On Rational Series in One Variable over certain Dioids

Transcription

1 INSTITUT NATIONAL DE RECHERCHE EN INFORATIQUE ET EN AUTOATIQUE On Rational Series in One Variable over certain Dioids Stéphane GAUBERT N 16 Janvier 1994 PROGRAE 5 Traitement du signal, automatique et productique apport de recherche ISSN

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3 On Rational Series in One Variable over certain Dioids Stephane GAUBERT Programme 5 Traitement du signal, automatique et productique Projet ETA Rapport de recherche n16 Janvier pages Abstract: We give a characterization of rational series in one variable over certain idempotent semirings (commutative dioids) such as for instance the \(max; +) "semiring. We show that a series is rational i it is merge of ultimately geometric series. As a by-product, we obtain a new proof of the periodicity theorem for powers of irreducible matrices and also some more general auxiliary results. We apply this characterization of rational series to the minimal realization problem for which we obtain an upper bound. We also obtain a lower bound in terms of minors in a symmetrized semiring. Key-words: Rational series, Dioids, ax-plus algebra, inimal realization, Discrete Event Systems (Resume : tsvp) Stephane.Gaubert@inria.fr, (33 1) Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, LE CHESNAY Cedex (France) Téléphone : (33 1) Télécopie : (33 1)

4 Sur les series rationnelles en une indeterminee a coecients dans certains diodes Resume : Nous caracterisons les series rationnelles en une indeterminee sur certains semianneaux idempotents (diodes commutatifs), par exemple le semianneau \(max; +)". Nous montrons que les series rationnelles sont obtenues par embo^tement de series ultimement geometriques. On obtient comme consequence une nouvelle preuve du theoreme de periodicite des puissances de matrices irreductibles dans ces semianneaux, ainsi que des variantes plus generales. Comme autre application, on donne une borne superieure pour la dimension minimale de realisation. Une borne inferieure fait intervenir les mineurs dans un semianneau symmetrise. ots-cle : Series rationnelles, Diodes, Algebre (max; +), Realisation minimale, Systemes a Evenements Discrets

5 On Rational Series in One Variable over certain Dioids 1 Introduction def The traditional term \(max; +) algebra" refers to the semiring R max = (R[ f?1g; max; +), that is to the set R[ f?1g equipped with max as addition, denoted by (e.g. 3 = 3) and + as product (denoted by, e.g. 1 = 3). Some specic notation for the neutral elements is also useful: " def =?1 denotes the zero, such that x " = x; x " = " and e def = 0 denotes the unit (such that e x = x). This algebraic structure has been widely studied [9, 0, 1, 7]. It is known [31, 1] that an interesting subclass of Discrete Event Systems consists in causal (max; +) linear stationary operators, which can be represented by convolutions over the (max; +) algebra, that is u 7! y = h u : y(n) = h u(n) def = kn kn h(k) u(n? k) = sup [h(k) + u(n? k)] (1) (u; y are maps Z! R max, h : N! R max ). A case of particular interest arises when the transfer h admit a nite dimensional linear representation, that is, when there exists an integer p and three matrices A R pp max; B R p1 max; C R 1p max such that y(k) = kn CAk Bu(n? k) : () Here, as usual, concatenation denotes the matrix product induced by the semiring structure, that is def (UV ) ij = (U def V ) ij = U ik V kj k and consequently, A k stands for A : : : A (k times). This leads us to introducing the semiring R max [[X]] of formal series in a single indeterminate X over R max : the input-output behavior of the system is determined by its transfer series H = kn CAk B X k R max [[X]] : (3) This provides a motivation for the study of realizable series, that is, of series H for which there exists a nite dimensional triple (A; B; C) providing a representation of the form (3). Realizable series are classically related with rational series that we next dene after some notation. Given a series H, we shall write hh; X k i or H k for the coecient of H at X k, so that H writes H = kn H k X k = kn hh; X k ix k : Now, let s R max [[X]] with zero constant coecient, i.e. hs; X 0 i = ". The star of s which can be written formally s = in si is by denition the unique series such that 8k; hs ; X k i = in hsi ; X k i (this is indeed a nite sum: hs i ; X k i = " for i large enough due to the fact that s has zero constant coecient). Since series over the (max; +) algebra are naturally ordered, it is possible to dene alternatively s as the least upper bound of the innite set fs 0 ; s 1 ; : : :g, when it exists. We do not adopt this alternative convention here, but it leads to the same class of series (this in discussed in detail in the Appendix II).

6 Stephane GAUBERT Denition The semiring Rat of rational series is the least set of formal series containing polynomials and such that Rat Rat Rat Rat Rat Rat Rat Rat Since s is only dened for s with zero constant coecient, Rat Rat indeed means that (S) if s Rat and hs; X 0 i = ", then s Rat. The celebrated Kleene-Schutzenberger theorem [3] states that a series is rational i it is realizable. Therefore, the theory of systems of type () leads us to studying (max; +) rational series. In this paper, we present some characterizations of rational series over the (max; +) algebra and others idempotent semirings (dioids). As a by-product, we obtain some rather general cyclicity theorems for powers of matrices. Then, we present some bounds for the minimal realization problem. ost of these results are taken from the thesis of the author [16], with some generalizations. The characterization of rational series holds under a \weak stabilization" condition analogous to the stabilization condition already introduced by Dudnikov and Samborski. Weak stabilization requires the sum of two geometric sequences to be ultimately geometric. Then, rational series are obtained by merging some ultimately geometric series. This is a bit similar to the case of rational series over (R + ; +; ) (see [3], Chapter 5) and extends some results given by oller [9] for rational series over R max and by Cohen,oller,Quadrat and Viot [8] for rational series in the dioid of shift operators in timed event graphs (called ax in [[; ]]). Some related results motivated by logical problems have been obtained by Bonnier and Krob [4] for rational series over the tropical semiring (i.e. (N [ f+1g; min; +)). As an application, we obtain with these techniques some cyclicity theorem in rather general dioids. This is because there is a natural connection between rational series and sequences of powers of matrices (via generating series). Thus, the representation theorems proved for rational series automatically transfer to asymptotic theorems for matrices. In particular, we obtain another proof {with some relaxed assumptions{ of the cyclicity theorem [7, 1, 1], which states that an irreducible matrix A is projectively torsion, that is A n+c = A n for some integers n 0; c 1 and scalar. Next, we provide two bounds for the minimal realization problem: it consists in minimizing the size p of the linear representation of a rational series H (p is the size of A in (3)). The upper bound is a simple extension of a result of Cuninhame-Green that we mention here for the sake of completeness. The second one relies on a combinatorial identity (Binet-Cauchy formula) valid in a symmetrized semiring of R max. Indeed, the bound is more general since it is valid in any semiring (in the case of elds, it coincides with the traditional characterization in terms of the minors of the Hankel matrix [14]). We illustrate these two bounds by giving a few examples: we provide some equality cases, but we also show that the bounds can be arbitrarily coarse. We conclude the paper by showing how determinants can be computed in the (max; +) algebra in order to make the minor bound eective. Let us also mention that rational series with several noncommuting indeterminates over similar dioids have been considered by Hashiguchi, Simon, Krob, ascle,leung [1, 33, 3, 6, 5] and also by the author in [18, 17].

7 On Rational Series in One Variable over certain Dioids 3 1 Rational Series over Commutative Dioids 1.1 General Characterizations We rst characterize rational series over commutative dioids [1] (a dioid is a semiring whose addition is idempotent, i.e. a a = a). Of course, the main dioid that we have in mind for applications is R max, but the result that we give is more general Denition Let S denote a commutative semiring. We say that s S[[X]] is ultimately geometric if 9N N; c Nnf0g; R max such that This property admits an immediate algebraic characterization: n N ) hs; X n+c i = hs; X n i : (4) 1.1. Proposition Let S be a commutative semiring. The series s S[[X]] is ultimately geometric i there exists two polynomials p; q S[X], c Nnf0g, S such that s = p q(x c ) (5) Proof (4) ) (5). This follows from s = nn?1 hs; X n ix n N +c?1 n=n hs; X n i! (X c ) : (5) ) (4). We have, for n > deg p, hs; X n+c i = = = hq(x c ) ; X n+c i deg q k=0 deg q k=0 = hs; X n i : hq; X k ih(x c ) ; X n+c?k i hq; X k ih(x c ) ; X n?k i Consider S = B = f"; eg (the boolean semiring). A series s B[[X]] is rational i its support supp s def = fn N j hs; X n i 6= "g is a rational subset of N. It is well known ([13], Chapter V, Proposition 1.1) that a subset of N is rational i it is the union of a nite set and of a nite number of arithmetic progressions. The following fact, already noticed by oller in the case of R max ([9], ), extends this property Theorem Let S be a commutative dioid. A series s S[[X]] is rational if and only if it is a nite sum of ultimately geometric series.

8 4 Stephane GAUBERT Proof Let G denote the set of nite sums of series of the form (5) and R denote the set of rational series. Since S[X] G R, it is enough to show that G is closed under addition (this is obvious), product, and star. Let a; b S[[X]] such that ha; X 0 i = hb; X 0 i = ". The following well known rational identities [4, 8, 16] hold in commutative dioids In particular, for a = X c (with S, c 1), (a b) = a b (6) (ba ) = e b(a b) (7) 8k 1; a = (e a : : : a k?1 )(a k ) : (8) (b(x c ) ) = e bb (X c ) : (9) Hence, G G G ) G G. It remains to show that G G G. Indeed, it is enough to check that for all c; d Nnf0g; ; S, we have t def = (X c ) (X d ) G : (10) Dene c 0 ; d 0 by cc 0 = dd 0 = lcm(c; d). Then, the assertion (10) follows from (by (8)) together with (by (6)). c 0?1 d 0?1 t = ( i X ci )( c0 X lcm(c;d) ) ( j X dj )( d0 X lcm(c;d) ) (11) i=0 j=0 ( c0 X lcm(c;d) ) ( d0 X lcm(c;d) ) = (( c0 d0 )X lcm(c;d) ) In order to characterize rational series, we introduce the following notion, which already appears in the theory of rational positive series [3]. The merge of k series s (0) ; : : :; s (k) is obtained by taking alternatively the coecients of these series. ore precisely: Denition (erge of Series) We say that s is the merge of the series s (0) ; : : :; s (k?1) S[[X]] if 8i f0; : : :; k? 1g; 8n 0; hs; X nk+i i = hs (i) ; X n i : For instance, the series hs; X p i = p; hs; X p+1 i = e is the merge of (X) and X. ore generally, it is easily realized that a series s is the merge of ultimately geometric series i the following periodicity property holds 9N N; 9c Nnf0g; 9 0 ; : : :; c?1 R max ; 8i f0; : : :; c? 1g; 8n N; hs; X nc+i+c i = i hs; X nc+i i : (1) We shall also need the two following conditions (the second one is borrowed to Dudnikov and Samborski, [11], Condition.) Denition (Weak and Strong Stabilization) The dioid S satises the weak stabilization condition if for all a; b; ; S, there exists c; S and N N such that n N ) a n b n = c n : If this property holds with = (as soon as a; b 6= "), we say that S satises the strong stabilization condition.

9 On Rational Series in One Variable over certain Dioids 5 The scope of these conditions should become clear after a few examples Example The following dioids satisfy the strong stabilization condition: 1. R max and its subdioids (e.g. (N [ f?1g; max; +)).. The completed dioid of R max, i.e. R max def = (R[ f1g; max; +). 3. (R n [f"g; max; +) (obtained by adjoining a zero element to (R n ; max; +), i.e. "x def x) = x; "x def = 4. (N [f"g; lcm; ), where " is also a zero and lcm is seen as a binary law (e.g. 64 = lcm(6; 4) = 1). 5. The set of (not necessarily bounded) closed intervals of R, equipped with AB = conv(a[b), A B = A + B (conv denotes the convex hull, + the vector sum). 6. Let (D; _; ^) be any distributive lattice with universal lower bound? and upper bound > (i.e.? _ x = x; > ^ x = x; 8x). Then, setting def = _ and def = ^, D becomes a dioid which satises trivially the strong stabilization condition, for n = ^ : : : ^ = for n 1 and a n b n = (a ^ ) _ (b ^ ) = c ^ ( _ ) = c( ) n = cst for n 1, with c def = (a _ b) ^ (a _ ) ^ ( _ b) due to the distributivity. Given a dioid D, let Dc n denote the set D n equipped with componentwise 1 sum and product ((u def def v) i = u i v i ; (u v) i = u i v i ). The following obvious fact is worth noticing: Observation If D satises the weak stabilization condition, then so does D n c Example For n, R max n ;c = ((R[ f?1g) n ; max; +) satises the weak stabilization condition (by the preceding proposition) but not the strong one. For instance, for n =, let a = = (e; e), b = (e; "), = ("; 1). We have a n b n = (e; e) n (e; ")("; 1) n = (e; e); 8n : There does not exist c such that this sum of geometric series reduces to c( ) n = c (e; n) Example The weak stabilization condition fails for the following dioids: 1. The set of nite subsets of N, equipped with [ as addition and + as product (consider f0g n f1g n ).. The set of compact convex subsets of R k (with k ), equipped with a b def = conv(a [ b), a b = a + b (let o def = f(0; 0)g; i def = f(1; 0)g; j def = f(0; 1)g R, and consider o n j(o i) n ). 3. S max, the symmetrized semiring of R max [30, 16] (consider e n ( e) n ). We are now in position to state the main theorem which extends a periodicity result given by Cohen,oller,Quadrat and Viot [8] (Theorem 1) for a particular dioid of shift operators. The periodicity theorem of [8] is essentially equivalent to our periodicity result restricted to the subclass of nondecreasing series in R max [[X]] (a series is nondecreasing if n m ) hs; X n i hs; X m i). 1 We use the notation D n c in order to distinguish D c from the free symmetrized dioid D to be dened in x..

10 6 Stephane GAUBERT Theorem Let S be a commutative dioid which satises the weak stabilization condition. Then a series s S[[X]] is rational i it is a merge of ultimately geometric series. Proof 1/ The merge of the rational series s (0) ; : : :; s (k?1) is clearly equal to s = k?1 i=0 X i s (i) (X k ) (13) where s (i) (X k ) denotes the series obtained by substitution of X k to X in s (i). Hence it is rational. / Conversely, let s be a rational series. It follows from Theorem that s is a nite sum s = p r q i ( i X c i ) : (14) i=1 After replacing c i by lcm(c 1 ; : : :; c m ) as in (11), we may assume that c i = c (constant). Then, it is enough to show that the series s (j) def = nn hs; X nc+j ix n ; 0 j c? 1 are ultimately geometric, since s is the merge of s (0) ; : : :s (c?1). Clearly, we have from (14) s (j) = p 0 j r qij( 0 i X) : (15) i=1 for some polynomials p 0 j ; q0 ij. We show that such a sum is ultimately geometric. Consider for a; b N,; ; 6= S: t = X a (X) X b (X) : (16) Then, for n max(a; b), we have ht; X n i = n?a n?b = ( max(a;b)?a ) n?max(a;b) ( max(a;b)?b ) n?max(a;b) The weak stabilization condition implies that there exists N N and c; S such that n? max(a; b) N ) ht; X n i = c n?max(a;b) : Thus, t is ultimately geometric and the proof of Proposition 1.1. yields t = u m(x) (17) where u is a polynomial of degree < N + max(a; b) and m = c N X N +max(a;b). Applying inductively the rewriting rule (16)!(17) to the sum (15), we get s (j) = v w( 0 X), where v is a polynomial, w a monomial, and 0 S. By Proposition 1.1., this shows that s (j) is ultimately geometric. 1. Application to Cyclicity Theorems for Powers of atrices Since there is a close connection between sequences of powers of matrices and rational series, the above periodicity results given for rational series automatically provide some periodicity results for matrices. We rst state the most general one. For a diagonalizable matrix A (over a eld) with eigenvalues 1 ; : : :; k, we have obviously 8ij; 9 1 ; : : :; k ; 8n N; A n ij = kx r=1 r n r : (18) An analogous property holds in general commutative dioids, but we have to take into account the cyclicity and the transient behavior.

11 On Rational Series in One Variable over certain Dioids Theorem Let S be an arbitrary commutative dioid and A S pp. Then, there exists c 1, N N such that for all ij, for all l f0; : : :; c? 1g, there exists a nite family of scalars 1 ; 1 ; : : :; k ; k such that 8n N; A nc+l ij = k r=1 r n?n r : (19) In other words, the sequence fa n ijg nnultimately coincides with a merge of sums of geometric series. When c = 1 and N = 0, (19) reduces to the familiar (18). Proof The sequence A n ij = CAn B where B k = kj,c k = ki (Kroneker's delta) is realizable, hence by the Kleene-Schutzenberger theorem [3], the \generating series" s ij = n A n ij X n = n CA n BX n = (AX) ij S[[X]] is rational. An application of Theorem gives s ij = P k Q r ( r X cr ) : (0) r=1 where P and Q k are polynomials and k are scalars. Perhaps after changing the Q k according to (8), we may assume that c k = c = cst (and even that this constant does not depend on ij). Then, the property follows from hq k ( k X c ) ; X l+nc i = which is meaningful as soon as Thus (19) holds for all N such that ml (mod c) mdeg Q k hq k ; X m ih( k X c ) ; X l+nc?m i = ml (mod c) mdeg Q k 8m l (mod c); m deg Q k ) n + (l? m)=c 0 : hq k ; X m n+(l?m)=c ik ; cn max(deg P + 1; max deg Q k ) : k Under weak stabilization, we have the following more precise property: 1.. Proposition Let S be a commutative dioid satisfying the weak stabilization condition and let A S pp. For all i; j f1; : : :; pg, there exists c Nnf0g, 0 ; : : :; c?1 S, N N, such that 8l f0; : : :; c? 1g, n N ) A n+l+c ij = l A n+l ij : (1) Proof As shown in the proof of the preceding theorem, the sequence A n ij = CAn B is rational. It remains to apply Theorem together with Property (1). This extends the well known periodicity property for irreducible matrices in the (max; +)-algebra (see e.g. [1]) if A is irreducible, there exists c; such that A n+c = c A n ()

12 8 Stephane GAUBERT for n large enough. Dudnikov and Samborski[1, 11] have proved that this result holds in dioids with cancellative product which satisfy the (strong) stabilization condition. Rational series provide an alternative proof under extended assumptions. Let us recall that a semiring has no divisors of zero if ab = " ) a = " or b = " (in the case of dioids, this property is weaker than cancellativity) Theorem (Cyclicity) Let S be a commutative dioid without divisors of zero and satisfying the strong stabilization condition. If A is irreducible, then there exists S, c 1; N N such that n N ) A n+c = A n : Proof From Theorem 1.1.3, we have for each ij a rational expression of the form (AX) ij = kf ij u kij X lkij ( kij X c ) ijk ; u ijk S ; (3) where c can chosen independent of ij (by (8)) and where for all ij, F ij is a nite set. By Proposition 1.1., it is enough to show that up to a polynomial term, it is possible to take ijk = (independent of ijk) in (3). The following Lemma is central: 1..4 Lemma Assume that the strong stabilization condition holds. Let m; n N, p 1, a; b S n f"g; ; S. If m n (mod p), there exists a polynomial P, c S and an integer q m (mod p) such that ax m (X p ) bx n (X p ) = P cx q (( )X p ) : (4) Proof of the Lemma. We shall use the following obvious identity: 8r 1; = e : : : r?1 r : (5) Let us assume for instance that m n. Then, there exists r such that n + rp = m, and applying (5) to = X p, we get bx n (X p ) = P b r X m (X p ) for some polynomial P. Since the strong stabilization implies that for a certain c, the Lemma is proved. We return to the proof of the Theorem. ax m (X p ) b r X m (X p ) = cx m (( )X p ) 1/ We rst assume that A is primitive, i.e. as in the Perron-Frobenius theory [, 8] that all the entries of A K are dierent from " for a certain K. A fortiori, it is possible to choose k satisfying the following condition: 8l f0; : : :; c? 1g; 8i; j; 9m k; m l (mod c) and A m ij 6= " : (6) The graphical interpretation [8, 1] should make this statement intuitive: due to the primitivity, there is a path of length at most k and of arbitrary remainder modulo c between any nodes of the graph associated with A.

13 On Rational Series in One Variable over certain Dioids 9 The identity (AX) ij = 0mk 1udim A (AX) m iu (AX) uj : (7) is the projection on the entry (ij) of the following obvious rational identity: (AX) = (Id AX : : : (AX) k?1 )(AX) : (8) It remains to substitute the development (3) of (AX) uj in (7) and to apply Lemma 1..4 to obtain (AX) ij = P ij Q ij (X c ) (9) where def = i;j kf ij kij : and P ij ; Q ij are polynomials. The fact that S has no divisors of zero is used here in a critical way (in loose terms, this implies that all the kij appear in (7)). Then, Proposition 1.1. applied to (9) gives the periodicity property for A n ij with a common rate and cyclicity c. /Non primitive case Let us now consider an irreducible matrix A with cyclicity c, with the following Frobenius normal form (see e.g. [8]) Hence, where the diagonal blocs A = 6 4 " A 1 " : : : " " " A.. " "... " A c?1 A c " " A c = diag(a 1 ; A ; : : :; A c ) def def A 1 = A 1 : : : A c ; A = A A 3 : : :A c A 1 ; : : : are primitive. The rst part of the proof (primitive case) shows that (A 1 ) n+k = k (A 1 ) n for some k 1 and n large enough. Since (A ) n = A : : : A c (A 1 ) n?1 A 1 ; we conclude that (A ) n is also periodic with the same and k. Since an analogous argument applies to the others diagonal blocs of A c, we are done Example We show that when A is not irreducible, the sequence A n ij ; n N can admit several distincts rates, so that Corollary 1.. cannot be rened. A = 6 4 " e " e " " " 1 " " " 1 " " e " " "? e " " " " " C = ; B = h 6 4 " " " " e ; e " " " " : i (30)

14 10 Stephane GAUBERT Let s n = CA n B. We have s 0 = s 1 = ", 8n 1; s n+1 = n? 1; s n =?4n + 4. This is the merge of two geometric series with distinct rates 0 =?4 et 1 = Example Strong stabilization cannot be replaced by weak stabilization in Theorem Consider for instance " # (e; e) (e; ") A = (R (e; ") (e; 1) max;c ) (see for the denition of the dioid R max;c, which satises the weak stabilization but not the strong one). The matrix A is obviously irreducible, but it is plain that the sequence " # (e; e) (e; ") A n = (e; ") (e; n) does not satisfy Bounds for the inimal Realization Problem.1 Upper Bound via Weak Rank L The Hankel matrix associated with s = khs; X k ix k is the N N-matrix: 3 hs; X 0 i hs; Xi hs; X i : : : hs; Xi hs; X i hs; X 3 i : : : H = hs; X i hs; X 3 i In the classical theory of realization of rational series over elds [14], the rank of the Hankel matrix provides the minimal dimension of realization. For rational series over dioids, there are some weaker notions of rank which only provide bounds for the minimal realization problem and that we discuss now. Indeed, in the case of dioids, the complete solution of the minimal realization problem remains open and seems almost as dicult as in the case of positive rational series [15]. In the following, we shall speak of moduloids which are dened over dioids in a way similar to modules over rings [34, 16]..1.1 Denition (Weak Rank) The weak dimension of a moduloid is the minimal cardinal of a generating family. The weak column rank of a matrix A denoted by rg w A, is the weak dimension of the moduloid generated by the columns of A. Here, we shall just deal with Hankel matrices which are symmetric and we shall not need the notion of weak row rank (dened dually). The notions of rank and linear dependence over dioids have been previously studied by Cuninghame-Green [9, 10], oller [9] and Wagneur [34]. inimal generating families can by obtained by standard residuation techniques [4, 5, 9, 10, 16]. Assume that s is the merge of the ultimately geometric series s (0) ; : : :; s (k?1), i.e. that (13) holds. Let H (i) denote the Hankel matrix associated with s (i) (X k ). The following upper bound is a straightforward generalization of a result given by Cuninghame-Green in [10] and valid for the subclass of ultimately geometric series

15 On Rational Series in One Variable over certain Dioids Proposition (Upper Bound) There exists a realization of s of dimension P k i=1 rg w H (k). Hence, by Theorem , this proposition provides an upper bound for the minimal realization of a rational series over a dioid satisfying the stabilization condition (such as R max ). Proof Recall that if two series u; u 0 admit two realizations (C; A; B), (C 0 ; A 0 ; B 0 ) of respective size n and n 0, then, u u 0 admit a realization of size n + n 0, namely C 00 = [C; C 0 ]; A 00 = " # A " ; B 00 = " A 0 " B B 0 # Hence, it is enough to show the theorem when s = s (i) (X k ). i.e. when s is ultimately geometric, which is the case considered by Cuninghame Green [10]. For the sake of completeness, we recall an alternative realization algorithm which is classical [] and a bit simpler than that of [10]. Let H ;i1 ; : : :; H ;ir be a minimal generating family of the moduloid spanned by the columns of H (hence, r = rg w H). Thus, there exists a r r-matrix A such that Similarly, there exists B S r such that [H ;i1 +1; : : :; H ;ir+1 ] = [H ;i1 ; : : :; H ;ir ]A : H ;0 = [H ;i1 ; : : :; H ;ir ]B (we number the columns of the Hankel matrix from 0). It remains to set C = [H 0;i1 ; : : :; H 0;ir ]: (C; A; B) is a realization of s of size rg w H. We note that this realization is eective in the case of R max since A; B can be obtained by residuation of nite matrices[4, 5, 9, 10, 16].. Lower Bound via inor Rank We next provides a lower bound for the realization problem. We shall need the notion of symmetrized semiring, which has been introduced in [30] and more completely studied in [16]. We say that (S; ; ; ) is a symmetrized semiring if (S; ; ) is a semiring and if is an unary operator which satises the three following properties: (a b) = ( a) ( b) ( a)b = a( b) = (ab) a = a : A map ' is a morphism of symmetrized semiring if it is a morphism of semiring which satises '( x) = '(x) : Rings equipped with the usual minus sign are obvious examples of symmetrized semirings. A natural problem which extends the usual symmetrization of N by Zconsists in embedding a semiring S into a symmetrized semiring S 0. This problem is studied in [16]. Here, we shall only need the free symmetrized semiring S, that we dene now. We consider S equipped with the following operations (a 0 ; b 0 ) (a 00 ; b 00 ) = (a 0 a 00 ; b 0 b 00 ); (a 0 ; b 0 ) (a 00 ; b 00 ) = (a 0 a 00 b 0 b 00 ; a 0 b 00 a 00 b 0 ); (a 0 ; a 00 ) = (a 00 ; a 0 ) : :

16 1 Stephane GAUBERT The null element is ("; ") and the unit is (e; "). We shall as usual write a b for a b. We have an injective morphism of semirings S! S ; x 7! (x; ") ; hence, S provides a symmetrization of S, and we shall identify S to the subsemiring of S composed of the elements of the form (x; "), writing x = (x; "). With this convention, an element x of S admits a unique decomposition x = x 0 x 00 ; x 0 ; x 00 S : (31) We dene the determinant of a matrix A with entries in a symmetrized semiring by det A = sgn no i=1 A i(i) (3) where the sum is taken over the permutations of f1; : : :; ng and sgn = e if is even and e is is odd. We next extend some well known properties of determinants to symmetrized semirings. We rst dene the balance relation r -which will replace to some extent the equality relation-: (a 0 ; a 00 ) r (b 0 ; b 00 ) () a 0 b 00 = a 00 b 0 : Observe that r is not transitive in R max. Now, let S be a commutative semiring, and let ' denote the unique morphism of symmetrized semirings (N[X 1 ; : : :; X k ])! (S[X 1 ; : : :; X k ]) such that 8i; '(X i ) = X i. The following transfer principle states that combinatorial identities valid in commutative rings involving +; ;? also hold in symmetrized commutative semirings, provided that equalities are replaced by r. This is just a formalization of a fact already noticed by Reutenauer and Straubing [3]...1 Transfer Principle Let P + ; P? ; Q + ; Q? N[X 1 ; : : :; X k ]. Assume that P +? P? = Q +? Q? ; holds in Z[X 1 ; : : :; X k ] : Then, '(P + ; P? ) r '(Q + ; Q? ) holds in S [X 1 ; : : :; X k ] : Proof immediate from the denition of r. This allows translating the well known Binet-Cauchy formula. Let us denote by A [IjJ] submatrix (A ij ) ii; jj. the.. Proposition Let S be an arbitrary commutative semiring. Let A S nr, B S rp, I f1; : : :; ng, J f1; : : :; pg with #I = #J = k. The following identity holds in S : det(ab) [IjJ] r K det A [IjK] : det B [KjJ] ; (33) where the sum is taken over all the subsets K f1; : : :; rg of cardinal k (this sum is empty, conventionally equal to " if k > r).

17 On Rational Series in One Variable over certain Dioids 13 Proof Let us rst consider the entries a ij ; b kl of the matrices as indeterminates. By applying the transfer principle to the classical Binet-Cauchy formula in Z[a ij ; b ij ], we see that the identity holds formally in the symmetrized semiring (S[a ij ; b kl ]). Hence it is true for all values of a ij ; b kl. It is also possible to provide a combinatorial proof along the lines of Zeilberger [35]. We say that an element x S is balanced if x r ", unbalanced otherwise. Then, we dene the minor rank of a matrix...3 Denition (inor Rank) The minor rank of a matrix is by denition the maximal size of a square submatrix with unbalanced determinant...4 Theorem (Lower Bound) Let S be an arbitrary commutative semiring and s S[[X]]. The dimension of any realization of s is at least equal to the minor rank of its Hankel matrix. Observe that in the case of elds, this bound coincides with the minimal dimension of realization [14]. Proof This follows from H = OC, where O = [C; CA; CA ; : : :] T and C = [B; AB; A B; : : :] are the usual observability and controllability matrices. Consider a minor extracted from H, det H [IjJ]. The Binet-Cauchy formula states that det H [IjJ] r K det O [IjK] det C [KjJ] : This sum is empty as soon as the cardinal of I is greater than the size p of the realization. Hence, det H [IjJ] r " for all I J submatrix of H of size > p...5 Remark This bound is easily extended to rational series with several noncommuting indeterminates. Let S be a commutative semiring, the free monoid over a nite alphabet and consider a series s Shhii. The Hankel matrix of s [14] is the def matrix H: H u;v = hs; uvi. Let ; ; be a linear representation of dimension p of s, i.e. S 1p ; S p1, is a morphism! S pp, and hs; wi = (w). We have H = OC with O S p, O u = (u), and C S p, C v = (v). The same argument shows that the minor rank of H is a lower bound for the dimension p of the linear representation of s. 3 Some Examples We next exhibit a few examples which should made it clear when the lower and upper bounds for the minimal realization problem are accurate. 3.1 An Example where Weak Rank and inor Rank Coincide Consider the realization 6 A = 4 4 " " " 5 " " 0 " ; B = " ; C = h 0 " 0 i

18 14 Stephane GAUBERT with truncated Hankel matrix H [0::6j0::6] = (the other columns of H are proportional to the the 6-th one). We have " # 16 0 det H [1j56] = det = = = (41; 40) 6r " : 0 5 Hence, the dimension of realization is at least. Indeed, let us apply the realization algorithm sketched in the Proof of Proposition.1.. It is easily seen that the columns of indices 0 and 5 are a minimal generating family of the moduloid of columns of H. Hence, we obtain the following -dimensional realization " # 4 5 A 0 =?0 5 " # 0 B 0 = " The lower bound shows that it is minimal. C 0 = h A Series with Large Weak Rank but Small inimal Realization Consider the following triple 6 A = 4 5 " 0 " 4 0 " " 3 with truncated Hankel matrix: H [0::5j0::5] = ; B = i ; C = h i Since det H [0;1;j0;;5] = r ", H has minor rank 3 and (A; B; C) is a minimal realization. However, the above mentioned residuation techniques show that the columns of indices 0,1,,4 are a minimal generating family of the set of columns of the Hankel matrix. Hence, the upper bound via weak rank is 4 which is greater to actual minimal dimension of realization. Indeed, the situation can be much worse. Let us consider more generally the following matrix: h i C = 0 r s : 3 7 5

19 On Rational Series in One Variable over certain Dioids 15 Let H def = C(AX) B. We have H = (5X) r(4x) s(3x) : 3..1 Proposition The weak rank of the Hankel matrix of H can be arbitrarily large for properly chosen r; s. First, we have hh; X N i = max(5 N; r + 4 N; s + N) = N 5 rn 4 sn 3 = N 3 (N r p s)(n ( s r ^ ps)) (recall that p s in the (max; +) algebra stands for s= in the usual algebra). Let us assume r > s, then, the map k 7! hh; X k i = H k;0 is the max of the three ane functions drawn in Figure 1, with the following two distinct corners: n 1 = r > n = s r : Let us further suppose that s; r N and s > r which implies that s r = s? r N. Then H ni = (ni) 3 (ni r)(ni s r ) : (34) 3.. Lemma Under the foregoing assumptions, if 1 s r r, the columns H ;0; : : :; H ; s r belong (up to a scaling) to any generating family of the moduloid of columns of H. This result might seem unnatural to the reader who is not familiar with rank theory in moduloids: it is important to note that minimal generating families of moduloids of nite type over R max are unique (up to a permutation and a scaling) [9, 34], due to the fact that the only invertible matrices are products of permutation and diagonal matrices. Thus, the elements of minimal generating families are akin to the extremal directions of (conventional) convex cones. The above lemma exhibits a subset of 1 s = s? r + 1 \extremal columns" of the Hankel matrix, which implies that r the weak column rank of H which is at least s? r + 1 can be arbitrarily large with respect to the minimal dimension of realization (at most 3). It should be intuitively clear by looking at Figure 1 H ;i H ;0 H N 0 = N 5 rn 4 sn 3 s r r N Figure 1: A series with large weak rank that the columns H ;i (for 0 i s? r) are extremal, i.e. that there does not exist a relation of the form H ;i = jj j H ;j ; (35) where J is a nite subset of N which does not contain i. Indeed, j must be such that the map associated to the column j H ;j lies below H ;i. It should be geometrically clear that the part of the graph of H ;i between s? i and r? i (shaded part) will never be attained by the max at the right r hand of (35). A more precise (but technical) proof of this Lemma is provided in Appendix III.

20 16 Stephane GAUBERT 3.3 A Rational Series with Innite Weak Rank The weak rank of the Hankel matrix of a rational series which is not ultimately geometric can be innite. This is why we have to consider all the Hankel matrices H (0) ; : : :; H (k?1) in Proposition.1.. For instance, let us consider the series i.e. s = (X ) 1X((1X) ) R max [[X]] ; hs; X i i e if i even = 1 i = i 1 = i if i odd. We claim that the moduloid generated by the columns of the Hankel matrix of s does not admit a nite generating family. Indeed, let fh ;i1 ; : : :; H ;ik g be a nite generating family of the moduloid of columns of H. We have for all i N a linear combination of the form k H ;i = il H ;il : (36) (i) If i an i l are not of the same parity, then il = ". This follows from il ^ jn H j;i H j;il = ^ l=1 e 1 j+i l jn\(i+z) 1 0 A ^ jn\(i+z+1) 1 j+i e 1 A = " : (ii) We have for all i and l, il e. Indeed, from (i), we are reduced to the case where i and i l have the same parity. Then il H ii H iik = e e = e : These two remarks together with (36) imply that the rst row of H is bounded above: a contradiction. 3.4 An Irrational Series with Finite inor Rank Proposition Consider the series s = L k s kx k, with " s k = e if 9p N; k = 3 p otherwise Although s is not realizable, the minor rank of its Hankel matrix is nite. Proof Plainly, s k is not a merge of ultimately geometric series, hence it is not realizable. We show that the large minors taken from the Hankel matrix are equal to e e (hence, they are balanced). Let B = f"; eg denote the boolean dioid, which is a subdioid of R max. The key point in the proof is the following observation which shows that boolean minors with too many e entries are balanced Proposition Let B nn. If det 6r ", then, at least n(n?1) entries of are equal to ".

21 On Rational Series in One Variable over certain Dioids 17 Proof For a matrix, this is clear. After multiplying by a permutation matrix N (which changes the sign of det but not the fact that it is balanced or not), we may assume that i ii = e. If the ij entry is nonzero, then, then the ji entry must be zero (otherwise, we apply the proposition already proved for matrices, and we get by minor expansion det = det [ijjij]e : : : = e e : : : = e e). Thus, at most one half of the outdiagonal entries of can be nonzero. It should be intuitively clear that since s k takes asymptotically very few zero values, due to this proposition, the large minors of the Hankel matrix are balanced. The precise proof seems less immediate, and the proof that we oer relies on the following perhaps articial trick Denition We say that the matrix A has no lower triangles if the following situation does not occur A ij = "; A lk = "; A max(i;l);max(j;k) = " pour i 6= l et j 6= k. This is illustrated by Figure : A ij = " A lk = " A max(i;l);max(j;k) = " Figure : Lower triangle Lemma The Hankel matrix of the series s has no lower triangles. Proof If i + j = 3 p, l + k = 3 q, we have max(i; l) + max(j; k) 3 max(p;q). If H admits a lower triangle, we must have max(i; l) + max(j; k) = 3 r with r > max(p; q), hence 3 r 3 max(p;q) : a contradiction. It remains to show that a matrix without lower triangles cannot have too many zeros, which together with Proposition will show that the minor rank of H is nite. Indeed, we have the following more precise result Proposition Let C(n; p) denote the maximal number of zeros of a n p matrix without lower triangles. We have C(n; p) = n + p? 1 : Proof First, C(n; 1) = n, C(p; q) = C(q; p). Let r denote the number of zeros of the last column of A. We have the following dynamic programming equation: C(n; p) = max [C(n? r + 1; p? 1) + r] : (37) 1rn

22 18 Stephane GAUBERT Indeed, let us assume that the zeros on the last column of A are in position (i 1 ; p); (i ; p); : : :; (i r ; p) with i 1 < i < : : : < i r, i.e. 1 : : : p? 1 p i 1 " A = i + : : : + " 6 4 i r + : : : + " Then, the entries that are both on the p?1 rst columns and on the rows i ; : : :i r of A are necessarily non zero (these entries are written \+"). Hence, if r is chosen, we are reduced to maximizing the number of zero entries of the submatrix of size (n? r + 1) (p? 1), A(i : : : i r jp) (that is A without the rows marked + and the last column). This gives the induction (37) from which Proposition easily follows. Thus, the number of zeros of an n n minor taken from H is at most C(n; n) = n? 1. Since n(n?1) > n? 1 (for n 5), Proposition 3.4. shows that all the minors of H of size greater than 5 are balanced, hence, the minor rank of H is nite (at most 4) : Appendix I Eective Computation of Determinants In this appendix, we provide an O(n 3 ) algorithm to decide if the determinant of an n n matrix A with entries in R max is balanced. Another specic algorithm should be provided to compute the rank of matrices more eciently than by minors inspection, but it is beyond the scope of this paper. Let us dene for a = (a 0 ; a 00 ) R max, jaj = a0 a 00. Then, O perm A def = A i(i) = j det Aj i is the well known permanent function. As already noticed by Butkovic and Cuninghame-Green [6], computing perm A is equivalent to a standard assignment problem, i.e. with conventional notation perm A = max X i A i(i) Hence, computing perm A is O(n 3 ). Since j det Aj = perm A, the case perm A = " which implies that det A = " is trivial, and we shall assume that perm A 6= ". Hence, there is a permutation such that N i A i(i) = perm A 6= ". Consider the matrix and let C ij = Ai(i) if j = (i) " otherwise Since det B = det C?1 det A, we are reduced to the following Canonical Form: B = C?1 A (38) B Id; perm B = e (39)

23 On Rational Series in One Variable over certain Dioids 19 Given a cycle c = (i 1 ; : : :; i k ), the weight of c with respect to the matrix B is by denition w B (c) = B i1 i : : :B ik i 1 : We shall write w(c) instead of w B (c) when the matrix will be clear from the context. I.1 Lemma For a matrix B satisfying (39), for all circuit c, w(c) e. Proof Let us rst notice that all the diagonal elements of B are equal to e, for 8i; e B ii B ii O j6=i B jj perm B = e : Next, consider a circuit c and let be the cyclic permutation associated with c. We have, since B ii = e for all i, O B i(i) = w(c) perm B = e : Let us denote by F the \out-diagonal" part of B, i.e. i B = Id F; F ij = Bij if i 6= j " if i = j. : (40) The following notation is standard [0, 9]: A + = A A A 3 : : : From Lemma I.1, we get that B + converges (this is a well known theorem about stars of matrices in the (max; +) algebra [0], Chapter 3,x.3). I. Proposition Let B = Id F as in (40), with perm B = e. The following assertions are equivalent 1. det B r ". det B = e e 3. There exists a circuit of F of weight e and even length. 4. tr(f ) + = e. The following algorithm is an immediate consequence of this proposition. I.3 Algorithm Let A R nn max. We decide whether det A is balanced. 1. Compute perm A using an assignment algorithm (e.g. [0], Hungarish Algorithm (p. 157)). If perm A = ", then det A = " and we are done. Otherwise, we get a permutation such that perm A = O i A i(i) 6= " :. Dene B and F by (38,40). 3. If tr(f ) + = e, then det A r ", otherwise det A 6r ".

24 0 Stephane GAUBERT ore precisely, the algorithm shows that if tr(f ) + = e, and if tr(f ) + 6= e, det A = perm A perm A det A = sgn () (perm A t) for some t < perm A. Indeed, this result would write more simply in the symmetrized semiring S max [30, 16, 1]. det A = sgn ()(e tr(f ) + )perm A Since the computation of (F ) + using the Jordan algorithm is O(n 3 ) (see e.g. [0], Chapter 3, x4) and the standard assignment algorithms are O(n 3 ), checking if det A r " is O(n 3 ). Proof of the Proposition. (1),() follows immediately from j det Bj = perm B = e and the denition of r. (3),(4) follows from the fact that tr(a + ) is equal to the sum of the weights of the circuits of A. (3) ) (). Write det B = x 0 x 00 as in (31). Since B Id, det B det Id = e, hence, x 0 e. Since j det Bj = x 0 x 00 = perm B = e, we get x 0 e; x 00 e, hence x 0 = e. Let c = (i 1 ; : : :; i p ) be a circuit of length p and weight e, and introduce the permutation such that (j) = j if j 6 fi 1 ; : : :; i p g and (i k ) = i k+1 (convention i p+1 = i 1 ). Clearly, sgn () = e. Hence, det B O i B i(i) = Op l=1 B il i l+1 = Op l=1 F il i l+1 = e for B j(j) = B jj = e if j 6 fi 1 ; : : :; i p g and B j(j) = F j(j) otherwise. This implies that x 00 e, hence x 00 = e and det B = e e. () ) (4). Consider the decomposition of a permutation = c 1 : : : c k as a product of disjoint cycles of L respective lengths l 1 ; : : :; l k, and dene the weight of by w() = w(c 1 ) : : :w(c k ). Since det B = sgn ()w() = e e, there exists a permutation such that sgn ()w() = e. Since sgn () = e, admits at least one cycle of even length. oreover, e = w() = w(c 1 ) : : :w(c k ) with w(c i ) e for all i implies that w(c 1 ) = : : : = w(c k ) = e. Hence, there is a circuit of F with even length and weight e. I.4 Example To illustrate this algorithm, let us consider the matrix A = 6 4 e 1 " " " e 1 " " " e 1?3 " " e Since there are only two permutations with non " weight (namely, Id and (1; ; 3; 4), with weight e and opposite sign), it is immediate that det A = e e. Let us check this by the algorithm I.,(4). Here, we have perm A = e and A = Id F with F = 6 4 " 1 " " " " 1 " " " " 1?3 " " " : :

25 On Rational Series in One Variable over certain Dioids 1 By denition, (F ) + is the limit (reached in a nite time) of the sequence X 1 = F ; X n+1 = F X n F. We have 3 0 " " " 0 " X = 6 4 7? " 0 " 5 ; X 3 = X "? " 0 Hence, (F ) + = X and tr(f ) + = e, which yields a second verication that det A r " via I.,(4). I.5 Remark It should also be possible to compute determinants in another symmetrized semiring, namely the semiring S max studied in [30, 16]. Some additional simplication rules hold in S max, e.g. 3 = 3. The adaptation of the above algorithm to S max is straightforward: the resulting computations in S max are essentially equivalent although a bit simpler due to the simplication rules. II Equivalence of the Dierent Possible Denitions of Rational Series We briey discuss here another possible convention for the denition of rational series (used e.g. in [1]), and show that it is essentially equivalent to the one used in this paper. In the introduction, following the tradition [3], we only dened s when s has zero constant coecients (Axiom (S)). This denition which is the simplest one works in general semirings. In the case of dioids [1], it is possible to dene s for more general series due to the underlying ordered structure. We briey show that this alternative convention provides the same class of rational series under mild conditions. Let us recall that a dioid is canonically ordered by Let X S. We say that the innite sum a b () a b = b : (41) xx is well dened if X admits a least upper bound. Then, we set xx x x def = sup X : (4) The dioid of rational series has been dened as the closure of the dioid of polynomials for the following operations: sum and product of series and star of a series with zero constant coecient (Axiom (S)). Since s exists under more general conditions, we dene an a priori more general dioid of formal series Rat 0 as follows: II.1 Denition Rat 0 is the least set of formal series containing polynomials and such that Rat 0 Rat 0 Rat 0 Rat 0 Rat 0 Rat 0 (S 0 ) If s Rat 0 and s = L k s k is well dened according to (4), then s Rat 0. This amounts to replacing the discrete ultrametric topology on S[[X]] (see [3]) by some order topology [19].

26 Stephane GAUBERT Indeed, we just have replaced (S) by (S') in Denition We next show that under natural assumptions, the two conventions are equivalent: Rat = Rat 0. ore precisely, we shall assume that the following innite distributivity property holds We have: If X S and xx x exists; then 8y S; xx xy exists and xx xy = ( xx x)y : (43) II. Proposition Let S be a commutative dioid which satises the innite distributivity (43). Then Rat = Rat 0. Proof Clearly, Rat Rat 0. We show that Rat 0 = Rat. It is enough to check that if s Rat and if the least upper bound s = L kn sk exists, then s Rat. Let us dene for an arbitrary series t, L [p] def t = kp tk. A simple computation shows that hs [p] ; X 0 i = hx; X 0 i [p] hence, the existence of s implies the existence in S of hs; X 0 i = hs ; X 0 i : L Setting s 0 = k1hs; X k ix k, we have s = hs; X 0 i s 0. Under the innite distributivity assumption (43), the identity (a b) = a b holds, hence, s = hs; X 0 i s 0 which belongs to Rat since hs; X 0 i is a scalar and s 0 Rat (for hs 0 ; X 0 i = "). III Proof of Lemma 3.. Let us assume a linear combination of the form (35). Observe that hh; X k i = s k 3 for k s=r. a/ If j < i, then j ( i j )3. Indeed, we get from (34) and that which shows a/. b/ If j > i, then j ( i j )5. This follows from H ni j H nj with n = 0 (44) H 0i = i 3 s j H 0j = j j 3 s; which holds for n large enough. H ni = (ni) 5 j H nj = j (nj) 5 ; c/ Let us choose an integer k such that s r < ki < r. We claim that L jj j H kj < H ki. In the case where j < i, we get via a/ j H kj ( i j )3 H kj = ( i j )3 (kj) 3 (kj r)(kj s r ) If j > i, we obtain by b/ = i 3 k 3 r(kj s r ) = r(ki)4 (kj s r ) ki < r(ki) 4 = H ki : j H kj ( i j )5 H kj = ( i j )5 (kj) 3 (kj r)(kj s r ) This concludes the proof of c/ and of the Lemma. = i5 j k3 (kj s r )kj = r(ki)4 ( ki r i j ) < r(ki)4 = H ki :

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