Conway, J.H. (1971). Regular Algebra and Finite Machines. Chapman and Hall.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with Tests expressions and a corresponding automatatheoretic presentation. One outcome of the theory is a coinductive proof principle, that can be used to establish equivalence of our Kleene Algebra with Tests expressions. 1 Introduction Kleene algebra (KA) is the algebra of regular expressions [2,4]. As is well known, the theory of regular expressions enjoys a strong connection with the theory of finite state automata. This connection was used by Rutten [12] to give a coalgebraic treatment of regular expressions. One of the fruits of this coalgebraic treatment is coinduction, a proof ....

Conway, J. H., "Regular Algebra and Finite Machines," Chapman and Hall, London, UK, 1971.

....factorial languages and introduce the notion of a canonical decomposition. Then we prove that for each factorial language, a canonical decomposition exists and is unique. 1 Introduction Catenation of languages is a natural operation, and language equations involving it attract much interest [4]. Nowadays the theory of solving them is not yet complete even for simple forms of equations, like the commutation equation X 1 X 2 = X 2 X 1 . It has non evident solutions even in the case of small nite sets [5] although the problem is completely settled for two element sets and some classes of ....

J. H. Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971.

....combinations of another series Cohn s theorem, 16] On the other hand, almost nothing was known on the commutation of sets of words. The centralizer e(X) of a set of words X is the largest set of words commuting with it, i.e. Xe(X) e(X)X. The following question has been proposed by Conway, [18], more than thirty years ago: is it true that the centralizer of any rational language is rational The answer to this question is still unknown up to date; as a matter of fact, much weaker question than Conway s problem are also open, such as: is it true that the centralizer of any finite ....

....[46] for some extensions. Nevertheless, there are several natural and apparently very difficult combinatoffal problems arising from this area. It was more than thirty years ago when Conway proposed such a problem, asking whether the largest set commuting with a given rational set is rational ([18]) The problem remained unanswered up to date; even worse, it is not known even if the centralizer of any rational set is recursively enumerable. Moreover, these questions are also unanswered for finite sets We discuss in this chapter the notion of centralizer of a set of words and prove that ....

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J.H. Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971.

....equation XL = LX . As it can be readily seen, the notion of centralizer of L is well de ned for any language L; as a matter of fact, C(L) is the union of all languages commuting with L. The best known problem with respect to the notion of centralizer is the intriguing question raised by Conway [9] more than thirty years ago. Conway s Problem: Is it true that for any rational language, its centralizer is rational Surprisingly enough, very little is known on Conway s problem. In fact, much weaker questions than Conway s are unanswered up to date: e.g. it is not known whether or not the ....

.... solving an old conjecture of Ratoandromanana [21] A similar characterization holds also for the commutation with periodic, binary, and as we proved here ternary sets of words, but not for languages with at least four words; this solves a conjecture of [13] The intriguing problem of Conway [9], asking if the centralizer of a rational language is rational, still remains far from being solved. We proved here that the centralizer of any rational code X is rational and in fact, C(X) X) an interesting connection between the notions of centralizer and primitive root. Except the ....

J.H. Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971.

....of such solutions can be extremely high. However, if we fix X and ask Y to be the maximal (but not finite) solution of (1) the situation changes essential. Indeed, it was asked more than 30 years ago by Conway whether such a solution is always rational for a given rational language X, cf. [5]. Obviously, such a maximal language, referred to as the centralizer of X, is unique. Thus we formulate: Conway s Problem. Is the centralizer of a rational language rational, as well Amazingly this problem is still unanswered, or in fact it is not even known whether the centralizer of a ....

J. H. Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971.

..... The same result holds if one considers iterated shuffle defined by: x : 1 x x x Omega x Omega x : But there is nothing surprising here, Theorem 7. 2 follows from the fact that the regular sets (over the one letter alphabet) do not have a finite axiomatization, cf. 19] [9]. The proof of Theorem 7.2 using this fact is standard. 8 OPEN PROBLEMS 11 There are other operations of interest. In contrast with the star operation, the equational laws of reversal(mirror image) can be captured by the involution axioms relative to any variety considered above: x y) x y ....

J. Conway. Regular Algebra and Finite Machines. Chapman & Hall, London, 1971.

....does not match the expressive power of timed automata which requires both renaming and intersection. We will use the following shorthands: a = #a# 0 ; # ; # = # # ; #i = ## i times Operations #, and # satisfy well known properties of Kleene algebra (see [Con71] We state some simple additional algebraic properties involving absorbing concatenation. Proposition 1 (Algebraic Properties of Absorbing Concatenation) The # operation satisfies the following equalities: # distributivity: # ) # = # # # and # ( # ) # # # ....

John H. Conway. Regular Algebra and Finite Machines. Chapman & Hall, London, 1971.

....a power language. We also prove that two codes commute if and only if they have the same primitive root. Both these results extend elementary properties of words to codes. This fact is remarkable since the set of codes is not even closed under product. A related problem was proposed by Conway ([6]) 30 years ago, asking if the maximal set commuting with a regular set R, i.e. its centralizer (denoted in this paper by C(R) is regular. Up to date, it is only known that the answer is positive for sets of cardinality at most three ( 4] 9] and that the complement of the centralizer of any ....

J.H.Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971.

....the commutation XY = Y X , is poorly understood. On the other hand, it proposes several natural and apparently very difficult combinatorial problems. It was almost 30 years ago when Conway proposed such a problem, asking whether the maximal set commuting with a given rational set is rational, see [6]. The problem remained unanswered up to date, even for finite sets. Even worse, it seems to be unknown whether the centralizer of a finite set is recursive, or even recursively enumerable. A related problem asking whether any decomposable rational language L, i.e. a rational language having the ....

....of a finite set is recursive, or even recursively enumerable. A related problem asking whether any decomposable rational language L, i.e. a rational language having the decomposition L = XY for some languages X;Y 6= f1g, is decomposable via rational languages, is much simpler, as shown in [6], cf. also [8] 12] and [3] Another related problem is to search for a characterization of all languages commuting with a given rational or finite set. In the case of multisets, i.e. polynomials over noncommuting variables and with rational coefficients, this problem has an elegant solution due ....

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J. H. Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971.

....i.e. we consider X;Y satisfying XY = Y X: 3) We recall that for a given X there exists the unique maximal Y satisfying (3) Such a Y is the centralizer of X, in symbols C(X) Problem 8. For a given nite X is its centralizer rational This is a problem posed by Conway in [Co] (in a slightly general form assuming that X is rational) In some special cases the answer is known to be armative: this is the case if X is a pre x set, cf. Ra] or card(X) 3 cf. CKO] and [KPe] On the other hand, in the general case it is only known (and not very dicult to see) that C(X) is ....

....unions of powers of V are replaced by one variable polynomials on V . This is a deep result of 7 Bergman, see [Be] as well as the explanation for the abbreviation BTC: Bergman s Type of Characterization. Historical remarks. Problem 8 for rational languages was proposed already in 1971, cf. [Co]. After recent attacks on the problem even a much easierlooking problem, namely Problem 9, has turned out to be challenging, cf. CKO] An attention to the BTC condition was drawn in [MSY] while Problems 10 and 11 as concrete examples of such problems were proposed in [KPe] 8 Matrix problems ....

Conway, J. H., Regular Algebra and Finite machines, 1971, Chapman & Hall.

....monoid closed under the rational operations but our favorite interpretations are P(A ) P(A Theta B ) Rat(A ) and Rat(A Theta B ) 3 A rediscovered property In this section we consider the most basic Fatou property of rational languages stated in the next Theorem. It was first proved in [5] by means of quotients, later reproved among other things in [9] and finally explained in [12] through a direct construction based on automata. Theorem 1 Assume that L is rational and has the decomposition L = X Delta Y where X;Y A . Then there exist two rational languages X and Y such that ....

J. H. Conway. Regular Algebra and Finite Machines. Chapman Hall, 1971.

....an answer yet. It is straightforward to verify that given a subset of the free semigroup, there exists a unique The authors acknowledge the support of the Academy of Finland under grant #44087 1 maximal subset which commutes with it, called its centralizer. The question was raised by Conway in [6], whether or not the centralizer of a rational subset of a free semigroup is itself rational. To our knowledge this question is still open. Our result can be viewed as a solution of Conway s problem for two element sets. Actually, Conway s problem was originally formulated for free monoids and ....

J. H. Conway. Regular Algebra and Finite Machines. Chapman Hall, 1971.

.... 1987; Kuich and Salomaa 1986] and the design and analysis of algorithms [Aho et al. 1975; Iwano and Steiglitz 1990; Kozen 1991] Many authors have contributed to the development of Kleene algebra [Anderaa 1965; Archangelsky 1992; Backhouse 1975; Bloom and Esik 1993; Boffa 1990; Cohen 1994a; Conway 1971; Gorshkov 1989; Kleene 1956; Kozen 1981; 1990; Author s address: Computer Science Department, Cornell University, Ithaca, NY 14853 7501; email: kozen cs.cornell.edu. The support of the National Science Foundation under grant CCR 9317320 is gratefully acknowledged. Permission to make digital hard ....

Conway, J. H. 1971. Regular Algebra and Finite Machines. Chapman and Hall, London, U.K.

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Conway, J.H. (1971). Regular Algebra and Finite Machines. Chapman and Hall.

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J. H. Conway. Regular Algebra and Finite Machines. Chapman Hall, 1971.

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Conway, J. H. (1971). Regular Algebra and Finite Machines. Chapman and Hall Ltd., London.

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J. H. Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971.

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J.H. Conway, Regular algebras and finite machines, Chapman & Hall, 1974.

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J.H. Conway (1971), Regular algebra and finite machines. Chapman and Hall, London.

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