Corresponding to automaton with the output M, we c n build the language transformatio schema equivalent to M, which also recognizes the same language pair with the initial automato.. Thi
Trang 1T,!-p chi Tin tioc vaDi'eu khifln hoc, T.17, S 2 (2001), 39-4
NG UYEN VAN DINH
Abstract In accordance with the concept of automato with the output we have built the la guage tran s f o r-mati o schema E (see [6]) In this paper we stu y the relatio between the aut o mata co mplexity g(E), the number of essential v rtices l EI an the depth of o e atio s 5o a langua e tra sf o rmati o s chema. The esti-matio of the automata complexity of alanguage transformation schema that h lds operatio s with restricted degree is also given
Tom t~t Du'a tren kh ai n iern 0 to mat co loi ra, ta xay du g oU"<!C1U"<)"00 bien oo'i ngori ngir (BD N) [6]
Khi cac 1U"<!c00BDNN chira cac phep l~p khOng han che thl oi? phire tap 0 to mat cua no kho g p u thuoc
vao so phep l~p v a oa du'o'c danh gia b ·i [6]' nhirng khi so phep l~p, cl g nhu· mot so phep to t n khac n u·
phep lly cac c~p tjr chRn, lly cac c~p tir le, ph ep bo' sung , co b~c bi hari cM thl oi? phire tap 0 to mat cda
cac hro'c 0 BDNN se ph u th uoc vao oi? sau o~t dilu (b~c) cd a cac phep toano
Bai nay trinh bay moi lien h~ gira oi?phu'c tap 0 to mat so dinh cot yeu va oi? sau o~t dfiu cac ph ep toin cila mot lu'o'c 00 BDNN co chira cic phep toin co b~c ducc han che
1 INTRODU C TI O N
Using n tation £ = X X Y , wherein X = {Xl , X2 , , X n } is input alphabet of source (original) lang age, and Y = {Yl, Y 2, · Y m } isoutput alphabet of target (final) language An automaton with the output, M = (S, X, Y , 50 , 5, A, F), recognizes a language on the input alp abet, symb lized T g ,
and transforms it into another language on the output alphabet, that is symbolized Tv This lan uage transformatio isdue to automata mappings, on which the len ths of words are completely preserved (See [6] In brief, the automaton M recognizes a language pair R = (T x, T y ) Corresponding to automaton with the output M, we c n build the language transformatio schema equivalent to M, which also recognizes the same language pair with the initial automato (See[6] When the lang age transformatio s hema holds unrestricted repetitions, its automata complexity does not depend o the number of the repetito s and has been appraised by [6].However, if the number of the repetito s,
a well a an ther o erations: creat ing even word pairs, odd word pairs or complement have limited degrees, the a tomata complexity ofthe language tran s formation schema will be reliant o the depth of ope at ons. This paper isto analyze evaluate the automata complexity ofa lan uage transformation schema that contains operations with restricted degree
2 PRELIMINAR Y C ON C EPT S
Some preliminary concepts ab ut the lan uage transformatio schema presented in [6] are sum-marized in th is part In additio , new relating concepts will be separately described The language
pair (Tx, T y ) on the dual alphabet £ = X x Y, denoted R, and R' " ispair of repeated lan uage with degree m ofTx and Tv The subset ofR , containsword pairs with eiher even or odd lengths, denoted er R) or 0(R) , respectively The set of all initial parts of word pairs, of which lengths are
at most 5, den ted ){ (R , s ).
The langua ge transformatwn graph o the dual alphabet £ = X x Y is a directed grap G, in which, a particu lar vertex called e ntry vertex, denoted a( G), and a nonempty set of vertices c lled the set of f n l ve r ti ces, denoted F ' (G) , an we denote the set of all vertces as A( G). Initial vertex an
Trang 2fnal vertex of an edge a are denoted a ( a ) and , 8 ( a ) , respectively On each edge a of the graph Gis
la ele by a gro p of word p irs o th dual alpha et c , this group is denoted M c ( a ) Suppose that
to edge a, (1 : i :: ;t) , then word pairs Tl T 2 T t are considered is created b this path We denote th set of all word pairs gen rated by path initiating from vertex a and ending at vertex ,8 as NG(a, , 8).
The set N r , (a, , 8) is inductively defined as the following prnciples:
1 A= ( c , c ), wherein e is a empty word of X* a d Y*
2 I X , Yare word pairs, wherein X E Nn (a , /, ) , and " t = a (a) , ,8 = , 8 (a) , Y E M n (a) then
X Y E Nr.(a, , 8 )
D e finitio n 2.1 The language transformation schema ~ on the dual alphabet c =X x Y is a range
of th la g age ta sformation graphs o c:
and on this ran e, we build a functio mB( a ) , d fn d on the set of all e ges of all graphs Gi
(1 ::;i::; n) and satisfie the following conditions:
1 For a certain edge a of C, (1 ::;i ::;n ) , the functio mB(a ) is satisfie one of followin standards:
1.1 mB( a ) = A and Mn, ( a ) = A, the a is calle a empty e ge
a is calle essential edge
1 3. mB(a) = G) (1 ::;J ::; n - 1) and M n, ( a ) = C( N (G ) )) ' th n e ge a is stated to d p n ing
o graph G) and is called a e en edge
1.4 mB( a ) = Gk (1 ::;k ::;n - 1) and MG , (a) = 0 (N(Gk )) , the edge a is state to depending
o graph Gk and is caled an odd edge
1 5. mB ( a ) =C; (1 : :; r: :; n - 1) and M e, ( a ) =N S ( Gr) , th n e ge a is state to d p nding on
graph G; and is calle re eated e ge with degree s
1.6 mB( a ) = Gt (1 ::;t ::; n - 1) and M r., ( a ) = N(N( G t } , s), the edge a is state to de ending
o graph Gt and it is called complement edge with degree s
2 Each graph Gl,G 2, , Gn -l has one and only one e g of graph Gn d p nding o Graph Gn
calle a base graph
G n isc lle the base graph of the la guage transformation schema If ~ contains only one graph,
this uniqu graph is also sig e ~, and c lled a simple language ta sformation schema The set of
word p irs N ( Gn ) is considere to be create by ~, a d d n ted N(~). We at times use N x (~)
and N y (~) to d note separately the set of origin and the fnal words define by~ Obviously,
N ( ~ ) ~ Nx ( ~ ) x N y ( ~ )
The v rte a of graph G is calle a essential v rtex if it is entere by at least one essential
e ge The number of essental v r ces of graph G is sig e IGI The number of essental v rtices of
all graph belong to ~ is signe I~I We state that graph C, depends on graph G) if it contains some edge depen ing o graph G).
D e finition 2.2. For th la guag ta sformation schema, th depth of operations, denoted l(~),
and d termin d as follow:
Let a is a unintention l e ge of graph G; (1 ::; r ::; n) , th n th d pth of operatio s of a is
sig e l(a), and we d fine th depth of o eratio in G by the formula:
LtG) =m a xl( a ) ,
< Er.
wherein l( a ) is inductively d termined as follows:
1 If a is an e e edge, odd e g , re eate edge with d gree n or complement e ge with degre s
and a d pends on graph G) (1 ::;j ::;i ) , and l( G) has been d fined, th n:
Trang 3THE AUTOMATA COMPLEXITY OF THE LANGUAGE TRANSFORMATION SCHEMA 4
2 For the remaining cases, l(a) =O
The equatio l(~) = I(Gn) is admitted
Definition 2.3 Possible minimum number of states of an weak deterministic finite automaton with o tput (see [6]) which recognizes N (~) , is c lled the automata complexity of the language transformation schema ~, and denobe g(~).
3 THE RESULTS
In this part, we will prove a theorem so that to appraise the automata complexity g ( ~ ) of the
language transfor;mation schema ~ which depends on the number of essential vertices I~I and the
depth ofoperatio s.
Firstly, we prove some lemmas
Lemma 3.1 For every simple language transformation s c hema ~ J th e re exi s ts an weak d e t e rmi n i s ti c
T(A) = N(~) and I A I :S 21EI +1
Pr o of As ~ is a simple language transform schema, thus, according to [6]'it is possible to build an automato with output, M = (5, X, y , so , 8,A, F) as follows:
- 5: the set ofstates of automaton M , includes all vertex sig s of ~
- Entry vertex sign of ~ isregarded a initating state S o of automaton M
- The set of final vertices of ~ is admitted to be a set of final states of M
-State transiional function 5 : 5 x ~ - +5 ofautomaton M is determined: Vs E 5, Vx E X then
8 ( , x ) = { S il, S i 2 , , s id {} V i with 1: SJ' : St then: zE N x [s, S i J)' S iJ E5
- The output function A: 5 x X - + Y of automato M is determined: Vs E 5, Vx E X then
A ( s, x ) =Y E Y, wherein y isan element corresp nding to x in the pair ( x, y ) which is written in the
edge ( s, 5 ( s, x) ) ofthe simple lan uage transformatio schema ~
- The input and output alphabet X , Y of the simple language transformation schema ~ are
considered as the input and o tput alp abet, respectively, of automaton M
In this way, it is obviously that Tx ( M ) = Nx (~) and Ty (M) = N y(~ ). Indeed, we have build
an automaton M that recognizes the same language pair with the simple language transformation
schema ~.'
With automaton M, using algorithm and determinazing automata program [7],we can build an
the same set of word pairs with automato M). In additio , as the results in [2], hen:
T(A) = T(M) = N(~) and I A I :S 2 MI + 1= 2 EI +1
Lemma 3.2 For eve r y s imp l e l anguage t r an s formation s ch e ma ~J th e re e xi s t s another si mpl e sc h e ma
~ ' J s uch that :
N(~/) = [(N(~)) and I~/ I :S 21~ ·
P r oof 1 Builds sdiema ~ : This schema includes two vertices Q oan Ql ' Vertex Q ois entry as wel
as final vertex of ~ From Qo to Ql and conversely from Ql to Q o , there are just right n edges on each of which, one ofsymbols from the dual alphabet C = X x Y iswritten, and two different edges are written with two different pairs ofsymbols Thus, the schema ~o just right two essential vertices,
i.c, I ~ l = 2, an N (~ o ) contains word pairs whose lengths are even
2 Build schema ~/: We regard ~' as an intersecto ofschemas ~o and ~, as ~' is the intersection
of simple schemas, according to the results of [6]'the conclusion is:
N (~/) = N (~ o ) n N ( ~ ) and I ~/ I :S l~ l.I 1 = 2·1~1·
Obviously, N (~/) is a set of word pairs possessing even lengths
Trang 4N GU YEN VAN DINH
L e mma 3 3 Fo r eve r s imple language tran s formation schema E, th e re exist s another s imple s chema
E ' , s u c h t h at :
P r oo ] 1 Builds a schema E1 : This schema is structured the same as Eo, except ao is input vertex,
different edges are written wih two different pairs ofsymbols Hence, the schema El has just right
two essential vertices, obvio sly, IEl = 2, and N(Ed contains word pair whose lengths are odd only
2 Build up a schema E': Weregard E' a anintersectio ofschemas El and E, a E' is the intersection
of simple schemas, according to the results of 1 ]'the conclusion is:
N (E') = N(Ed n N(E) and IE'I ::;IEllIEI = 2.IEI
Lemma 3 4. For every s impl e languag e tran s formati o s chema E and with any int e ger s > 0 , there
e xi s t s an o th e s i mple s ch e ma E' , s uch that :
are structured as same as schema E Nevertheless, their vertices are symbolized vario sly Each final
vertex ofih-s hema (0 ::;i ::;s - 1) has an empty edgelinking with the entry vertex ofi+ 1th-schema
The entry vertex of the first schema isconsidered as the entry of E', and the set of final vertex
Lemma 3 5. For every s imple language tran s formation s chema E and with any integer m > 0 , there
e xi s t s an o th e s imple s ch e ma E' , s uch th a t:
Pro o f 1 Build up a schema Err" including m + 1vertice : a , al, , a m-l, am From ai to ai+l
c are written From am -l to am, there are n edges on which different pairs of signs from c are
written The vertex ao is regarded as the entry, and,ao, al, ,arn-2, am-l are acknowledged as
whose len ths are bounded at m and they have just right m essential vertices
ofword pairs whose lengths are at most m, and IE'I ::;m.IEI
L em ma 3.6. For any t h e lan g uage t r ans f o rmati o s h e ma E, of whi c h the d e pth of op e at i o s
re s t r ic t ed at s, there e xi s t s a s impl e sc h e ma E', s uch that :
P r oo f The lemma is proved by mathematical induction on to the depth of operations on the schema
E
Trang 5THE AUTOMATA COMPLEXITY OF THE LANGUAGE TRANSFORMATION SCHEMA 43
2 Suppose that, ~ = (GI, G 2, ,Gm-l) and al, a2, , a p are even edges, odd edges or complement
edges with degree t, repeated edges with degree u on the graph G n. As the depth of operations not exceed s, thus t :S sand u.:Ss.
Suppose that, a; (1 :S i :Sp) depends on the graph Gki. For every i (1 :S i :Sp) it is possible to use
all the graphs on which Gki depends, including G ki , to build schema ~ki. Then we have:
N(~kd =N(Gkd·
According to the definition of the depth of operations, there is a conclusion:
l(~) = I(G n) ~ l (a;) +1= I(Gki) +1= l(~k;) +1
Hence:
To match the definition of induction, for each i (1 :S i :Sp) , there exists a simple schema ~L,such that:
N(~~i) = N(~kd and I~~il :S l~ki l ·sl(Bk ; :S l~kil·sl(B)-l.
Ifc; is an even edge (or an odd edge), then, in accordance with Lemma 3.2 (or Lemma 3.3) we can build up a simple schema 6.i, such that:
, N(6 ;) = [(N(~~ill = [(N(L;k;)) = {(N(Gkill
(or N(6 i) = O(N(~~ ; )) = O(N(~ki)) = O(N(Gk ; )), respectively) with:
16.,-1< 21~1·1k, <- 21~ ·1.s1(B)-1k, <- I~ ·1 tci · s1(B)
If a, (1 :S i :S p) is the complement edge with degree r (or repeated edge with degree
according to Lemma 3.5 (or Lemma 3.4) we can build a simple schema 4i, such that:
r), then
N(6 ;) = ){(N(~~ ; ), r) = ){(N(~kd, r) = ){(N(Gki) , r)
(or N(6 i) =N;(~~ ; = N;(~kd = N;(Gk;), respectively) with:
l6.il:S r.I~~il:S rl~kilsl(B)-l:s l~ki l sl(B).
Replace a, (1:S i :Sp) on G n with schema 6.i, in accordance with the definitions of substitution (seeI6]), we have a simple schema ~I, such that:
N(~I) = N(~)
with:
I~II= IGnl+ I: l6.il
l:Si:Sp
:SIGnl+ I: I~ki Is1(B)
l:Si:Sp
:S IGnlsl(B) + I: 1~'lilsl(B)
l:Si:Sp
= (IGnl + I: I~ki I)sl(B)
l: i:Sp
Theorem For any language transformation schema ~, on which the depth of operations restricted
at s, then:
Proof 1 For the language transformation schema ~ as above, using Lemma 3.6, we can build up a simple schema ~I, such that:
2 For this simple scherr.a ~I, using Lemma 3.1, we can build up an weak deterministic finite au-tomaton with output A, such that:
Trang 6T(A) = N(~I) =N(~) , and:
g(~) = I A I < ZIL'1 +1: zlLlal(El +1
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U nit e d Nation s Inte r national School-Hanoi