1 { REGULAR LANGUAGES DEFINED BY A LIMIT OPERATOR. Keywords: 1-regular language, 1-acceptor, transition graph, limit operator.

Transcription

1 Mathematica Pannonica 7/2 (1996), 215 { { REGULAR LANGUAGES DEFINED BY A LIMIT OPERATOR Karel Mikulasek Department of Mathematics, Technical University, Brno, Technicka 2,Czech Republic Received: May 1995 MSC 1991: 68 Q 45 Keywords: 1-regular language, 1-acceptor, transition graph, limit operator. Abstract: Finite deterministic 1-acceptor accepting both nite and innite words over a nite alphabet is introduced. It is shown that 1-regular languages can be dened as sets of 1-words accepted by an 1-acceptor. A limit operator on regular languages is used to dene a special class of 1-regular languages. In 1-acceptors accepting these languages incidence relations between the sets used for acceptance are determined. 1. Introduction The paper deals with special classes of 1-regular languages. If a nite alphabet is given, then by an1-regular language over we understand the union of a regular and an!-regular language over. In this paper we show that it is possible to dene such languages by means of a single nite-state device. A deterministic machine capable of constructing both nite and innite sequences is rst introduced in [7]. In [4] the structure of the sets constructed in [7] is investigated and in [5] a non-deterministic version of such machines is shown. Another generalization can be found in [3] where the notion of a k-machine is introduced. In [6] generalized non-deterministic acceptors accepting sequences of both nite and innite length are introduced. Limit operators on regular languages are usefull tools for investigating relations

2 216 K. Mikulasek between regular and!-regular languages. In [8] and [9] they are also used for studying topological properties of!-languages. The operator lim used in this paper appears e.g. in [1], [2], [6] and [9]. A generalization of some ofthe results of [6] gives rise to a question of how the structure of 1-regular languages dened by inclusions between a limit closure of their words and their!-words is reected in the incidence of sets used for accepting words and!-words. 2. Preliminaries In this paper, all the numbers are non-negative integers if not specied otherwise. By! we denote the least innite ordinal number. A non-empty nite set is called alphabet and its elements letters. denotes the set of all nite sequences of. The empty word is denoted by. Its elements are called words. For a u 2,wewrite u = u 1 u 2 :::u n.for two non-empty words u = u 1 u 2 :::u m, v = v 1 v 2 :::v k, we dene their catenation u:v = u 1 u 2 :::u m v 1 v 2 :::v k.weput: :u = = u: = u, u 2. Thecatenation of k identical words w 2 is denoted by w k.! denotes the set of all innite sequences over. If w 2!, then we write w = w 1 w 2 :::.These sequences are called!-words. For w 2, w! denotes the!-word w:w: : : :. Weput 1 = [!.The elements of 1 are called 1-words. Wedene the catenation of two 1-words u v by usingthe above denition for u v 2 and putting u:v = u 1 u 2 :::u n v 1 v 2 :::,ifu 2 v 2!.For u 2!,the catenation is not dened. For a word w = w 1 w 2 :::w n, n 1, wedene its length as j w j= n and for w 2! weput j w j=!. Finally weput j j= 0. For a w 2! wedene the setofleft fractions of w: If (w) =fu 2 j w = u:v, v 2! g. Asubset of is called a language, asubset of! an!-language and asubset of 1 an 1-language. For an innite sequence q = q 0 q 1 q 2 ::: of elements of a nite set Q we dene In(q) asthe set of all the elements of Q which occur innitely many times in q. In the sequel, we will use the following evident assertion: For an arbitrary sequence s 1 s 2 ::: s k, k 1 of elements of In(q), there are indices i 1 <i 2 <:::<i k such that s j = q ij, 1 j k.

3 1-regular languages 217 A graph is a triple D =(U H) where U is a nite setofnodes, H is a nite set of arcs and : H! U U is an incidence mapping. A graph D 0 =(U 0 0 H 0 )isasubgraph of a graph D =(U H) if U 0 U H 0 H and 0 : H 0! U 0 U 0 is a restriction of on H. For V U, we say that D V =(V 0 H 0 )isthe subgraph of D induced by V if H 0 is a maximum subset of H such that (h) 2 V V for every h 2 H 0 and 0 is a restriction of on H 0. A trace in a graph D =(U H) isanalternating sequence of nodes and arcs u 0 a 1 u 1 a 2 ::: a n u n, n 1where (a i )=(u i;1 u i ), 1 i n. We say that a graph D =(U H) is strong if, for every u v 2 U, there is a trace from u to v. In proofs wewillusethe followingobvious assertion: D =(U H) is strong i, for every u v 2 U, there is a trace in D starting in u, ending in v and containing all nodes of U regular languages Denition 3.1. We say that ave-tuple A =(Q q 0 F)isanite deterministic acceptor or, shortly, anacceptor if Q is a nite set of states, is an alphabet, : Q! Q is a transition function, q 0 2 Q is the initial state and F Q is a set of nal states. For w 2 2 w = a 1 :a 2 :::a n, n 0, a sequence q(w) = q 0 q 1 q 2 ::: q n is called the run of A on w if q i = (q i;1 a i ), 1 i n. Weput (w) = = q n.if (w) 2 F then we say that A accepts w. The set of all words accepted by A is denoted by L(A). A language L is called regular if L = L(A) for an acceptor A. Denition 3.2. We say that ave-tuple A =(Q q 0 ) is a - nite deterministic!-acceptor or, shortly, an!-acceptor if Q is a nite setofstates, is an alphabet, : Q! Q is a transition function, q 0 2 Q is the initial state and 2 Q is a system of subsets of innitely many times occurring states. For w 2!, w = a 1 :a 2 ::: a sequence q(w) = q 0 q 1 q 2 ::: is called the run of A on w if q i = = (q i;1 a i ), 1 i. Weput! (w) = In(q(w)). If! (w) 2 then we say that A accepts w. The set of all words accepted by A is denoted by L(A). An!-language L! is called!-regular if L = L(A) for an!-acceptor A. Note 3.3. Clearly, for a w 2! the run of an acceptor on w is also dened and soisthe runofan!-acceptor on a w 2.

4 218 K. Mikulasek Denition 3.4. An 1-language L 1 is called 1-regular if a regular language L F and an!-regular language L!! exist such that L = L F [ L!. Denition 3.5. We say that a six-tuple A =(Q q 0 F ) is a nite deterministic 1-acceptor or, shortly, an1-acceptor if Q is a - nite set of states, is an alphabet, : Q! Q is a transition function, q 0 2 Q is the initial state, F Q is a set of nal states and 2 Q is a system of subsets of innitely many times occurring states. For w 2 1 wedene the run of A on w usingeither Denition 3.1 if w 2 or Denition 3.2 if w 2!. Similarly, weput 1 (w) = = (w) or 1 (w) =! (w) depending onwhether w 2 or w 2!. If 1 (w) 2 F [, then we say that A accepts w. The set of all 1-words accepted by A is denoted by L(A), the set of all!-words accepted by A is denoted by L! (A) andthe set of all words accepted by A is denoted by L F (A). Denition 3.6. Let A =(Q q 0 F ) be an 1-acceptor. We say that G(A) =(Q H) isthe transition graph of A if H = (q i a q j ) qi q j 2 Q a 2 q j = (q i a) (q i a q j )=(q i q j ): For a P Q, wedenote by G(A P )the subgraph of the transition graph of A induced by P. Theorem 3.7. Let L 1 be an1-regular language. Then there exists an 1-acceptor A such that L = L 1 (A). Proof. We have L = L F [ L! where L F is regular and L!! is!-regular. This means that L F = L(B) L! = L(C) where B =(P p 0 G) is an acceptor and C =(R r 0 ) is an!- acceptor. Let us construct an 1-acceptor A =(Q q 0 F ) by putting Q = P R, q 0 =(p 0 r 0 ), ((p r) a)=((p a) (r a)) and F = f(p r) 2 Q j p 2 Gg. To construct let us consider w 2 L!.Let (1) p(w) =p 0 p 1 p 2 ::: be the runofb on w and (2) r(w) =r 0 r 1 r 2 ::: the runofc on w. Put (3) q(w) =(p 0 r 0 ) (p 1 r 1 ) (p 2 r 2 ) ::: and dene =fin(q(w)) j w 2 L! g. is nite since, for every w 2 L!, In(q(w)) Q. Let w 2 L. For a w 2 L F wehave (w) 2 G, which means that 1 (w) 2 F and thus w 2L F (A). If w 2 L!,then, using the notation (1), (2) and (3), we see that In(q(w)) 2 andthus, since clearly 1 (w) = In(q(w)), we getw2l! (A).

5 1-regular languages 219 To prove the inverse inclusion L 1 (A) L let us rst take aw 2 2L F (A). This gives us a run ofa on w: (p 0 r 0 ) (p 1 r 1 ) (p 2 r 2 ) ::: ::: (p k r k ) k 0, where p k 2 G. Then, of course, p 0 p 1 p 2 ::: p k is a run of B on w and w 2 L(B) = L F. If, on the other hand, w 2 L! (A), we getthe runof B on w: p(w) = p 0 p 1 p 2 :::, the run of C on w: r(w) = r 0 r 1 r 2 ::: and the runof A on w: q(w)= =(p 0 r 0 ) (p 1 r 1 ) (p 2 r 2 ) ::: with 1 (w) 2. This means that there is a w 0 2L(C) such that (4) In(q(w)) = In(q(w 0 )) where again p(w 0 ) = p 0 p 0 1 p0 2 ::: is the run of B on w, r(w0 ) = = r 0 r 0 1 r0 2 ::: is the run of C on w and q(w0 ) = (p 0 r 0 ) (p 0 1 r0 1) (p 0 2 r0 2) ::: is the runofaon w. Let s 2 In(r(w)). This means that s occurs innitely many times in r(w) andthus a p i exists such that (p i s) occurs innitelymany times in q(w) or(p i s) 2 In(q(w)) and (4) gives us (p i s) 2 In(q(w 0 )). Then s must occur innitely many times in r(w 0 ), which implies s 2 In(r(w 0 )) and In(r(w)) In(r(w 0 )). However In(r(w 0 )) In(r(w)) can be proved in muchthe sameway. The equality In(r(w)) = In(r(w 0 )) then implies w 2L(C) =L!. 4. Limit 1-regular languages Denition 4.1. Let L.PutlimL = fw 2! j card(lf(w) \ L) = =!g. Theoperator lim maps the set of all languages over into the setofall!-languages over. Note 4.2. In [1] this operator is denoted by L and in [8] it is called -limit. Weusethe notationasintroduced in [2] and [6]. Lemma 4.3. Let w 2! and L. Then w 2 liml iasequence w 1 w 2 ::: of words from L exists such that j w i j<j w j j i<j w i 2 lf(w) i 1: Proof. The proof follows directly from Def Denition 4.4. We say that an1-acceptor A =(Q q 0 F ) is concise if L F (A) 6= L! (A) 6= and, for any A 0 =(Q q 0 F 0 ), A 00 =(Q q 0 F 00 )where F 0 F, 00, we have L F (A 0 ) L F (A), L! (A 00 ) L! (A): Lemma 4.5. Let A =(Q q 0 F ) be aconcise 1-acceptor. Then the following conditions are equivalent: 1. liml F (A) L! (A) 2. if, for an S Q, G(A S) is strong and S \ F 6=, then S 2.

6 220 K. Mikulasek Proof. Let the rst condition hold and let S Q be such that G(A S) is strong and S \ F 6=. Let f 2 S \ F. Since A is concise, there is a word u 2L F (A) such that 1 (u) =f. G(A S) being strong and f 2 S implies that there is a trace in G(A S) c 1 (c 1 a 1 c 2 ) c 2 (c 2 a 2 c 3 ) ::: (c k;1 a k;1 c k ) c k k>1 such that it contains all the nodes of G(A S) and c 1 = c k = f. Letus now consider a sequence of words w 1 w 2 ::: where w i = u:(a 1 :a 2 ::: :::a k;1) i and an!-word w = u:(a 1 a 2 :::a k;1)!. It is easy to see that w i 2L F (A) w i 2 lf(w), i 1and j w i j<j w j j i < j,which, by Lemma 4.3, implies w 2 liml F (A). It is also obvious that fc 1 c 2 ::: ::: c k g is exactly the setofstates occurring innitely many times in the runofaon w and thus 1 (w) =S. However, by the assumption, w 2L! (A) andthus S 2. Let the second condition hold and w 2 liml F (A). By Lemma4.3, we get a sequence of words w 1 w 2 ::: w i 2 lf(w), i 1such that j w i j<j w j j, i < j. Let us consider an innite sequence of states from F (5) f 1 f 2 ::: f i = 1 (w i ) i 1: Since F is nite, there is an f which occurs innitely many times in (5). Denote by (6) q(w) =q 0 q 1 q 2 ::: the runofaon w. Since, clearly, (5) is a subsequence of (6), we have f 2 In(q(w)). G(A In(q(w)) is strong and j w i j<j w j j implies that it has at least one arc and sowegetin(q(w)) 2 bythe assumption and nally w 2L! (A). Lemma 4.6. Let A =(Q q 0 F ) be aconcise 1-acceptor. Then the following conditions are equivalent: 1. liml F (A) L! (A) 2. F i \ F 6= for every F i 2. Proof. Let the rstcondition hold and let F i 2. As A is concise, there exists a w 2L! (A) such that 1 (w) =F i. By the assumption then w 2 liml F (A) and, by Lemma 4.3, there exists a sequence of words w 1 w 2 ::: w i 2 lf(w) w i 2L F (A) i 1 such that j w i j<j w j j i< j.thus we have 1 (w i ) 2 F i 1and since F is nite, there mustbeanf 2 F which occurs innitely many times in the sequence (7) 1 (w 1 ) 1 (w 2 ) ::: : Let

7 1-regular languages 221 (8) q(w) =q 0 q 1 q 2 ::: be the runofaon w. It is easy to seethat (7) is a subsequence of (8), which means that f 2 In((q(w)) = 1 (w) =F i. Therefore F \ F i 6= and the second condition holds. If the second condition holds and w 2L! (A), then 1 (w) =F i 2 2. Since F i \ F 6=, there exists an f 2 F i \F which occurs innitely many times in the runofa on w and thus there is a sequence w 1 w 2 ::: w i 2 lf(w) w i 2L F (A) i 1 such that j w i j<j w j j i <j.then, by Lemma 4.3, w 2 liml F (A). Denition 4.7. Let D =(U H) be a graph. For every v 2 U, we dene a system of subsets of U: (D v) =fv U j v 2 V ^ G(D V ) is strongg. For a W U, weput (D W) = S (D v). v2w Theorem 4.8. Let A =(Q F ) be aconcise 1-acceptor. Then the following conditions are equivalent: 1. liml F (A) =L! (A) 2. = (G(A) F). Proof. Let the rst condition hold and F i 2. Then G(A F i ) is strong since A is concise. By Lemma4.6wehave F i \F 6= with anf 2 F i \F such that F i 2 (G(A) f). This means that F i 2 (G(A) F). For an F i 2 (G(A) F), G(A F i ) is strong and F i \ F 6=. Thus, by Lemma 4.5, we get F i 2. If = (G(A) F), then, for every S Q such that G(A S) is strong and S \ F 6=, S 2 (G(A) F)=and, by Lemma 4.5, liml F (A) L! (A). Next if F i 2, then F i 2 (G(A) F) implies F i \ F 6= and, by Lemma 4.6,wegetL! (A) liml F (A). References [1] DAVIS, M.: Innitary Games of Perfect Information, in: Advances in Game theory, Princeton Univ. Press, Princeton N. J. 1964, 89{101. [2] EILENBERG, S.: Automata, Languages and Machines, Volume A, New York, Academic Press [3] GRODZKI, Z.: The k-machines, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 18/7 (1970), 399{402. [4] KWASOWIEC, S.: Generable Sets, Information and Control 17 (1970), 257{ 264.

8 222 K. Mikulasek: 1-regular languages [5] MEZNIK, I.: G-machines and Generable Sets, Information and Control 20 (1972), No 5, 499{509. [6] NOVOTN Y, M.: Sets Constructed by Acceptors, Information and Control, 26 (1974), 116{133. [7] PAWLAK, Z.: Stored Program Computers, Algorytmy 10 (1969), 7{22. [8] STEIGER, L.: Research inthe Theory of!-languages, J. Inf. Process. Cybern. EIK 8/9 23 (1987), 415{439. [9] STEIGER, L.: Hierarchies of Recursive!-languages, Elektron. Inf. Verarb. Kybern. EIK 5/6 22 (1986), 219{241.