is a right disjunctive domain. 10/28/2020 醫學影像處理實驗室 29

Properties of Right Disjunctive Domains n n n Cor. If L is a right disjunctive domain, then for any finite subset F L, L F is a right disjunctive domain. Cor. There is no minimal (right) disjunctive domain. Rem. right disjunctive domain contains infinitely many primitive words. 10/28/2020 醫學影像處理實驗室 30

-Languages n n An -word over a finite alphabet A is an infinite sequence of letters of A. By A we denote the set of infinite words ( -words). Subsets of A are called -languages. There are three syntactic congruences for languages, i. e. , P , L, IL and OL. 10/28/2020 醫學影像處理實驗室 31

The Congruence P , L An analogue of the principal congruence when considering relations on A* introduced by an -language L A is the equivalence relation P , L , defined by: for u, v A*, u v (P , L ) ( x A* A , xu L iff xv L). n P , L is a congruence and L A is called P-disjunctive if P , L is the equality. n 10/28/2020 醫學影像處理實驗室 32

The Congruence IL n n n Let L A. An equivalence relation IL on A* introduced by L is defined by: for u, v A*, u v (IL ) ( x, y A*, x(uy) L iff x(vy) L). IL is a congruence on A*, which has been called an infinitary syntactic-congruence. L A is called I-disjunctive if IL is the equality. 10/28/2020 醫學影像處理實驗室 33

The Congruence OL n n n Let L A. A compatible equivalence relation OL on A is defined by: for , A , (OL ) ( x A*, x L iff x L). OL is a congruence on A , which is called the -syntactic congruence of L. L A is called O-disjunctive if OL is the equality. 10/28/2020 醫學影像處理實驗室 34

P-Dense and P-Discrete An -language L is P-dense if u A*, x A* A s. t. xu L. n An -language L is P-discrete if u v A*, {xu , xv } L lg(u) lg(v), where x A* A . n Prop. P-discrete P-dense -language is P-disjunctive. n 10/28/2020 醫學影像處理實驗室 35

I-Dense and I-Discrete n An -language L is I-dense if u A*, x, y A* s. t. x(uy) L. n I-disjunctive -language is I-dense. n I-dense -language is infinite. n An -language L is I-discrete if u v A*, {xu , xv } L lg(u) lg(v), where x A*. 10/28/2020 醫學影像處理實驗室 36

O-Disjunctive and -Dense n n L A is O-disjunctive 1 2 A , x A* s. t. x 1 L x 2 L, or vice versa. The complement of an O-disjunctive language is O-disjunctive too. O-disjunctive -languages are also called separative -languages. L A is called -dense if A , x A* s. t. x L. 10/28/2020 醫學影像處理實驗室 37

P-Disjunctive -Languages Prop. Let | A| 2 and L A. Then the following are equivalent: (1) L is P-disjunctive; (2) u, v A*, lg(u) = lg(v), u v (P , L) u = v ; (3) p, q Q, lg(p) = lg(q), p q (P , L) p = q. n Let A+ ={x 1< x 2< x 3< } be ordered by the standard total order and let i = xiai, a A. n = 1 2 A. { } is P-disjunctive. n 10/28/2020 醫學影像處理實驗室 38

I-Disjunctive -Languages 1 Prop. Let | A| 2 and L A. Then the following are equivalent: (1) L is I-disjunctive; (2) u, v A*, lg(u) = lg(v), u v (IL) u = v ; (3) p, q Q, lg(p) = lg(q), p q (IL) p = q. n Prop. I-discrete I-dense -language is I-disjunctive. n 10/28/2020 醫學影像處理實驗室 39

I-Disjunctive -Languages 2 n n Th. An -language is I-dense it contains an I-disjunctive -language. (For any I-dense -language, one must construct an I-disjunctive subset. ) Prop. L A is I-disjunctive PL is I-disjunctive for any finite prefix code P A +. 10/28/2020 醫學影像處理實驗室 40

O-Disjunctive -Languages n n Prop. Let | A| 2 and L A* an infinite prefix code. Then L' ={uw | u L, A , w Alg(u), w <p } is an O-disjunctive -language. Prop. Let P A+ be a finite prefix code. Then L A is O-disjunctive PL is an O-disjunctive language. 10/28/2020 醫學影像處理實驗室 41

Families of Disjunctive -Languages 1 An -language is said to be I-closed if for x, u, v A+, xunv L for every n 1 xu L. n Prop. I-closed I-disjunctive -language is P-disjunctive. n Prop. O-disjunctive -language is P-disjunctive. n Prop. P-disjunctive -language which is not O-disjunctive. 醫學影像處理實驗室 10/28/2020 42 n

Families of Disjunctive -Languages 2 n n n Prop. O-disjunctive -language is I-disjunctive. Prop. I-disjunctive -language which is not O-disjunctive. Prop. P-discrete -language which is not I -discrete I-discrete -language which is not P-discrete. 10/28/2020 醫學影像處理實驗室 43

Quasi-Completely O-Disjunctive -Languages n n An O-disjunctive -language L is called quasi-completely O-disjunctive if -dense subset of L is O-disjunctive. Prop. Let {a, b} A and L={aibw | A , w Ai, w <p }. Then L is a quasi-completely O-disjunctive -language. 10/28/2020 醫學影像處理實驗室 44