Parallel Composition of Words with Re-entrant Symbols - CiteSeerX

Parallel Composition of Words with Re-entrant Symbols Alexandru Mateescu Arto Salomaa

Turku Centre for Computer Science TUCS Technical Report No 15 May 1996 ISBN 951-650-757-3 ISSN 1239-1891

Abstract We introduce and investigate an operation for words and languages, referred to as the re-entrant product. This operation is related to parallel composition of concurrent processes that contain re-entrant routines. Intuitively the reentrant product of two words contains all traces of the parallel execution of these words assuming that the atomic actions are re-entrant routines. This paper investigates language-theoretic properties of re-entrant products.

TUCS Research Group

Mathematical Structures of Computer Science

1. Introduction

Let  be an alphabet. The set of all words over  is denoted by  and  is the empty word. A word w = a a : : : an 2 , ai 2 , 1  i  n, can be considered as the trace of a process, where ai, 1  i  n, are atomic actions. If one considers all the traces of the parallel execution of two processes with traces u and v, u; v 2 , then one obtains the shue of u and v, denoted uttv, and de ned formally by: 1 2

(auttbv) = a(uttbv) [ b(auttv);

and

(utt) = (ttu) = fug; where u ,v 2  and a ,b 2  . The shue operation is extended in a natural way to languages : the shue of two languages L and L is : 1

L ttL = 1

2

2

[

u2L1 ;v2L2

uttv:

It is well known that the shue operation is a commutative and an associative operation with the unit element . The shue operation is not suitable to describe the existence of re-entrant routines. A re-entrant routine, see [1], [3], is a routine that can be executed concurrently by more than one user (process). Throughout this paper the reader may consult the monograph [15] for all unexplained notions of formal languages. For general results concerning the theory of semirings, the reader may consult the monographs [5] and [9]. Related problems of re-entrant routines are studied also in [6] and [10]. Consider the following version of the Latin product, denoted by . If u = a a : : :an and v = b b : : :bm are words over , then 1 2

(

1 2

b u  v = aa aa :: :: :: aanbb bb :: :: :: bbm;; ifif aan = n m n 6= b 1

2

2 3

1

1

2

1 2

1

By de nition, u   =   u = u. It is easy to observe that the ordered system M = (; ; ) is a monoid, called the Latin monoid. Let  be the embedding of  to M, i.e.,  :  ?! M, (a) = a, for all a 2 . Clearly,  can be extended to a unique morphism of monoids

fl :  ?! M: 1

The set of at-words over , denoted Fl() is the set of all xed points of the morphism fl, i.e.,

Fl() = fw 2  j fl(w) = wg: For instance, if  = fa; bg, then aba, baba are in Fl(), but ab a, b a b are not in Fl(). Note that the set of at-words, Fl(), is exactly the set of all words free of square letters. The at-image of an word w is by de nition the word fl(w). For instance, fl(ab a) = aba, fl(b a b) = bab. If w is in Fl() the the at-image of w is w. De nition 1. Let w = ai1 ai2 : : :ainn be a nonempty word over , where ai 2 , 1  i  n and ij  1, for all j , 1  j  n, and, moreover, ak is dierent of ak , for all 1  k  n ? 1. The power of w is the vector p(w) = (i ; i ; : : :in). 3

3

2 2

2 2

1

2

+1

1

2

De nition 2. The re-entrant order or r-order, denoted r, is de ned as u r v i fl(u) = fl(v) and p(u)  p(v); where u; v 2  . (Here p(u)  p(v) means the componentwise order between vectors of nonnegative integers.) By de nition,  r w, for any w 2 . +

Remark 3. The relation r is a partial order relation on  with the

rst element .

Example 4. Note that, for instance ba b r b a b, whereas ba b and ba 2

2 3

2

are not comparable.

The proof of the following proposition is omitted.

Proposition 5. (i) If ui r vi, 1  i  n, then u u : : : un r v v : : :vn. (ii) If u r v v : : :vn, then there are ui, 1  i  n, such that u = u u : : : un and ui r vi, 1  i  n. (iii) If u u : : :un r v, then there are vi, 1  i  n, such that v = v v : : :vn and ui r vi, 1  i  n. 1

2

1 2

1

2

1

2

1 2

2

1 2

2. The re-entrant product

In this section we introduce the main operation of this paper, called the re-entrant product. De nition 6. Let L be a language. The downr mapping is de ned as:

downr (L) = fu j u r v; for some v 2 Lg: Note that downr is a closure operator. De nition 7. Let u and v be words over . The re-entrant product of u with v, denoted u ./ v is:

u ./ v = downr (uttv): The above operation is extended in the natural way to languages. If L and L are languages, then: 1

2

L ./ L = 1

2

[

u2L1 ;v2L2

u ./ v:

Note that

L ./ L = downr (L ttL ): The re-entrant product is a monotonous operation in each argument. 1

2

1

2

Theorem 8. The re-entrant product is a commutative and associative operation. Proof. Obviously, ./ is a commutative operation. Now assume that x 2 u ./ (v ./ w), where x; u; v; w 2 . It follows that there is a word t = u0 1u1 : : : n un such that x r t, where u0u1 : : : un = u and 1 2 : : : n 2 v ./ w. Hence, there exists z 2 v ./ w such that 1 2 : : : n r z. From Proposition 5, (iii), it follows that z = z1z2 : : : zn, such that i r zi, 1  i  n. Note that y = u0z1u1 : : :zn un is in utt(vttw) and thus in the set (uttv)ttw. Moreover, x r t = u u : : : n un r u z u : : : znun = y 2 (uttv)ttw: 0 1

1

0 1

1

Thus, x 2 (uttv) ./ w. Since (uttv) ./ w  (u ./ v) ./ w we conclude that x 2 (u ./ v) ./ w. Note that ./ is distributive over union, but ./ does not have unit element. Thus we obtain: 3

Corollary 9. Let  be an alphabet. The quadruple Hr = (P (); [; ./; ;) is a commutative hemiring (i.e., a semiring without unit element for the multiplication operation, see [5], [9]).

De nition 10. Let L be a language. The shue closure of L, denoted by Ltt , is de ned by [ Ltt = Lk tt ; (

)

k0

where Lk tt denotes the shue product of L with itself (k ? 1)-times, if k  2. L tt = fg and L tt = L. Analogously, the re-entrant closure of L is denoted by L./. Thus, [ L./ = Lk ./ ; (

)

0(

)

1(

)

(

)

k0

where Lk ./ denotes the re-entrant product of L with itself (k ? 1)-times, if k  2. Moreover, L ./ = fg and L ./ = L. (

)

0(

)

1(

)

In the next sections we will show that there are some signi cant dierences between the shue closure operator and the re-entrant closure operator.

3. Some properties of the re-entrant product

Here we present some properties of the re-entrant product and of the re-entrant closure operator. Proposition 11. Let L be an arbitrary language. (i) L./ = (downr (L))./: (ii) L./ = (L)./: (iii) L./ = (Ltt )./: Proof. Since the operator downr is distributive over union and idempotent, we obtain the equality (i). Now observe that L  L  L./ and thus, L./  (L)./  (L./)./ = L./. Similarly, one can show the equality (iii).

The proof of the following proposition is omitted. Proposition 12. Let L; L ; L be languages. 1

2

4

(i) L./ ./ L./ = L./: (ii) (L./ ./ L./)./ = L./ ./ L./: 1

2

1

2

Proposition 13. Let L and L be languages. 1

2

(i) (L [ L )./ = L./ ./ L./. 1

2

1

2

(ii) If  2 L \ L , then (L L )./ = L./ ./ L./. 1

2

1

2

1

2

Proof. (i) Note that (L1 [ L2 )  L./1 ./ L./2 . Hence (L1 [ L2 )./  (L./1 ./ L./2 )./. From Proposition 12, (ii) it follows that (L1 [ L2)./  L./1 ./ L./2 . Obviously, the converse inclusion does hold. (ii) Observe that since  2 L1 \ L2, it follows that L1L2 = L1 [ L2 [ L1L2. Therefore, (L1 [ L2)./  (L1L2)./. For the converse inclusion, it is easy to see that for all k  0,

(L L )k tt  (L [ L ) k tt : 1

2

(

)

1

2

2 (

)

Hence, (L L )./ = (L [L )./ and using the equality (i) we obtain the equality (ii). 1

2

1

2

Proposition 14. Let A; B; C be languages. Then, (AB C )./ = (ABC )./: +

Obviously, (ABC )./  (AB +C )./. Now observe that AB +C = S Proof. k k1 AB C . One can show that

downr (AB k1 C tt : : : ttAB kn C )  downr ((ABC ) k1 (

:::+kn )(tt) ):

+

Therefore the equality holds.

Proposition 15. Let A be a full AFL. (i) A is closed under the operator downr . (ii) If A is closed under shue, then A is closed under the re-entrant product, ./.

5

Proof. (i) Let L   be a language in A. We de ne a GSM , M = (Q; ; ; ; q0; F ) such that L(M ) = downr (L). Q = fq0g [ fqa j a 2 g, F = Q, (q0; a) = f(qa; a)g, for all a 2 . (qa; a) = f(qa; a); (qa; )g and if a 6= b, then (qa; b) = f(qb; b)g. One can verify that L(M ) = downr (L). (ii) It follows from (i) and the closure of A under shue.

Corollary 16. The families of regular languages and of recursively enumerable languages are closed under the re-entrant product. Remark 17. The family of context-free languages is not closed under the re-entrant product. Consider the languages: L = f(ab)n(cd)n j n  1g and L = f(ef )m(gh)m j m  1g. Let L be the language L ./ L . Observe that the language L0 = L \ (ab) (ef ) (cd) (gh) is not context-free and 1

2

+

+

+

+

1

2

hence L ./ L is not context-free. 1

2

4. Some properties of the re-entrant closure

In this section we prove some properties of the re-entrant closure. We will show that in many respects the re-entrant closure is dierent from the shue closure, although, apparently, they seem to be closely related: L./ = downr (Ltt ). Remark 18. The shue closure of a word can be a non-context-free language. Let w be the word abc and consider the language L = wtt = (abc)tt. Since L \ a b c = fanbncn j n  0g: we conclude that L cannot be a context-free language. If w = ab, then wtt is the Dyck language over fa; bg which is a context-free language but not a regular language. In general, one can show that the shue closure of a word w is a regular language if and only if w is a word over a one-letter alphabet. However, this is not the case for the re-entrant closure of a word. Theorem 19. Let w be a word over some alphabet . The re-entrant closure of w, w./ , is a regular language. Proof. Firstly, we will show the property for some xed words and then we will give the general construction. Assume that w = ab and  = fa; bg. We will prove that (ab)./ = ab [ fg: () Obviously, (ab)./  ab [fg. For the converse inclusion, note that  is in (ab)./ and let u = avb be a word from ab. The proof is by induction on + + +

6

the length n of v. If n = 0, then u = ab 2 (ab)./. Assume that the property holds for words v of length n and let v be a word of length n +1. Then v = av0 or v = bv0 for some word v0 of length n. By the inductive assumption, av0b 2 (ab)./. Hence there is a k  0 such that av0b 2 downr ((ab)k tt ). Therefore, there is t = at0b 2 (ab)k tt such that av0b r at0b. Now observe that z = aat0bb 2 (ab) k tt and that aav0b r aat0bb. Hence avb = aav0b 2 (ab)./. For the second case, i.e., v = bv0, de ne the word z as being z = aabt0b and note that again abv0b r z and z 2 (ab) k tt . Hence the equality () holds. Assume that w = abc and  = fa; b; cg. A similar proof shows that: (

(

( +1)(

)

)

)

( +1)(

)

(abc)./ = a bbc [ a bc [ fg: +

+

+

()

+

In general, assume that w = a a : : : an is a word over some alphabet , where ai 2 , 1  i  n. We start by de ning recursively the languages Li, 2  i  n ? 1 and Kj , 2  j  n ? 1 as follows: L = a a , K = a, Li = (Li ai ) (K [ : : : [ Ki)ai , Ki = (Liai ), where 2  i  n ? 2. Similarly, we de ne Rn? = an? an , Jn? = an, Rn?i? = an?i? (Jn?i [ : : : [ Jn? )(an?iRn?i ) , Jn?i? = (an?i Rn?i ), where 1  i  n ? 3. One can verify the following claims: (i) If uai 2 Pref (wtt ) and j u jai =0, then uai 2 Li, for all 2  i  n ? 1. (ii) If aj v 2 Suf (wtt ) and j v jaj =0, then aj v 2 Rj , for all 2  j  n ? 1. Now consider the set An? ; = Ln? R . Observe that if t is a word from wtt such that the rst (leftmost) occurrence of an? precedes the last (rightmost) occurrence of a , then t 2 An? ; . Now we consider the set Bn? ; such that this set contains all the words t from wtt such that the rst (leftmost) occurrence of an? does not precede the last (rightmost) occurrence of a , i.e., all occurrences of a precede all occurrences of an? . The set Bn? ; is obtained from the set An? ; = Ln? R = (: : :)an? a (: : :) by replacing the expression an? a with the expression a ;n? an? , where  ;n? = fa ; : : :; an? g. We continue the above process for all pairs of indices (i; j ), with i > j , until we obtain the language Bk ;k = a a : : : an , for some 2  k  n ? 1. Let M be the language obtained as the union of An? ; with all these languages Bj;i. Claim: w./ = downr (M ). Proof of Claim. One can easily see that wtt is a subset of M and therefore w./ = downr (wtt )  downr (M ). For the reverse inclusion consider a word u 2 M . We omit the details of an inductive argument based on the length of u (see the example for  = fa; bg 1 2

+

+1

2

+1

+

1

+ 1 2

2

1

1

+1 +

1

2

1

1

1

12

1

2

1

2

12

12

1

2

2

1

12

12

1

3

2

3

1

2

2

+ 2

1

3

2

+ + 1 2

+1

+

12

7

2

+

2

1

1

and  = fa; b; cg), showing that there is a word z 2 wtt such that u r z. Hence, u 2 downr (wtt ) = w./. Finally, note that M is a regular language and from Proposition 15, (i), we conclude that downr (M ) = w./ is a regular language.

De nition 20. A language L is strictly bounded i L  aa : : :an for some letters a ; a ; : : : ; an. A language L is bounded i L  ww : : :wn for 1 2

1

2

1

2

1

some words w ; w ; : : :; wn .

2

Theorem 21. Let L be an arbitrary strictly bounded language. (i) The language downr (L) is a regular language. (ii) The language L./ is a regular language. Proof. (i) Without loss of generality we can assume that L  a+1a+2 : : :a+n . (Otherwise L is a nite union of languages, each of them being a subset of some a+j1 a+j2 : : :a+jm , where 1  jf  n, 1  f  m.) If L is a nite language, then downr (L) is a nite language and thus a regular language. If L is a an in nite language, then let i1; i2; : : :; ik be all those indices in f1; 2; : :+: ; ng with the property that L contains an in nite number of subwords from ait , 1  t  k. Moreover, if ir is an index not in fi1; i2; : : : ; ikg, then let mr be the largest power of a word in a+ir that is a subword in L. One can verify that downr (L) = ap11 ap22 : : : apnn ; where ps = + if ps 2 fi1; i2; : : : ; ik g and ps = ms, otherwise. Therefore, from Proposition 15, (i), downr (L) is a regular language. (ii) Firstly assume that L  a+1 a+2 : : : a+n . From (i) we deduce that

downr (L) = ap1 ap2 : : : apnn ; where ps = +, if ps 2 fi ; i ; : : :; ik g and ps = ms, otherwise. Now using Proposition 14, we obtain that 1

1

2

2

L./ = (aq1 aq2 : : :aqnn )./; where qs = 1, if ps = +, and qs = ps , otherwise. Observe that the language F = aq1 aq2 : : : aqnn is nite, say F = fv ; v ; : : :; vmg. From Proposition 13, (i), it follows that: L./ = F ./ = v./ ./ v./ ./ : : : ./ vm./: 1

2

1

1

2

2

1

8

2

By Theorem 19 and from Corollary 16 we conclude that L is a regular language. Now consider the general case, i.e., L = L [ L [ : : : [ Ln , where each Li is a subset of some aj1 aj2 : : :ajm . By Proposition 13, (ii), L./ = L./ ./ L./ ./ : : : ./ L./n. From the above proof and using Corllary 16, we conclude that L./ is a regular language. +

+

1

+

2

1

2

Note that in the above Theorem 21, the language L is not necessarily a recursively enumerable language and still downr (L) and L./ are regular languages. Moreover, observe that although L is a strictly bounded language, the language L./ can be even non-bounded. The results from Theorem 21 are not true for bounded languages. For instance, the language L from Remark 17 is a bounded language. Observe that downr (L ) = L and, obviously, L is not a regular language. Let L , L be the languages from Remark 17 and consider the language L = L [ L . Note that L is a bounded context-free language but L./ is not a context-free language (see the argument from Remark 17). Now we will extend the result of Theorem 19 to a certain family of regular languages. Notation. If L is a language, then L = L [ fg. De nition 22. The family of strong -regular languages, denoted R, is the smallest family of languages such that: 1

1

1

1

1

2

1

2

(i) If F is a nite language, then F is in R. (ii) The family R is closed under union, catenation and Kleene star operations.

Remark 23. Note that if a language L is in R, then L is regular. The

converse is not true. Moreover, there are regular languages that contain  and which are not in R. For instance the language L = a b [ fg is a regular language that contains  but L is not in R. + +

Theorem 24. If L 2 R, then L./ is a regular language. Proof. Let L be a language in R. The proof is by induction on the structure of L. If L = F for some nite language F = fu ; u ; : : : ; ung, then from Proposition 13, (i), it follows that:

L./ = u./ ./ u./ ./ : : : ./ u./n: 1

2

9

1

2

By Theorem 19 and from Corollary 16 we conclude that L is a regular language. Now consider the inductive step. Firstly, assume that L = L [ L , for some L , L in R. From Proposition 13 (i) we deduce that L./ = L./ ./ L./. By the inductive assumption L./ and L./ are regular languages. Since the re-entrant product of regular languages is a regular language, it follows that L./ is a regular language. Assume that L = L L , for some L , L in R. Note that  2 L \ L . Hence, by Proposition 13, (ii), L./ = L./ ./ L./. By the inductive assumption L./ and L./ are regular languages and thus L./ is a regular language. Finally, consider that L = L for some L in R. From Proposition 11, (ii), we deduce that L./ = L./. Therefore, by the inductive assumption, L./ is a regular language. 1

1

2

1

1

1

2

2

2

1

1

1

2

2

1

2

2

2

1

1

1

However, the following problem remains open: Open Problem. Let L be a regular language. Is L./ a regular language? We state one more property of the re-entrant product. Theorem 25. Let L be a language over an alphabet . There exists a nite language F such that L ./  = F ./ . Proof. From Higman's Theorem, it follows that there is a nite language F such that Ltt = F tt. Hence, L ./  = downr (Ltt) = downr (F tt) = F ./ .

5. Conclusions

The re-entrant routines are used in many operating systems like for instance in the kernel of UNIX, LINUX, etc. An algebraic development of processes was initiated in [13], namely CCS . An important contribution on this line is the paper [4], see also [2], that is the starting point for a more algebraic theory of concurrent processes. Here we model some aspects of parallel compositions of concurrent processes that contain re-entrant routines. For other versions of the shue operation the reader may consult [6], [7], [8], [12]. The in ltration operation considered in [14], see also [11], is an alternative product for modeling some aspects of the use of re-entrant routines in parallel execution of processes. We hope to continue this study in a forthcoming paper.

10

7. References 1. T. Axford; Concurrent Programming Fundamental Techniques for Real Time and Parallel Software Design, John Wiley & Sons, New York, 1989. 2. J.C.M. Baeten, W.P. Weijland; Process Algebra , Cambridge Tracts in Theoretical Computer Science 18, Cambridge University Press, Cambridge, 1990. 3. M. Ben-Ari; Principles of Concurrent and Distributed Programming, Prentince Hall international series in computer science, C.A.R. Hoare, Series Editor, Prentice Hall International, 1990. 4. J.A. Bergstra, J.W. Klop; Algebra of Communicating Processes , in Math. and Comp. Sci., ed. J.W.Bakker, M.Hazewinkel, J.K.Lenstra, North-Holland, Amsterdam, 1986, 89-138. 5. J.S. Golan; The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science , Longman Scienti c and Technical, Harlow, Essex, 1992 (Pitman Monographs and Surveys in Pure and Applied Mathematics 54). 6. J. S. Golan, A. Mateescu, D. Vaida; Semirings and Parallel Composition of Processes, submitted. 7. M. Kudlek, A. Mateescu; Distributed Catenation and Chomsky Hierarchy. Fundamentals of Computation Theory, FCT'95, Dresda, Lecture Notes in Computer Science, LNCS 965, Springer Verlag, 1995, 313-322. 8. M. Kudlek, A. Mateescu; Rational and Algebraic Languages with Distributed Catenation. Developments in Language Theory, DLT'95, Magdeburg, 1995, to appear in World Sci. Publ. 9. W. Kuich, A. Salomaa; Semirings, Automata, Languages, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, Berlin, 1986. 10. M. Lipponen, T. Harju, A. Mateescu; Flatwords and Post Correspondence Problem, to appear in Theoretical Computer Science, TCS. 11. M. Lothaire; Combinatorics on Words , Addison-Wesley Publ.Comp., Reading Mass., Encyclopedia of Math. and Appl., vol. 17, Gian-Carlo Rota, Ed., 1983. 11

12. A. Mateescu; On (Left) Partial Shue, in: Results and Trends in Theoretical Computer Science, Graz, June, 1994, LNCS 812, SpringerVerlag, (1994) 264-278. 13. R. Milner; A Calculus of Communicating Systems, LNCS 92, SpringerVerlag, Berlin, 1980. 14. J. Sakarovitch, I. Simon; Subwords, in Combinatorics on Words, M. Lothaire, Addison-Wesley, Read. Mass., 1983. 15. A. Salomaa; Formal Languages, Academic Press, New York, 1973.

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