A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer, Berlin, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which conditions there exists a sequential function mapping a given rational language onto another have been devised, cf. 7] and [9] In [6] the same problem is considered where bijective rational transductions are used in place of sequential functions. 2 Preliminaries We refer to [4] 2] and [8] for a more detailed exposition of the main notions briefly recalled here. 2.1 Free monoids Given a set A (an alphabet) consisting of letters, A denotes the free monoid it generates. The elements of A are words or strings. The length of a word w is denoted by jwj. The empty word, denoted ....

A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer, 1978.

....the language variables (nonterminals) as numerical variables, interpret the terminals by the given valuation, and finally interpret the catenation as multiplication and the union as addition. In the following, we assume that the reader is familiar with some basics of formal power series, see e.g. [29, 23]. One of the connections of valuations and formal power series is the next. Lemma 4. Any valuation fi : Sigma n (0; 1) induces a semiring morphism fi : 0; 1] Sigma n AE [0; 1] by s; w w 7 s; w fi(w) To the set of productions P of a contextfree grammar G = X; Sigma n ; ....

A. K. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. New York: Springer, 1978.

....we list some classes of generating functions: 2) 2) 2) 2) 3) 3) 2) 2) 2) 2) 3) 3) 3) 3) a) b) 2) 4) Figure 1: The rst levels of two equivalent generating trees. R is the set of rational generating functions of integer sequences (Z rational functions, using the notation in [SS]) R is the set of rational generating functions of positive integer sequences; REG is the set of generating functions of regular languages; S is the set of rational generating functions of succession rules; F is the set of generating functions of nite succession rules. Summarizing ....

....the set of rational generating functions of positive integer sequences; REG is the set of generating functions of regular languages; S is the set of rational generating functions of succession rules; F is the set of generating functions of nite succession rules. Summarizing the results in [SS], FPPR] we obtain the following scheme: REG F S The classes R, REG, and F are decidable, while R is not decidable. In [FPPR] is conjectured that F = S, i.e. every rational rule is equivalent to a nite one. This note proposes a simple tool to pass from a rational generating ....

A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer-Verlag, 1978.

....recognition applications are typically acyclic weighted automata, however our algorithm is general and applies to all weighted automata. A weighted automaton is a directed weighted graph in which each edge or transition has a label a phoneme or a word in the case of phone or word lattices [9, 10, 11, 12]. The weights are often interpreted as negative log of probabilities, but in general they may correspond to some other measured quantity. They are added along each path and the weight of a string x is the weight of the minimum weight of a path labeled with x. In what follows, we will thus assume ....

Arto Salomaa and Matti Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer-Verlag: New York, 1978.

....generated by the hybrid systems. The synthesis problem, then involves determining a controller such that the series representation of the continuous state trajectory satisfies specified safety conditions. The resulting series representations of the controlled systems, by way of the Schutzenberger [50] representation theorems, could then be used to extract finite automata generating the desired switching logic for the hybrid system. This continualized approach was first discussed in [48] 80] and that work combines concepts from logic, automata theory, and differential geometry to attempt to ....

A. Salomaa, M. Soittola, Automata Theoretic Aspects of formal power series, Springer-Verlag, Berlin, 1973.

....algorithm when used with an acyclic graph weighted with real valued numbers. The Generic Topological Single SourceShortest Distance algorithm is useful among other things for computing the coefficients of a rational power series represented by a weighted automaton or a weighted transducer [5, 6, 10, 36, 23]. An e#cient implementation of the Generic Single Source Shortest Distance algorithm with various queue disciplines including the generic topological order can be found in the latest version of the FSM library [30] The complexity of the Generic Topological Single Source Shortest Distance ....

A. Salomaa, M. Soittola, Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag, New York, 1978.

....sup ( G) A symbol X 2 V is Pb if fXg is Pb. The Parikh supremum is the maximum number of symbols which can occur in a word of L. We write sup (L) a) for sup (L) fag) sup (L) a) can be considered as the supremum of the corresponding component over all Parikh vectors for L. See [9] for the de nition of Parikh vectors. De nition 3.3. Let and be nite alphabets. A substitution : 2 is Parikh bounded for a language L if for each a 2 which is not Parikh bounded (a) fag holds. The projection (L) is a Parikh bounded projection if n contains Parikh ....

A. Salomaa and M. Soittola. Automata theoretic aspects of formal power series. Springer, 1978.

....of a language X A # , denoted X,is the power series associating 1 with the words belonging to X and 0 with the words not belonging to X. In the following we will often identify X with its characteristic series without stating it explicitly. Some classical references to formal power series are [4, 9, 20]. Definition 2.1 A language X A # factorizes T A # if there exists Y that X Y = T.IfX factorizes A # ,thenX A # is called factorizing. In other words, X factorizes A # if there exists a language Y A # such that any word w A # has a unique factorization w = xy,withx X and ....

A. Salomaa and M. Soittola, Automata-theoretic aspects of formal power series, Springer, Berlin Heidelberg New York (1978).

....from simpler ones. Let K be a commutative semiring and let T 1 and T 2 be two weighted transducers de ned over K such that the input alphabet of T 2 coincides with the output alphabet of T 1 . Then, the composition of T 1 and T 2 is a weighted transducer T 1 T 2 de ned for all x; y by [3, 9, 28, 15]: T 1 T 2 ] x; y) z T 1 (x; z) T 2 (z; y) There exists a general and ecient composition algorithm for weighted transducers [26, 22] States in the composition T 1 T 2 of two weighted transducers T 1 and T 2 are identi ed with pairs of a state of T 1 and a state of T 2 . Leaving ....

.... By de nition of the multiplication of power series in the tropical semiring: min u 0 u k = u 0 ) u k ) a 0 ; b 0 ) a n ; b n ) c( a i ; b i ) c( is a rational power series as the closure of the polynomial power series [28, 4]. Thus, by the theorem of Sch utzenberger [29] there exists a weighted automaton A de ned over the alphabet and the semiring T realizing . A can also be viewed as a weighted transducer T with input and output alphabets . Figure 5 shows the simple nite state transducer T realizing ....

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Arto Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

....the morphism M : T R ThetaR max , given by M(a) sr = 1 if s = r; r 62 R(a) u(a; r) Gamma l(a; s) if r 2 R(a) s 2 R(a) 0 otherwise. 3) This is the specialization to the (max, semiring of the classical notion of automaton with multiplicity [14] or recognizable series [25], 4] Classically, in automata theory, only the map y A is said to be recognized. It is convenient here to extend the definition of recognizability. THEOREM II.5. Let H = T ; R; R; l; u) be a heap model. The (max, automaton (R; 1 R ; 1R ; M) recognizes the upper contour xH and the ....

A. Salomaa and M. Soittola. Automata Theoretic Aspects of Formal Powers Series. Springer Verlag, Berlin, 1978.

....the CFG case and then move on to file case of definite grammar schemes. CFGs, Algebraic Systems, and the GGNK The most powerful transformation methods existing for contextfree grammars are algebraic ( matrix based [8] ones relying on the concepts of formal power series and algebraic systems (see [5, 9]) Using such concepts, u context free grammar such as: a 1 ala l a I [1 is refonuulated into the algebraic system: al ala2 a2 l a = V] which represents a fixpoint equation in the variables (or nonterminals ) a,a on a certain algebraic structure (u non commntative semiring) of formal power ....

Salomaa, A. and M. Soittola. 1978. Automata Theoretic Aspects of Formal Power Series. New York: Springer Verlag.

....the set of graphs, which represent the same series. In this way, the series of star height 0 are associated with acyclic graphs and are limited to polynomials. In this sense, the star height can be interpreted as a measure of the loop complexity of series. We know, from Soittola s theorem (see [14,12]) that an N rational series in one variable is at most of star height 2. This result has been obtained independently by Katayama, Okamoto and Enomoto ( 8] by different methods. Preprint submitted to Elsevier Preprint 5 July 1996 In this paper, we establish a property of series of star height 1 ....

....series is that they have an analytic characterization which is given by Soittola s theorem. The proof of this result allows us to also obtain a bound for the star height. 3. 1 Further definitions and results To begin with, we mention some definitions and results about rational series (see [2] [14] ) A sequence r = r n ) n0 of elements of a semiring K is called K rational if there exists a triple (l; M; c) with l 2 K ; M 2 K c 2 K (n being an integer greater than 1) such that This triple is called a linear representation of the sequence r. One can note that the matrix M is the ....

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A. Salomaa and M. Soittola. Automata theoretic aspect of formal power series. Springer-Verlag, Berlin, 1978.

....[8] and already used by R. Adler, D. Coppersmith and M. Hassner in [1] to construct some nite state codes with sliding block decoders for constrained channels. A variant of the problem consists in replacing the enumerative sequence of leaves by the enumerative sequence of all nodes. Soittola ([15]) has characterized the series which are the enumerative sequences of nodes in a rational tree. We give a characterization for rational k ary trees in the last part of the paper. We prove that a series t is the enumerative sequence of a k ary rational tree if it sati es the following conditions: ....

A. Salomaa and M. Soittola. Automata-theoretic Aspect of Formal Power Series. Springer-Verlag, Berlin, 1978. 25

.... r for all solutions r of (2.1) where is the natural order on R M) Clearly, a least solution, if it exists, is unique. The following theorem shows that in some conditions, any algebraic system has a least solution and it gives a recurrent procedure to compute it. Theorem 2.3. 1 ( 43] 45] [71]) If R is an co continuous semiring, then the least solution of any algebraic system (2.1) exists in R n and it is approximated by the sequence (pn(O) I n N) A series r R( M) is called R algebraic if it is a component of the least solution of an R algebraic system. The family of R algebraic ....

....if it is a component of the least solution of an R algebraic system. The family of R algebraic series is denoted by For further details on formal power series, as well as for connections between the context free languages and the algebraic series, we refer to [8] 21] 43] 45] 69] and [71]. Families of semilinear formal power series The semilinearity is a central notion in the theory of Formal Languages that until now has not been considered per se for formal power series. We discuss in Chapter 3 this notion. We define two subfamilies of rational series, both natural ....

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A. Salomaa, M. Soittola, Automata-theoretic Aspects of Formal Power Series, Springer-Verlag, 1978.

....b, c if b 0 for some k # 0 is NP hard. 2. Zeros in linear recurrent sequences Theorem 1. 1 has an immediate consequence for the long standing problem of determining when an integer linear recurrent sequence has a zero coefficient; a problem that is known as Pisot s problem, see e.g. [2,5,20,24,25]. In 1935, Mahler showed that the set of indices of zero coefficients in a recurrent sequence is the union of a finite set and of a finite number of arithmetic progressions. For linear recurrent sequences of order 3, these sets of indices can be constructed effectively (see [29] and so Pisot s ....

A. Salomaa, M. Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer, New York, 1978.

.... in image compression (Culik II and Kari [CK93] Hafner [Haf99] Katritzke [Kat01] Jiang, Litow and de Vel [JLdV00] and in speech to text processing (Mohri [Moh97] MPR00] Buchsbaum, Giancarlo and Westbrook [BGW00] For theoretical background on formal power series, we refer the reader to [SS78, KS86, BR88, Kui97b]. In this paper, we wish to extend Thatcher and Wright s and Sch utzenberger s approaches to weighted automata on trees, for short: weighted tree automata. In the case where the weight semiring is a eld, Berstel and Reutenauer [BR82] showed that the recognizable series again coincide with the ....

A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Texts and Monographs in Computer Science, Springer Verlag, 1978.

....question arises: What generative capacity do both models imply In the following sections we will answer this question. 2 Basic De nitions In this section we give a brief presentation of some basic concepts regarding formal grammars and automata. For details, the reader is referred to [3] and [5]. A context free grammar (CFG) is a four tuple G = I; T ; P; S) where I and T are nite disjoint sets of nonterminals and terminals, respectively; S 2 I is the axiom and P is a nite subset of I (I [ T ) the set of productions. For (A; 2 P we write A . For f being the name of A ....

....f . Again, the fact that G is free guarantees the existence of all the x . Note, that if l = 1 also x b = x e = since we assume l to be maximal and thus, it is not possible to generate x b and x e . If we set l : i j 1, these derivations imply the following set of equations (see [5] for details) A = v b B l x e w; B = v e Ax b v 1 v 2 : v i x 1 x 2 : x j : 1) If we now solve the set of equations (1) we obtain the statement of the lemma. Remark 3 In the case k = 1 our pumping lemma leads to the well known pumpinglemma for regular languages ....

A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer, 1978 25

....the basic structure of the curve while allowing local changes. 2 Introduction and Basics Obviously, syntactic recognition of fractal sets is inspired from the techniques used in their generation. A course on L systems is the well known book by Rozenberg [1] more advanced results can be found in [4] while short reviews on the subject are included in [5] The basic relationship between sequences of symbols and self similar curves is discussed in [6, 7] while [8] presents some exotic models for space filling curves. Topological notions of D0L systems are studied in [9, 10] and generalizations ....

Arto Salomaa and Matti Soittola. Automata-theoretic aspects of formal power series. Springer-Verlag, 1978.

....and whose zero terms exactly correspond to the solutions to Post s correspondence problem. This result can be obtained by construction the rational series whose coe#cients are the square of the coe#cients of the series constructed on Paterson s matrices. Such a construction is described, e.g. in [SS78]. We provide a slightly di#erent proof that has the advantage that the resulting matrix is of dimension 6 rather than 9. In a second step, we reduce the cardinality of the alphabet to two at the expense of an increase of the state dimension. In a third step, we modify this series so as to obtain a ....

A. Salomaa and M. Soittola. Automata-theoretic aspects of formal power series. Texts and Monographs in Computer Science. Springer-Verlag, New York, 1978. 19

....n if b T A k c = 0 for some k # 0 is NP hard. 2. ZEROS IN LINEAR RECURRENT SEQUENCES Theorem 1. 1 has an immediate consequence for the long standing problem of determining when an integer linear recurrent sequence has a zero coe#cient; a problem that is known as Pisot s problem, see e.g. [2, 5, 25, 24, 20]. In 1935, Mahler showed that the set of indices of zero coe#cients in a recurrent sequence is the union of a finite set and of a finite number of arithmetic progressions. For linear recurrent sequences of order 3, these sets of indices can be constructed e#ectively (see [29] and so Pisot s ....

A. Salomaa and M. Soittola, Automata-theoretic aspects of formal power series, Springer, New York, 1978.

....(0; 1) in the de nition of string acceptance does not change the class Stochastic [Paz, 1971] Rabin [Rabin, 1963] showed that Stochastic contains nonregular languages. For examples of other nonregular stochastic languages, including unary languages, see [Dwork and Stockmeyer, 1990, Paz, 1971, Salomaa and Soittola, 1978]. The class Stochastic was initially de ned for 1pfa s only, but Kaneps [Kaneps, 1989] building on work of Turakainen [Turakainen, 1969] showed that 2pfa s accept exactly those languages accepted by 1pfa s. Macarie [Macarie, 1998] showed that the question of whether a given string x is accepted ....

Salomaa, A. and Soittola, M. (1978). Automata-theoretic aspects of formal power series. Texts and Monographs in Computer Science. Springer-Verlag, New York. 23

....regular and C(X) X 0 . 2 Commuting with a code For basic notions and results on the theory of codes we refer to [2] and [11] For details on the de nition of the centralizer and commutation problems, we refer to [4] 9] 12] and [13] Also, for notions of formal power series we refer to [14]. Throughout this paper, we will consider nite alphabets only. For an alphabet , we denote by ( resp. the set of nite (right in nite, left in nite, resp. words over . For a nite word u 2 and a language L, 1 we say that v 1 : vn is a factorization of u over L ....

A.Salomaa, M.Soittola, Automata-Theoretic Aspects of Formal Power Series, SpringerVerlag,

....fu 2 Sigma Phi j (r; u) 6= 0g is called the support of r. The subset of KhhMii consisting of all series with finite support is denoted by KhMi and its elements are referred to as polynomials. For further definitions and results in the theory of formal power series, we refer to [5] 6] and [9]. We denote the set of nonnegative integers by N. For a formal power series r 2 Nhh Sigma Phi ii, we say that r has bounded coefficients if there is K 2 N such that for all u 2 Sigma Phi , r; u) K. For two formal power series r; s 2 Nhh Sigma Phi ii, we say that r is smaller than s, ....

A. Salomaa, M. Soittola, Automata-theoretic Aspects of Formal Power Series,

....word w. If instead of sets we consider multisets, then we have k spectrum with multiplicity, denoted by SXM (w; k) where X 2 fF; Sg. We can also define the full k spectrum as S X (w; k) k S i=1 SX (w; i) where X 2 fF; S; FM;SMg. We represent multisets as formal power series, see e.g. [SS]. The coefficient of f in the power series S F (w; k) denoted hS F (w; k) fi, is the number of occurrences of factor f in the word w, if jf j = k, and 0, otherwise. Sometimes we say X spectrum and by Definition 3 it is clear what type of spectrum we have in mind. If we shall talk about a ....

Salomaa, A., Soittola, M., Automata-theoretic aspects of formal power series, Springer-Verlag, New York, 1978.

....Parikh s theorem will lead us to a different case, as we will prove that the multiplicities are enough to make distinction between the context free and rational languages. 2 Preliminaries We will recall first some definitions of formal power series needed here. For more details, one can see [SaSo] or [KuSa] Let M be a monoid and A a semiring. Mappings r of M into A are called formal power series on M with coefficients in A. The values of r are denoted by (r; w) where w 2 M , and r is written as a formal sum r = X w2M (r; w)w: 1 The collection of all formal power series defined ....

....i = p i ; i = 1; n (1) where p i 2 Nh Sigma Phi [ Z i. The system (1) is termed proper iff, for each i and j, p i ; 1) 0 and (p i ; z j ) 0. It is important to restrict ourselves to proper system in order to asure the uniqueness of the solution of such a system. It is proved in [SaSo] that this is indeed the case: Lemma 2.1. For each proper algebraic system (1) there exists exactly one solution oe = oe 1 ; oe n ) satisfying the condition (oe i ; 1) 0, for 1 i n. Definition 2.1. A quasiregular series in Nhh Sigma Phi ii is termed N algebraic iff it is a ....

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A. Salomaa, M. Soittola, Automata-theoretic aspects of formal power series, Springer-Verlag, 1978.

....semantically equivalent for both, the top down and the bottom up case. 1 Introduction In [Kui99] a restricted class of top down tree transducers was equipped with costs which are taken from some semiring (A; Delta; 0; 1) The idea of adding costs to automata goes back to [Sch62] cf. also [Eil74, SS78, BR88, KS86, Kui97]) While tree transducers compute a tree transformation, the generalized tree transducers of [Kui99] computes transformation of type T Sigma AhhT Delta ii, where AhhT Delta ii is the set mappings from trees in T Delta to elements of the semiring A. Recently, in [FV01] the cost model of ....

A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Monographs in Theoretical Computer Science, An EATCS Series. Springer, 1978. 12

....paper by de Leeuw, Moore, Shannon and Shapiro [7] Rabin [22] considered (one way) probabilistic automata (1PFA) introduced the notion of recognition of a language by a 1PFA with cut point and understood the importance of isolation of the cut point. References to probabilistic automata are [20, 24]. A one way probabilistic finite automaton (1PFA) A over an alphabet # is a 4tuple A = Q,#, A(#) # # # ,# ,whereQ = q 1 ,q 2 , q m is a finite set of states; # is a 1 m stochastic vector representing the initial probabilities; ## # #, A(#) represents the transition probabilities ....

....word. It is important to observe that, as shown by Gill, a probabilistic automaton that accepts a language with # isolated cut point # corresponds to a probabilistic automaton that computes with (# #) bounded error probability [14] Another important result on 1PFA s is Turakainen s Theorem [25, 24], stating that the language w (#,w) 0 for all rational power series # is stochastic. In other terms it proves that 1PFA s recognize exactly the positive parts of R rational On 2PFA s and the Hadamard quotient 167 series, showing therefore that stochastic languages can be defined in a purely ....

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A. Salomaa and M. Soittola, Automata-theoretic aspects of formal power series, Springer (1978).

.... Controlled Formal Power Series Henning Fernau 1 and Werner Kuich 2 Abstract Regulated rewriting is one of the classical topics in formal language theory, see [2, 3] This paper starts the research of regulated rewriting in the framework of formal power series, cf. [6, 7, 9]. More speci cally, we model what is known as free derivations and leftmost derivations of type 1 within context free grammars controlled by regular sets in the language case. We show that the class which is the formal power series analogue of controlled free derivations forms a semiring ....

A. K. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer, 1978.

....by M is the language L(M ) fw 2 Sigma j (fq 0 g; w) Q d 6= g. In [Sch62] finite state string automata were generalized by associating with every transition an element a 2 A of a semiring (A; Delta; 0; 1) assumed to be the cost (or multiplicity) of this transition (also cf. Eil74, SS78, BR88, KS86] In this approach, the transition function turns into a (Q Theta Q) matrix 2 (Ahh Sigmaii) Q ThetaQ where Ahh Sigmaii is the set of formal power series (in non commuting variables) i.e. mappings of type Sigma A. Intuitively, p;q (oe) 2 A is the cost of making a ....

A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Texts and Monographs in Computer Science, Springer Verlag, 1978.

....has exactly one positive real root, then there exists a polynomial P such that the product pP has one sign change. Theorem 7 has been extended to study the star height of one variable rational series. It is known that the star height of a one variable N rational series is at most two (see [56]) F. Bassino has used Theorem 7 to obtain a characterization of the series of star height one under the assumption that they have a unique pole of minimal modulus ( 6] 5] Much of the study of shifts of nite type is linked to that of positive matrices. Indeed, a shift of nite type is given ....

....eigenvalue is an eigenvalue such that j j for any other eigenvalue. Theorem 8 (Handelman [31] A matrix with integer coecients is conju39 gate to an eventually positive matrix i it has a dominant eigenvalue of multiplicity one. This result is very close to one due to M. Soittola (see [56]) characterizing N rational series in one variable among Z rational series. We quote it for series having a minimal pole i.e. a unique pole with minimal modulus. Theorem 9 (Soittola) A Z rational series with nonnegative integer coe cients f = X n 0 n z n having a minimal pole is ....

Artp Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series. New York, Springer-Verlag, 1978.

.... is the empty word, jwj is the length of the word w, and jwj a (resp. jwj ) is the number of occurrences of a (resp. letters of ) in w. Language theoretic issues and general background material can be consulted from [13, 3] facts about formal power series and rational sequences from [14, 5], and questions concerning Lindenmayer systems from [9, 14] The seminal paper in DNA computing is [1] whereas [8] is a general exposition and [10] underlines the signi cance of complementarity. Consider the letter to letter endomorphism hW of DNA de ned by hW (A) T ; hW (T ) A; hW ....

....w, and jwj a (resp. jwj ) is the number of occurrences of a (resp. letters of ) in w. Language theoretic issues and general background material can be consulted from [13, 3] facts about formal power series and rational sequences from [14, 5] and questions concerning Lindenmayer systems from [9, 14]. The seminal paper in DNA computing is [1] whereas [8] is a general exposition and [10] underlines the signi cance of complementarity. Consider the letter to letter endomorphism hW of DNA de ned by hW (A) T ; hW (T ) A; hW (G) C; hW (C) G: The morphism hW will be referred to as ....

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A. Salomaa, M. Soittola, Automata-Theoretic Aspects of Formal Power Series (Springer-Verlag, New York, 1978).

....us consider function d(t) t 0; with d(t) p(t 1) p(t) It can take values 1 or 0: d(t) gives the tth bit of the Watson Crick road of the second node. If d(t) would be a Z rational function, then 0s would occur in an ultimately periodic fashion among its values (Skolem Mahler Lech theorem, [11], pp. 58. Lemma 9.10) But this is not the case, thus, d(t) is not a Zrational function, which implies that p(t) is not Z rational either. Moreover, we can easily observe that p 1 (t) the string population growth function of the rst component is not Z rational either, but p 2 (t) the string ....

....used in Example 2, we can show that p(t) is not a Z rational function. Again, we consider function d(t) p(t 1) p(t) This function assumes only two values: 0 and r 1: If it were Z rational, then the 0 s would occur in an ultimately periodic fashion among its values (SkolemMahler Lech Theorem, [11], pp. 58. Lemma 9.10. But this is not the case if the Watson Crick road of the component is not ultimately periodic. 6 Remarks on black holes Communication in networks of Watson Crick D0L systems raises a lot of intriguing questions. Among them a particularly interesting problem is whether or ....

A. Salomaa, M. Soittola, Automata-Theoretic Aspects of Formal Power Series. Text and Monographs in Computer Science. Springer Verlag,

....growth function is not necessarily Z rational, 8] However, the known examples use alphabets bigger than the DNA alphabet. For standard Watson Crick D0L systems, the decision problems problems mentioned in Theorem 1 are open and, indeed, closely linked, 8, 14, 15] with the well known problem, [16, 6, 10], of deciding whether or not a given Z rational sequence consists of nonnegative integers. 3 DNA systems We now come to the basic notion investigated in this paper. Briefly, a DNA system is a Watson Crick D0L system, where the underlying DNAlike alphabet has two barred and two non barred ....

....be defined in many ways. Intuitively a simple way is to consider a square matrix M with integer entries and read the sequence from the upper right corners of the powers M i , i = 1; 2; 3; Further discussion about Z rational sequences and their different representations can be found in [16, 10, 6]. While examples of standard Watson Crick D0L systems that are also weird have been presented earlier, 8, 14, 15] we will show in Section 6 the existence of such DNA systems. It is easy to give examples of regular or standard DNA systems that are stable, by making sure that a word in the ....

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Salomaa, A. and Soittola, M. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag, Berlin, Heidelberg, New York, 1978.

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A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer, Berlin, 1978.

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Salomaa, Arto and Matti Soittola. 1978. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag, Texts and Monographs in Computer Science.

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A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Texts and Monographs in Computer Science, Springer-Verlag, 1978.

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A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag, 1978.

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A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

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A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

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Salomaa, A., and Soittola, M. Automata-Theoretic Aspects of Formal Power Series. Springer, 1978. Texts and monographs in computer science.

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Arto Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

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Arto Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

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A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

No context found.

Arto Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

No context found.

Arto Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

No context found.

A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

No context found.

A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

No context found.

A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag: New York, 1978.

No context found.

A. Salomaa and M. Soittola. Automata-theoretic aspects of formal power series, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1978.

No context found.

Arto Salomaa and Matti Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer-Verlag: New York, 1978.

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