structured/0709.4117.structured.json

{"paper_meta":{"paper_id":"arxiv:0709.4117","title":"0709.4117","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0709.4117v1 [cs.CC] 26 Sep 2007\nDeciding Unambiguity and Sequentiality starting\nfrom a Finitely Ambiguous Max-Plus Automaton\nInes Klimann, Sylvain Lombardy, Jean Mairesse,\nand Christophe Prieur∗\nJuly 4, 2004\nAbstract\nFinite automata with weights in the max-plus semiring are considered.\nThe main result is: it is decidable in an effective way whether a series that\nis recognized by a finitely ambiguous max-plus automaton is unambiguous,\nor is sequential. A collection of examples is given to illustrate the hierarchy\nof max-plus series with respect to ambiguity.\n1\nIntroduction\nA max-plus automaton is a finite automaton with multiplicities in the max-plus\nsemiring\nRmax = (R ∪{−∞}, max, +). Roughly speaking, it is an automaton\nwith two tapes: an input tape labelled by a finite alphabet Σ, and an output\ntape weighted in\nRmax. The weight of a word in Σ∗is the maximum over all\nsuccessful paths of the sum of the weights along the path.\nMax-plus automata, and their min-plus counterparts, are studied under var-\nious names in the literature: distance automata, finance automata, cost au-\ntomata. They have also appeared in various contexts: to study logical problems\nin formal language theory (star height, finite power property) [13, 23], to model\nthe dynamic of some Discrete Event Systems (DES) [10, 12], or in the context\nof automatic speech recognition [18].\nTwo automata are equivalent if they recognize the same series, i.e. if they\nhave the same input/output behavior. The problem of equivalence of two max-\nplus automata is undecidable [15]. The same problem for finitely ambiguous\nmax-plus automata is decidable [14, 25].\nThe sequentiality problem is defined as follows: given a max-plus automaton,\nis there an equivalent max-plus automaton which is sequential (i.e. determinis-\ntic in input). Let us give some motivations on why the sequentiality problem\nis important. In the case of a sequential automaton, the time complexity of\n∗LIAFA, CNRS (umr 7089) - Universit ́e Paris 7, 2, place Jussieu - 75251 Paris Cedex 5 -\nFrance. email: {klimann,lombardy,mairesse,prieur}@liafa.jussieu.fr\n1\n\ncomputing the output is roughly linear in the length of the input. This time\nefficiency is central in speech processing, see [18]. Consider now a DES mod-\nelled by a max-plus automaton. If the automaton is unambiguous, or a fortiori\nsequential, then one can compute the optimal, as well as the average behavior,\nof the DES, see [10, 11].\nSequentiality is decidable for unambiguous max-plus automata [18].\nIn the\npresent paper, we prove that sequentiality is decidable for finitely ambiguous\nmax-plus automata. To the best of our knowledge, it is not known if the finite\nambiguity of a max-plus series (defined via an infinitely ambiguous automaton)\nis a decidable problem. In particular, the status of the sequentiality problem\nis still open for a general max-plus automaton (even if the multiplicities are\nrestricted to be in\nZmax,\nNmax or\nZ−\nmax). To be complete, it is necessary to\nmention that in [18, §3.5], it is claimed that any max-plus automaton admits\nan effectively computable equivalent unambiguous one.\nIf that was true, it\nwould imply the decidability of the sequentiality for general max-plus automata.\nHowever, the statement is erroneous and counter-examples are provided in §3\nof the present paper1.\nThe sequentiality problem can be asked for automata over any semiring\nK.\nFor transducers, i.e. when\nK is the set of rational subsets of a free monoid\n(with union and concatenation as the two laws), the problem is completely\nsolved in the functional case (when, for every input, the output is a language of\ncardinality at most one) [3, 7, 8]. For a general transducer, the problem is wide\nopen. Observe that the semiring {a⩾n, n ∈\nN} = {ana∗, n ∈\nN} is isomorphic\nto\nNmin: ana∗+ ama∗= amin(n,m)a∗and ana∗· ama∗= an+ma∗. Similarly,\nthe semiring {a⩽n, n ∈\nN} is isomorphic to\nNmax (where a⩽n = {ε, a, . . ., an}).\nHence automata over\nNmax or\nNmin translate into transducers, but not functional\nones. Also the translation does not work for automata over\nRmax. Hence, the\nvast literature on transducers is of limited use in our context.\nIn the present paper, we work with\nRmax. Decidability and complexity should\nbe interpreted under the assumption that two real numbers can be added or\ncompared in constant time.\n2\nPreliminaries\n2.1\nMax-plus semiring and series\nThe free monoid over a finite set (alphabet) Σ is denoted by Σ∗and the empty\nword is denoted by ε. The structure\nRmax = (R ∪{−∞}, max, +) is a semiring,\nwhich is called the max-plus semiring.\nIt is convenient to use the notations\n⊕= max and ⊗= +. The neutral elements of ⊕and ⊗are denoted respectively\nby\n0 = −∞and\n1 = 0.\nThe subsemirings\nNmax,\nZmax, . . . , are defined in\nthe natural way.\nThe min-plus semiring\nRmin is obtained by replacing max\nby min and −∞by +∞in the definition of\nRmax. The results of this paper\ncan be easily adapted to the min-plus setting. Observe that the subsemiring\n1The version of [18] available on the author’s website has been correctly modified.\n2\n\nB = ({ 0,\n1}, ⊕, ⊗) is isomorphic to the Boolean semiring. For matrices A, B of\nappropriate sizes with entries in\nRmax, we set (A⊕B)ij = Aij ⊕Bij, (A⊗B)ij =\nL\nk Aik ⊗Bkj, and for a ∈\nRmax, (a ⊗A)ij = a ⊗Aij. We usually omit the ⊗\nsign, writing for instance AB instead of A ⊗B.\nConsider the set\nRmax⟨⟨Σ∗⟩⟩of (formal power) series (over Σ∗with coeffi-\ncients in\nRmax), that is the set of maps from Σ∗to\nRmax. We denote by ⟨S, u⟩\nthe coefficient of the word u in the series S. The support of a series S is the\nset Supp S = {u ∈Σ∗|\n⟨S, u⟩̸=\n0}. It is convenient to use the notation\nS = L\nu∈Σ∗⟨S, u⟩u = L\nu∈Supp (S)⟨S, u⟩u. Equipped with the addition (⊕) and\nthe Cauchy product (⊗), the set\nRmax⟨⟨Σ∗⟩⟩forms a semiring. The image of\nλ ∈\nRmax by the canonical injection into\nRmax⟨⟨Σ∗⟩⟩is still denoted by λ. In\nparticular, the neutral elements of\nRmax⟨⟨Σ∗⟩⟩are\n0 and\n1. The characteristic\nseries of a language L is the series\n1L such that ⟨1L, w⟩=\n1 if w ∈L, and\n⟨1L, w⟩=\n0 otherwise.\n2.2\nMax-plus automaton\nLet Q and Σ be two finite sets. A max-plus automaton of set of states (di-\nmension) Q over the alphabet Σ, is a triple A = (α, μ, β), where α ∈\nR1×Q\nmax ,\nβ ∈\nRQ×1\nmax , and where μ : Σ∗→\nRQ×Q\nmax\nis a morphism of monoids. The mor-\nphism μ is uniquely determined by the family of matrices {μ(a), a ∈Σ}, and\nfor w = a1 · · · an, we have μ(w) = μ(a1) ⊗· · · ⊗μ(an). The series recognized\n(or realized) by A is by definition S(A) = L\nu∈Σ∗(αμ(u)β)u.\nThis is just a\nspecialization to the max-plus semiring of the classical notion of an automaton\nwith multiplicities over a semiring [4, 9, 17].\nBy the Kleene-Sch ̈utzenberger\nTheorem [21], the set of series recognized by a max-plus automaton is equal to\nthe set of rational series over\nRmax. We denote it by Rat.\nA state i ∈Q is initial, resp. final, if αi ̸=\n0, resp. βi ̸=\n0. As usual a\nmax-plus automaton is represented graphically by a labelled weighted digraph\nwith ingoing and outgoing arcs for initial and final states, see e.g. Figure 6\n(the input or output weights equal to\n1 are omitted). The terminology of graph\ntheory is used accordingly (e.g. (simple) path or circuit of an automaton, union\nof automata, . . . ).\nA path which is both starting with an ingoing arc and\nending with an outgoing arc is called a successful path. The label of a path is\nthe concatenation of the labels of the successive arcs (so called transitions), the\nweight of a path is the product (⊗) of the weights of the successive arcs (including\nthe ingoing and the outgoing arc, need it be). We denote by weight (π) the\nweight of the path π. We use the following notations for paths in an automaton\nA = (α, μ, β):\np →q,\n→p →q,\np →q →,\np\nu|x\n−−→q,\nh\np\nu|x\n−−→q\ni\nA , if μ(u)pq = x in A .\nThe first example is a path (of any length) from p to q, the second also\nincludes an ingoing arc, the third an outgoing arc, in the fourth the weight and\nthe label are added and in the fifth the underlying automaton is recalled.\nAn automaton is trim if any state belongs to at least one successful path.\n3\n\nLet I be a finite set. The tensor product automaton of (Ai = (α⌞i, μ⌞i, β⌞i))i∈I,\ndenoted by ⊙i∈IAi, is defined as follows. It is the max-plus automaton (A, M, B)\nof dimension Q = Q\ni Qi, where Qi is the dimension of Ai, and such that\n∀p, q ∈Q,\nAp =\nO\ni∈I\nα⌞i\npi,\n∀a ∈Σ, M(a)p,q =\nO\ni∈I\nμ⌞i(a)pi,qi,\nBp =\nO\ni∈I\nβ⌞i\npi .\n2.3\nHeap model\nA heap or Tetris model [24], consists of a finite set of slots R, and a finite\nset of rectangular pieces Σ. Each piece a ∈Σ is of height 1 and occupies a\ndetermined subset R(a) of the slots. To a word u = u1 · · · uk ∈Σ∗is associated\nthe heap obtained by piling up in order the pieces u1, . . . , uk, starting with\na horizontal ground and according to the Tetris game mechanism (pieces are\nsubject to gravity and fall down vertically until they meet either a previously\npiled up piece or the ground). Consider the morphism generated by the matrices\nM(a) ∈\nRR×R\nmax , a ∈Σ, defined by\nM(a)ij =\n \n \n \n1\nif i, j ∈R(a),\n0\nif i = j ̸∈R(a),\n−∞\notherwise .\nLet x(u)i be the height of the heap u on slot i ∈R. We have ([5, 11, 12]):\nx(u)i =\n1M(u)δi, where\n1 = (1, . . . ,\n1) ∈\nR1×R\nmax and δi ∈\nRR×1\nmax is defined by\n(δi)j =\n1 if j = i and\n0 otherwise. In other words, the application x(·)i : Σ∗→\nRmax is recognized by the max-plus automaton (1, M, δi). We call (1, M, δ), δ =\nL\ni∈I δi, I ⊆R, a heap automaton (associated with the heap model). Among\nmax-plus automata, heap automata are particularly convenient and playful, due\nto the underlying geometric interpretation. Here, they are used as a source of\nexamples and counter-examples, e.g. Figures 3, 4 and 7.\nWe represent a heap automaton graphically as in Figure 1.\nb\na\nR(a) = {1, 2}\nR(b) = {2, 3}\nR = {1, 2, 3}\n(1, M, δ2)\nFigure 1: A heap automaton\n2.4\nAmbiguity and Sequentiality\nConsider a max-plus automaton A = (α, μ, β) of dimension Q over Σ.\nThe\nautomaton is sequential if there is a unique initial state and if for all i ∈Q, and\n4\n\nfor all a ∈Σ, there is at most one j ∈Q such that μ(a)ij ̸=\n0. In the case of\na Boolean automaton, we also say deterministic for sequential. The automaton\nA is unambiguous if for any word u ∈Σ∗, there is at most one successful path\nof label u.\nThe automaton is finitely ambiguous if there exists some k ∈\nN\nsuch that for any word u ∈Σ∗, there are at most k successful paths of label\nu.\nThe minimal such k is called the degree of ambiguity of the automaton.\nClearly, ‘sequential’ implies ‘unambiguous’ which implies ‘finitely ambiguous’.\nThe automaton is infinitely ambiguous if it is not finitely ambiguous.\nConsider a series S ∈Rat. The series is sequential (resp. unambiguous,\nfinitely ambiguous) if there exists a sequential (resp. unambiguous, finitely am-\nbiguous) max-plus automaton recognizing it. The series is infinitely ambiguous\nif there exists no finitely ambiguous max-plus automaton recognizing it. The\ndegree of ambiguity of a finitely ambiguous series is the minimal degree of ambi-\nguity of an automaton recognizing it. The sets of sequential, unambiguous, and\nfinitely ambiguous series are denoted respectively by Seq, NAmb, and FAmb.\nDefine FSeq = {S | ∃k, ∃S1, . . . , Sk ∈Seq, S = S1 ⊕· · · ⊕Sk}.\nConsider a total order on Σ∗. Given a series S ̸=\n0, define the normalized\nseries φ(S) by φ(S) = L\nu∈Σ∗(⟨S, u⟩−⟨S, u0⟩)u, where u0 is the smallest word\nof Supp S. The (left) quotient of a series S by a word w is the series w−1S\ndefined by w−1S = L\nu∈Σ∗⟨S, wu⟩u.\nA series S is rational if and only if the semi-module of series ⟨w−1S, w ∈Σ∗⟩\nis finitely generated, i.e. if there exists S1, . . . , Sk, such that:\n∀w ∈Σ∗, ∃λ1, . . . , λk ∈\nRmax, w−1S =\nM\ni\nλiSi.\nA series S is sequential if and only if the set of series {φ(w−1S), w ∈Σ∗} is\nfinite.\nProposition 1 A trim automaton A of dimension Q is infinitely ambiguous if\nand only if there exist p, q ∈Q, p ̸= q, and v ∈Σ∗, such that p\nv\n−→p, p\nv\n−→q,\nq\nv\n−→q. This can be checked in polynomial time.\nFor a proof, see [27] and the references therein. Observe that the (in)finite\nambiguity is independent of the underlying semiring.\nNext result is due to\nMohri [18] and is an adaptation of a classical result of Choffrut on functional\ntransducers, see [3, 7, 8] (for the decidability) and [2, 26] (for the polynomial\ncomplexity).\nTheorem 1 Let A be an unambiguous max-plus automaton.\nThere exists a\npolynomial time algorithm to decide whether S(A) is a sequential series.\nIf A is unambiguous and S(A) is sequential, a sequential automaton recog-\nnizing the series can be effectively constructed from A using an adaptation of\nthe subset construction of Boolean automata [1, 6, 18].\nIt is useful to detail Theorem 1. We need to introduce several definitions.\nGiven two words u, v ∈Σ∗, let u ∧v be the longest common prefix of u and\n5\n\nv, and define d(u, v) = |u| + |v| −2|u ∧v|. It is easy to check that d(., .) is a\ndistance on Σ∗. A series S is M-Lipschitz (M ∈\nR+) if:\n∀u, v ∈Supp S, |⟨S, u⟩−⟨S, v⟩| ⩽Md(u, v) ;\nand S is Lipschitz if it is M-Lipschitz for some M. The set of Lipschitz series is\ndenoted by Lip. Consider a trim max-plus automaton A of dimension Q. Two\nstates p, q ∈Q are twins if:\nh x0\n−→i\nu1|x1\n−−−→p\nu2|x2\n−−−→p,\ny0\n−→j\nu1|y1\n−−−→q\nu2|y2\n−−−→q\ni\n=⇒[x2 = y2] .\nIf all the states are twins, the automaton A is said to satisfy the twin property.\nWe denote the set of all such automata by Twin. The following implications\nhold:\nh\nA ∈Twin\ni\n=⇒\nh\nS(A) ∈Seq\ni\n=⇒\nh\nS(A) ∈Lip\ni\n.\n(1)\nFurthermore,\nh\nA ∈NAmb,\nS(A) ∈Lip\ni\n=⇒\nh\nA ∈Twin\ni\n.\n(2)\nThe twin property can be checked in polynomial time, hence Theorem 1 follows\nfrom the above implications.\n3\nHierarchy of Series\nThe examples in this section illustrate the classes of series on which we work.\nSeq ⊊(NAmb ∩FSeq)\n(§3.1)\n⊊\n⊊\nFSeq\n(§3.2)\nNAmb\n(§3.3)\n⊊\n⊊\nFAmb\n(§3.4)\n⊊\nRat\n(§3.5) ⊊Series\n(§3.6)\n3.1\nA Series in Seq ∩NAmb ∩FSeq\nAn example over a one-letter alphabet is provided in Figure 2. The recognized\nseries is\n⟨S, an⟩=\n(\n0\nif n is odd,\nn\nif n is even.\na|0\na|0\na|1\na|1\nFigure 2: Seq ∩NAmb ∩FSeq\nThe series is not Lipschitz, since |⟨S, an+1⟩−⟨S, an⟩| ⩾n, and consequently\nthe series cannot be sequential (see (1)). It is clear that it is an unambiguous\n6\n\nseries (the only successful path of label an is the right or left one depending\non the parity of n) and a sum of sequential series. In fact, any max-plus ra-\ntional series over a one-letter alphabet is unambiguous and a sum of sequential\nseries [16, 19].\na|1, b|0\na|0, b|1\na\nb\nFigure 3: FSeq ∩NAmb\n3.2\nA Series in FSeq ∩NAmb\nThe series ⟨S, u⟩= |u|a ⊕|u|b over the alphabet {a, b} is a sum of two sequential\nseries: the heap automaton of Figure 3 recognizes this series.\nAssume that S is unambiguous. The series S is 1-Lipschitz. So it has to be\nsequential, see (2) and (1). Consequently, there exist series S1,. . . Sk such that:\n∀u ∈Σ∗, ∃i, ∃λu ∈\nRmax\nu−1S = λu ⊗Si.\nBy the pigeon-hole principle, there must exist i ∈{1, . . . k} and two integers\nm < n such that\n∃λn, λm\n(an)−1S = λn ⊗Si,\n(am)−1S = λm ⊗Si.\nConsequently, we have\n⟨(an)−1S, bm+1⟩−⟨(an)−1S, ε⟩= ⟨(am)−1S, bm+1⟩−⟨(am)−1S, ε⟩.\nHowever\n⟨(an)−1S, bm+1⟩−⟨(an)−1S, ε⟩= ⟨S, anbm+1⟩−⟨S, an⟩= n −n = 0\n⟨(am)−1S, bm+1⟩−⟨(am)−1S, ε⟩= ⟨S, ambm+1⟩−⟨S, am⟩= m + 1 −m = 1.\nThis is a contradiction, consequently S is not sequential and thus cannot be\nan unambiguous series.\n3.3\nSeries in NAmb ∩FSeq\na) The first example is the series S given by the heap automaton of Figure 4 (a),\nor equivalently by the automaton of Figure 4 (b).\nConsider the series ̃S defined by ⟨ ̃S, w⟩= ⟨S, w⟩−|w|.\nAn automaton\nrecognizing ̃S can clearly be obtained from an automaton recognizing S by\nremoving 1 from each output weight. Hence S and ̃S are both sum of sequential\nseries or none of them is.\n7\n\na|1, b|0\na|1, b|1\na|1\na|1\nb\na\n(a)\nb|0\na|1, b|1\na|1\n(b)\nFigure 4: NAmb ∩FSeq\na|0\na|0, b|0\nb| −1\nFigure 5: NAmb ∩FSeq\nThe series ̃S is recognized by the automaton of Figure 5.\nSuppose that\n ̃S = S1 ⊕S2 ⊕· · · ⊕Sk, where k ∈\nN and the Si are sequential series.\nSince the Si are sequential series, they are Lipschitz. Let N be the maximal\nLipschitz coefficient of the Si. Let (Ni)i⩾0 be a sequence of integers such that\nN0 > N,\nN(Nk−1 + 1) < Nk −Nk−1 for all k ⩾1.\nThe coefficient of abNk in ̃S is −Nk, and it comes, for instance, from S1.\nThe coefficient of abNkabNk−1 is −Nk−1. We have:\nd(abNk, abNkabNk−1) = Nk−1+1 and |⟨ ̃S, abNk⟩−⟨ ̃S, abNkabNk−1⟩| = Nk−Nk−1.\nThe coefficient of abNkabNk−1 in ̃S does not come from S1, since\n|⟨S1, abNk⟩−⟨S1, abNkabNk−1⟩| ⩽N(Nk−1 + 1)\n< Nk −Nk−1 = |⟨S1, abNk⟩−⟨ ̃S, abNkabNk−1⟩|.\nIn the same way, we prove that any two words of the set\n{abNk, abNkabNk−1, . . . , abNkabNk−1 · · · abN0}\ncannot be recognized by the same Si. But this set has cardinality k + 1 and\nthus there is a contradiction.\nb) The second example is the series given by the automaton of Figure 6. The\nseries recognized by this automaton is:\n⟨S, am1bn1 · · · ampbnp⟩=\nX\nmieven\nmi,\n8\n\nwhere m1 ∈\nN, mk+1 ∈\nN −{0}, nk ∈\nN −{0} for 1 ⩽k ⩽p −1, and np ∈\nN.\nThe automaton is clearly unambiguous. Furthermore, it is not a finite sum of\nsequential series. To simplify notations, let us prove that S is not the sum of\ntwo sequential series. Suppose that S = S1 ⊕S2, with S1, S2 ∈Seq.\nThe series Si, i ∈{1, 2}, are sequential, so they are Lipschitz by (1). Let\nN be such that Si, i ∈{1, 2}, are N-Lipschitz. Let us consider words of the\nform arbnas, with n > 0. We discuss on the parity of r and s. The coefficient\nof the word a2p+1bna2q+1 in S, which is equal to 0, comes from one of the Si.\nFor instance\n⟨S, a2p+1bna2q+1⟩= 0 = ⟨S1, a2p+1bna2q+1⟩.\n(3)\nSet q > N. Since S1 is N-Lipschitz and d(a2p+1bna2q+1, a2p+1bna2q) = 1,\nwe have\n⟨S, a2p+1bna2q⟩= 2q = ⟨S2, a2p+1bna2q⟩.\n(4)\nFix q and n. Since S1 and S2 are Lipschitz, there exists an integer M such\nthat:\n∀u, v ∈Supp Si, d(u, v) ⩽2n + 4q + 2 ⇒|⟨Si, u⟩−⟨Si, v⟩| ⩽M.\n(5)\nWe have:\nd(a2pbna2q, a2p+1bna2q+1) = 2n+4q+2\nand\nd(a2pbna2q, a2p+1bna2q) = 2n+4q+1.\nSo, by Equation (5), we know that:\n– If a2pbna2q ∈Supp S1, then\n2p + 2q = |⟨S1, a2pbna2q⟩−⟨S1, a2p+1bna2q+1⟩| ⩽M,\nwhich is wrong for p large enough.\n– If a2pbna2q ∈Supp S2, then\n2p = |⟨S2, a2pbna2q⟩−⟨S2, a2p+1bna2q⟩| ⩽M,\nwhich is also wrong for p large enough.\nConsequently, S is not the sum of two sequential series.\nTo extend the re-\nsult to the sum of m sequential series, one has to consider words of the form\nar1bn1ar2 · · · arm−1bnm−1arm.\n3.4\nSeries in NAmb ∩FSeq ∩FAmb\na) Consider the heap automaton given in Figure 7 (a).\nThe corresponding\nseries is at most two-ambiguous since it is also recognized by the two-ambiguous\nautomaton of Figure 7 (b).\nIt cannot be unambiguous: on {a, b}∗, since it\ncoincides with the series of Figure 3 which is in NAmb. It cannot be a finite\nsum of sequential series: on {b, c}∗, it coincides with the series of Figure 4 which\nis in FSeq.\n9\n\ni\nj\na|0\na|0\na|1\na|1\nb|0\nb|0\nb|0\nb|0\nFigure 6: NAmb ∩FSeq\na|0, b|1, c|0\na|0, b|1, c|1\nb|1\nb|1\na|1, b|0, c|0\nb\na\nc\n(a)\na|0, c|0\na|0, b|1, c|1\nb|1\na|1, b|0, c|0\n(b)\nFigure 7: NAmb ∩FSeq ∩FAmb\nb) Another example is provided by the automaton A of Figure 8.\nDenote by S the series recognized by this automaton, by S1 the series recog-\nnized by the left part, say A1, of the automaton, and by S2 the series recognized\nby the right part, say A2.\nThe automaton A1 is the one introduced in Section 3.3 and the automaton\nA2 is the same one after permutation of the a’s and b’s in the labels. Recall\nthat A1 and A2 are unambiguous, so S is at most two-ambiguous.\nLet us prove that S is not a finite sum of sequential series. Denote by L the\nlanguage of words whose blocks of b’s have odd length. Let u be a word of L:\na|0\na|0\na|1\na|1\nb|0\nb|0\nb|0\nb|0\nb|0\nb|0\nb|1\nb|1\na|0\na|0\na|0\na|0\nFigure 8: NAmb ∩FSeq ∩FAmb\n10\n\nc|1\nc|1\na|1, b|0, c|1\na|0, b|1, c|1\n(a)\na\nb\nc\n(b)\nFigure 9: FAmb ∩Rat\nin A2, the b-blocks of u are always read in the upper part of the automaton, so\n⟨S2, u⟩= 0. Since the coefficient of u in A1 is at least 0, we have S⊙\n1L = S1⊙\n1L.\nSuppose that S is a finite sum of sequential series. Then so is S⊙\n1L and S1⊙\n1L.\nAnd this is false since one can choose an odd n in the proof of Section 3.3 for\nthe automaton of Figure 6.\nLet us prove that S is not unambiguous. Let M be the rational language\nof words whose a-blocks and b-blocks have even lengths and let u be a word\nof M.\nIn A1, the a-blocks have to be read in the lower part of A1 and so\n⟨S1, u⟩= |u|a. In the same way: ⟨S2, u⟩= |u|b. So we have S ⊙\n1M = S′ ⊙\n1M,\nwhere S′ is the series recognized by the automaton of Figure 3. Consequently,\nif S is unambiguous, so is S′ ⊙\n1M. We now apply the arguments of Section 3.2\nto show that S′ ⊙\n1M is not unambiguous.\nc) Besides, Weber has given examples of series which are k-ambiguous and\nnot (k −1)-ambiguous [25, Theorem 4.2].\n3.5\nSeries in FAmb ∩Rat\nConsider the series S recognized by the automaton of Figure 9. Assume that S\nis finitely ambiguous. Using the result of Corollary 1 below, S is recognized by\na finite union of unambiguous automata with the same support, say A1, . . . , Ak.\nDenote by Si the series recognized by Ai, for 1 ⩽i ⩽k, and by n the\nmaximal dimension of an automaton Ai. Observe that Supp S = Σ∗. Since all\nthe Si have the same support, we have Supp Si = Σ∗.\nNow, consider the word w0 =\n anbnc\n k. For any i, there is a single successful\npath labelled by w0 in Ai. Note that a path of length n contains necessarily a\ncircuit.\nSo, each automaton Ai contains a path of the form:\nπi :\na · · · a\na · · · a\nb · · · b\nb · · · b\nc\na · · · a\na · · · a\na · · · a\nb · · · b\na · · · a\nFor every j ∈{1, . . ., k}, we choose in the subpath labelled by the j-th factor\nan (resp. bn) a circuit that is called the j-th a-loop (resp. the j-th b-loop).\n11\n\nThe coefficient of a word in S is less than or equal to its lentgh, it is thus\nthe same for its coefficients in the Si. Consequently, the mean weights of the\nloops of πi are less than or equal to 1. Denote by av (πi, a, j) the mean weight\nof the j-th a-loop in the path πi, and define av (πi, b, j) similarly.\nSet j ∈{1, . . ., k}. For λ ∈\nN −{0}, consider the word\nwλ = (anbnc) · · · (anbnc) (an+λn!bn+λn!c)\n|\n{z\n}\nj-th block\n(anbnc) · · · (anbnc)\nThis word can be read on each path πi by turning into the j-th a- and\nb-loops, whose lengths are less than or equal to n and so divide n!.\nLet i ∈{1, . . . , k} be such that ⟨S, w0⟩= ⟨Si, w0⟩.\nWe have ⟨S, wλ⟩−\n⟨S, w0⟩= λn!, and so ⟨Si, wλ⟩−⟨Si, w0⟩⩽λn!.\nBut ⟨Si, wλ⟩−⟨Si, w0⟩=\n(av (πi, a, j) + av (πi, b, j))λn!, consequently\nav (πi, a, j) + av (πi, b, j) ⩽1.\n(6)\nConsider any u in {01, 10}k. For all p ∈\nN −{0}, let us define the word\nvp(u) = (an+λ1n!bn+μ1n!c) · · · (an+λjn!bn+μjn!c) · · · (an+λkn!bn+μkn!c),\nwhere (λj, μj) = (p, 0) if (u2j−1, u2j) = (1, 0) (we say then that the dominant\nj-th loop is the j-th a-loop) and (λj, μj) = (0, p) otherwise (the dominant j-th\nloop is the j-th b-loop), for any j ∈{1, . . . , k}.\nBy the pigeon-hole principle, for some i, there are infinitely many words of\nthe form vp(u) such that ⟨S, vp(u)⟩= ⟨Si, vp(u)⟩= k + nk + kpn!. Such words\nare read on the path πi. The dominant j-th loop in πi has then necessarily\nmean weight 1, and by Equation (6), the non-dominant j-th loop in πi has\nmean weight less than or equal to 0.\nConsequently, we have built an injection from the language {01, 10}k into\nthe set of paths {πi}. But the language has cardinality 2k and the set of paths\nhas cardinality k. So we have a contradiction.\n3.6\nRational and Non-Rational Series (Rat)\nA max-plus series is non-rational as soon as its support is a non-rational lan-\nguage. Here, we present a less trivial example of non-rational max-plus series.\nIn this paragraph, it is necessary to distinguish between\nRmin and\nRmax:\nfor R = Rat or NAmb, we use the respective notations\nRminR,\nRmaxR.\nIf\nS ∈\nRmax⟨⟨Σ∗⟩⟩, we identify S with ̃S ∈\nRmin⟨⟨Σ∗⟩⟩such that ⟨ ̃S, w⟩= ⟨S, w⟩if\nw ∈Supp S and ⟨ ̃S, w⟩= +∞if ⟨S, w⟩= −∞.\nClearly, we have\nRmaxNAmb =\nRminNAmb = NAmb.\nOn the other hand, it is easy to find S ∈\nRminFSeq ∩\nRminNAmb such that\nS ̸∈\nRmaxRat.\n12\n\nConsider for instance the series S = min(|w|a, |w|b) (recognized by the au-\ntomaton of Figure 3 seen as a min-plus automaton). Let us prove that S does\nnot belong to\nRmaxRat. If it does: let S1, . . . Sn be a minimal generating fam-\nily of ⟨u−1S, u ∈Σ∗⟩(see §2.4), we have: ∀u ∈Σ∗, ∃λ(u)\n1 , . . . λ(u)\nn , u−1S =\nL\ni λ(u)\ni\n⊗Si. The restrictions of the quotients of S to b∗are bounded, hence\nso are the restrictions of the Si. Let ki be such that: ⟨Si, bki⟩= maxk⟨Si, bk⟩.\nIt follows that for any word u: maxk⟨u−1S, bk⟩= maxki⟨u−1S, bki⟩. Consider\nk > maxi ki. Then arises a contradiction:\nmax\nl\n⟨(ak)−1S, bl⟩= k > max\nki ⟨(ak)−1S, bki⟩= max\ni\nki.\n3.7\nAmbiguity vs. sequentiality and Ambiguity vs. Lips-\nchitz\nHere are some examples of series that are in several classes described in Section 3:\nNAmb\nFAmb ∩NAmb\nFAmb\nSeq\na\nimpossible\nimpossible\nFSeq\n∩\nSeq\na|0\na|0\na|1\na|1\na\nb\nSec 3.2\nimpossible\nFSeq\nb\na\nSec 3.3\nb\na\nc\nSec 3.4\na\nb\nc\nSec 3.5\n13\n\nNAmb\nFAmb ∩NAmb\nFAmb\nLip\na\na\nb\nSec 3.2\na\nb\nc\nSec 3.5\nLip\nb\na\nSec 3.3\nb\na\nc\nSec 3.4\na\nb\nc\n4\nFrom Finitely Ambiguous to Union of Unam-\nbiguous\nWeber [25] has proved that a finitely ambiguous\nNmax-automaton can be turned\ninto an union of unambiguous ones.\nWe present a completely different and\nsimpler proof that holds in any semiring, in particular\nRmax.\nIn this section, we work on the structure of the automata. So we consider\nsimply Boolean automata.\nBelow, given a set S, we identify the vectors of\nBS with the subsets of S,\ni.e. x ∈\nBS is identified with {i ∈S | xi =\n1}.\nLet A = (α, μ, β) be a trim automaton. The past of a state p is the set of\nwords that label a path from some initial state to p. The future of p is the set\nof words that label a path from p to some final state. We write:\nPastA(p) = {w ∈Σ∗| (αμ(w))p =\n1},\nFutA(p) = {w ∈Σ∗| (μ(w)β)p =\n1}.\nLet A = (α, μ : Σ∗→\nBQ×Q, β) be an automaton. Let us recall the usual\ndeterminization procedure of A via the subset construction. Let R be the least\nsubset of\nBQ inductively defined by:\nα ∈R,\nX ∈R ⇒∀a ∈Σ, Xμ(a) ∈R.\nLet D = D(A) = (J, ν : Σ∗→\nBR×R, U) be the determinized automaton\nof A defined by:\nJ = {α},\nU = {P ∈R | Pβ =\n1},\nν(a)P,P ′ =\n1 ⇐⇒P ′ = Pμ(a).\nLemma 1 i) Let A be an automaton and D its determinized automaton. Then\nfor each state P of D,\nPastD(P) ⊆\n\\\np∈P\nPastA(p),\nand\nFutD(P) =\n[\np∈P\nFutA(p).\n14\n\np\nq\nr\npq\nqr\npr\nr\nq\np\np, {p, q}\nq, {p, q}\nq, {q, r}\nr, {q, r}\np, {p, r}\nr, {p, r}\nr, {r}\nq, {q}\np, {p}\na b\na\nb\nb\na\nb\na\nb\nb\na\nb\na\nb\na\nb\na\nb\na\nb\nb\na\na\nb\nb\na\nb\nb\na\nb\nb\nFigure 10: A (left), the determinized automaton (top) and the Sch ̈utzenberger cov-\nering\nii) Let A and B be two automata and A ⊙B their tensor product (cf. §2.4),\nthen, for all state (p, q) of A ⊙B,\nPastA⊙B(p, q) = PastA(p) ∩PastB(q),\nFutA⊙B(p, q) = FutA(p) ∩FutB(q).\nThe constructions and results given in Propositions 2 and 3 are inspired by\nSch ̈utzenberger [22]. They have been explicitely stated by Sakarovitch in [20].\nLet A be an automaton and D its determinized automaton. The trim part\nof the product A ⊙D is called the Sch ̈utzenberger covering S of A.\nProposition 2\nLet A = (α, μ, β) be a trim automaton, D its determinized\nautomaton and S its Sch ̈utzenberger covering.\ni) The states of S are exactly the pairs (p, P), where P is a state of D and\np ∈P. We call the set {(p, P) | p ∈P} of states of S a column (in gray on\nFigure 10).\nii) The canonical surjection ψ from the transitions of S onto the transitions\nof A induces a one-to-one mapping between the successful paths of S and A.\niii) Let P be a state of D. Then, for every p in P,\nPastS(p, P) = PastD(P),\nFutS(p, P) = FutA(p).\nThus, all the states of a given column have the same past.\nproof. i) A state (p, P) of S is initial if and only if p is initial in A (i.e.\np ∈α) and P is initial in D (i.e. P = {α}). Now, let (p, P) be a state of S\nsuch that p ∈P and (q, Q) a successor of (p, P) by a. Then, there exist two\ntransitions:\nh\np\na\n−→q\ni\nA\nand\nh\nP\na\n−→Q\ni\nD.\nBy definition of D, q belongs thus to Q.\n15\n\nConversely, let P be a state of D and p an element of P. For every w in\nPastD(P), w belongs to PastA(p) (Lemma 1). Therefore there is a path in S\nfrom an initial state to (p, P).\nii) Let π be a successful path of A, with label w = w1w2 · · · wn. Let θ be\nthe (unique) successful path with label w in D:\nπ =\nh\n→p0\nw1\n−→p1\nw2\n−→. . .\nwn\n−→pn →\ni\nA,\nθ =\nh\n→P0\nw1\n−→P1\nw2\n−→. . .\nwn\n−→Pn →\ni\nD.\nThere is a path in S: π′ =→(p0, P0)\nw1\n−→(p1, P1)\nw2\n−→. . .\nwn\n−→(pn, Pn) →.\nThe function π 7→π′ is obviously one-to-one.\niii) By results from Lemma 1:\nPastS(p, P) = PastA(p) ∩PastD(P)\n∀p ∈P, PastD(P) ⊆PastA(p)\n \n⇒∀p ∈P, PastS(p, P) = PastD(P).\nFutS(p, P) = FutA(p) ∩FutD(P)\n∀p ∈P, FutA(p) ⊆FutD(P)\n \n⇒∀p ∈P, FutS(p, P) = FutA(p).\n□\nDefinition 1 In S, different transitions with the same label, the same desti-\nnation and whose origins belong to the same column are said to be competing.\nLikewise, different final states of the same column are competing. A competing\nset is a maximal set of competing transitions or competing final states.\nLet U be an automaton obtained from S by removing all transitions except one\nin every competing set and by turning all final states of a column, except one,\ninto non-final states. The choice of the transition (or the final state) to keep in\na competing set is arbitrary.\nFor instance, the covering S of Figure 10 has two competing sets (drawn\nwith double lines); the first one contains two transitions with label b that arrive\nin (r, {r}), the second one contains the states (p, {p, r}) and (r, {p, r}) which are\nboth final. The above selection principle gives rise to four possible automata,\nthe automaton of Fig. 11 being one of them.\nProposition 3 Let S and U be two automata defined as above. Then,\ni) ∀P, ∀p ∈P, PastU(p, P) = PastS(p, P).\nii) Futures of states in a column of U are disjoint and\n∀P, ∀p ∈P,\n[\np∈P\nFutU(p, P) =\n[\np∈P\nFutS(p, P) .\nConsequently, the automaton U is unambiguous and equivalent to A.\nproof. i) The proof is by induction on the length of words. If (p, P) is\ninitial in S, it is still initial in U. Let wa be a word of PastS(p, P) and π a\n16\n\npath labelled by this word from an initial state to (p, P). We consider the last\ntransition of π:\nh\n(q, P ′)\na\n−→(p, P)\ni\nS.\nIf this transition does not belong to a competing set, it still appears in U and, by\ninduction, w ∈PastU(q, P ′), thus wa ∈PastU(p, P). If this transition belongs\nto a competing set, there exist q′ ∈P ′ and a transition\nh\n(q′, P ′)\na\n−→(p, P)\ni\nS\nwhich still appears in U, and by induction, since PastS(q, P ′) = PastS(q′, P ′),\nw ∈PastU(q′, P ′), so wa ∈PastU(p, P).\nii) We prove this by induction on the length of words. If there are several\nfinal states in a column of S, exactly one remains in U, so there is at most\none state whose future contains the empty word. Now let (p, P) and (p′, P ′) be\ntwo states in the same column such that the word au belongs to FutA(p) and\nFutA(p′):\nh\np\na\n−→q\nu\n−→t →\ni\nA\nh\np′\na\n−→q′\nu\n−→t′ →\ni\nA\nBoth transitions p\na\n−→q and p′\na\n−→q′ correspond to the same transition in D.\nThus q and q′ belong to the same column and, by induction, q = q′. Since there\nis no competing set in U, p = p′.\nObviously FutU(p, P) ⊆FutS(p, P). If au is in the future of a state (p0, P0) of S,\nthere exist a state (p1, P1) and a transition (p0, P0)\na\n−→(p1, P1), such that u is in\nFutS(p1, P1). By induction, there exists p′\n1 in P1 such that u is in FutU(p′\n1, P1),\nand there exists a transition (p′\n0, P0)\na\n−→(p′\n1, P1), thus au is in FutU(p′\n0, P0).\nLet w be a word accepted by A. For any factorization uv of w, there is\nexactly one column P of U such that, for every p in P, u is in PastU(p, P) and\nthere is exactly one state (p, P) in this column such that v is in FutU(p, P). This\ncharacterizes the only successful path with label w in U.\n□\nWe show now how the Sch ̈utzenberger covering can be used to convert a\nfinitely ambiguous automaton A into a finite union of unambiguous automata,\neach of them recognizing the same language as A.\nProposition 4 Let S be the Sch ̈utzenberger covering of a finitely ambiguous\nautomaton. Then, competing transitions of S do not belong to any circuit of S.\nThus a path of S contains at most one transition of each competing set.\nproof. Assume that a competing transition τ belongs to a circuit:\n→i\nu\n−→(p, P)\na\n−→\nτ\n(q, Q)\nw\n−→(p, P)\na\n−→\nτ\n(q, Q)\nv\n−→t →.\n17\n\na\nb\na\nb\nb\na\na\nb\nb\na\nb\nb\na\nb\nFigure 11: An unambiguous automaton equivalent to S\nHence, u(aw)∗is a subset of PastA(p). Let τ ′ be another transition that belongs\nto the same competing set: (p′, P)\na\n−→\nτ ′ (q, Q). From Lemma 1, u(aw)∗is a subset\nof PastA(p′). Thus, for every n, for every k in {0, . . . , n}, there exists a path:\n→i\nu(aw)k\n−→(p′, P)\na\n−→\nτ ′\nh\n(q, Q)\nw\n−→(p, P)\na\n−→\nτ\n(q, Q)\nin−k\nv\n−→t →.\nTherefore, there are at least n + 1 successful paths with label u(wa)nv in S,\nwhich is in contradiction with the finite ambiguity of S and A.\nIf there exists a path of S that contains two competing transitions τ and τ ′:\n(p, P)\na\n−→\nτ\n(q, Q)\nw\n−→(p′, P)\na\n−→\nτ ′ (q, Q),\nthen τ ′ belongs to a circuit, which is impossible.\n□\nAssume that A is finitely ambiguous. As a consequence of Proposition 4, for\nevery path in S (and thus for every path in A), one can compute an unambiguous\nautomaton U that contains this path. Consider the following algorithm.\nAs they do not belong to any circuit, competing sets of S are partially\nordered.\n– Compute C, the set of maximal competing sets of S (there is no path from\nany element of C to another competing set).\n– Let S1 and S2 be two copies of S. For every competing set X in C, let x\nbe an element of X;\n– if x is a transition, remove every transition of X\\{x} in S1 and remove x\nin S2;\n– if x is a final state, make every state of X\\{x} in S1 non-final and make\nx in S2 non-final.\n– Apply inductively this algorithm to S1 and S2.\n18\n\nThe result is a finite set of unambiguous automata. Each of them recognizes the\nlanguage of A and every path of S appears in at least one of these automata.\nNotice that the cardinality of this set may be larger than the degree of ambiguity\nof A. Denote by F the automaton obtained by taking the union of the automata\nin this set.\nAssume now that A is any automaton with multiplicities over an idempotent\nsemiring. Since there is a canonical mapping from the transitions (resp. initial\nstates, resp. final states) of the Sch ̈utzenberger covering S onto the transitions\n(resp. initial states, resp. final states) of A, one can decorate every transition\n(resp. initial state, resp. final state) of S with the corresponding multiplicity\nin A. This decoration can be carried out in the same way on the automaton F.\nObviously, since there is a one-to-one mapping between the successful paths\nof A and those of S, the series realized by S is equal to the one realized by A.\nFurthermore, as every path of S appears in F, the automaton F realizes the\nsame series as A. Notice that a path of S may appear several times in F, with\nno consequence since the semiring is idempotent.\nThe construction of F could be modified in order to get a one-to-one relation\nbetween paths of A and paths of F, but then the automata in the union would\nnot have the same support, which would be less convenient in the sequel.\nCorollary 1 A finitely ambiguous max-plus automaton can be effectively turned\ninto an equivalent finite union of unambiguous max-plus automata, all with the\nsame support.\n5\nThe Decidability Result\nIn this section, we show that a series, realized by a finite union of unambiguous\nautomata having the same support, is unambiguous if and only if a certain\nproperty denoted by (P) holds. Associated with Theorem 1 and Corollary 1,\nthis enables to prove Theorem 2, stated at the end of the paper.\nConsider a finite family of max-plus automata (Ai)i∈I with respective dimen-\nsions (Qi)i∈I. Set Ai = (α⌞i, μ⌞i, β⌞i). The corresponding product automaton P\nis an automaton with multiplicities in the product semiring\nRI\nmax, defined as\nfollows.\nSet Q =\nY\ni∈I\nQi and consider A, B ∈(RI\nmax)Q, M : Σ∗→(RI\nmax)Q×Q with\n∀p, q ∈Q, Ap = (α⌞i\npi)i∈I,\n∀a ∈Σ, M(a)p,q =\n(\n(μ⌞i(a)pi,qi)i∈I\nif ∀i, μ⌞i(a)pi,qi ̸=\n0\n(0, . . . ,\n0)\notherwise\nBp = (β⌞i\npi)i∈I .\nA state q ∈Q is initial if ∀i, (Aq)i ̸=\n0. A state q ∈Q is final if ∀i, (Bq)i ̸=\n0.\nThe trim part of (A, M, B) with respect to the above definition of initial and\nfinal states is the product automaton P.\n19\n\nClearly, if the automata (Ai)i∈I are unambiguous and all have the same\nsupport, then the product automaton P is also unambiguous and satisfies\n∀u ∈Σ∗, ∀i ∈I, ⟨S(P), u⟩i = α⌞iμ⌞i(u)β⌞i\n⇒\nM\ni∈I\n⟨S(P), u⟩i = ⟨\nM\ni∈I\nS(Ai), u⟩=\nM\ni∈I\nα⌞iμ⌞i(u)β⌞i.\nDefinition 2 Let θ be a simple circuit of P, whose weight is (x⌞i)i∈I. The set\nof victorious coordinates of θ, denoted by Vict (θ), is the set of coordinates on\nwhich the weight of θ is maximal, i.e. Vict (θ) =\n \ni ∈I | x⌞i = max\nj∈I {x⌞j}\n \n.\nThis definition is extended in a natural way to a strongly connected subgraph\nC of P: the set of victorious coordinates of C is the intersection of the sets of\nvictorious coordinates of the simple circuits of C. We also extend the definition\nto a path π of P: the set of victorious coordinates of π is the intersection of the\nsets of victorious coordinates of the strongly connected subgraphs of P crossed\nby π.\nLet us define the ‘dominance’ property (P):\nFor each successful path π of the product automaton P, the set of victorious\ncoordinates of π is not empty.\nObviously, the number of simple circuits is finite. Hence (P) is a decidable\nproperty.\nLet (Ai = (α⌞i ∈\nRQi\nmax, μ⌞i : Σ∗→\nRQi×Qi\nmax\n, β⌞i ∈\nRQi\nmax))i∈I be a finite family\nof unambiguous trim automata, all with the same support, and let P be the\nproduct automaton with set of states Q ⊆Πi∈IQi. We assume that P satisfies\nthe dominance property (P).\nLet N = |Q| and M = max( max\ni,a,p,q μ⌞i(a)p,q, max\ni,p β⌞i\np)−min( min\ni,a,p,q μ⌞i(a)p,q, min\ni,p β⌞i\np),\nwhere the minima are taken over non- 0 terms. In words, M is the difference\nbetween the largest and the smallest non-initial weights appearing in the au-\ntomata.\nWe use the following notations as shortcuts. For x = (x⌞i)i∈I ∈\nRI\nmax, set\nˇx = mini∈I{x⌞i | x⌞i ̸= −∞} and x = x −(ˇx, . . . , ˇx).\nSet I = {1, . . ., n}. We now define an automaton U that is shown to be\nunambiguous and to realize the series L\ni∈I S(Ai).\nThe states of U belong to\nRn\nmax × Q.\nInitial states.\nAll the initial states are defined as follows. If q = (q⌞1, . . . , q⌞n)\nis a tuple such that q⌞i is an initial state of Ai, and if we set α = (α⌞1\nq⌞1, . . . , α⌞n\nq⌞n),\nthen (α, q) is an initial state of U and the weight of the ingoing arc is ˇα.\nStates and transitions.\nIf (z, p) is a state of U, then for each transition in P\nof type: p\na|x\n−−→q such that x⌞i ̸= −∞for all i, there is a transition in U leaving\n20\n\np, labelled by the letter a, and that we now describe. Set t = z + x. Let V be\nthe set of victorious coordinates of the maximal strongly connected subgraph of\nq in P. Since P satisfies (P), the set V ∩{t⌞k ̸= −∞} is non-empty. Let j ∈V\nbe such that t⌞j = mink∈V {t⌞k | t⌞k ̸= −∞}, and let y ∈\nRn\nmax be defined by:\n∀i,\ny⌞i =\n(\n−∞\nif t⌞i < t⌞j −NM,\nt⌞i\notherwise.\nNow (y, q) is a state of U and we have the following transition:\nh\n(z, p)\na|ˇy\n−−→(y, q)\ni\nU .\nFinal states.\nAll the final states are defined as follows. If (z, q) is a state of\nU, and if q⌞i is a final state of Ai for all i, then (z, q) is a final state of U and\nthe weight of the outgoing arc is maxi∈I{z⌞i + β⌞i\nq⌞i}.\nLemma 2 The set of states of U is finite.\nproof. First, given a state (z1, q) of U, we show that there are finitely many\nstates of the form (z2, q) that can be reached from (z1, q).\nObserve that a path leading from (z1, q) to (z2, q) in U corresponds to a\ncircuit leading from q to q in P that can be fully decomposed into simple\ncircuits belonging to the strongly connected component of q. Let V be the set of\nvictorious coordinates of the strongly connected component of q. By definition\nof victorious coordinates, for all i ∈V the value of z⌞i\n2 −z⌞i\n1 is a constant, that\nwe denote by x, and for all i ̸∈V one has z⌞i\n2 ⩽z⌞i\n1 + x.\nLet C be the (finite) set of simple circuits of P. For a circuit θ ∈C, let the\nweight of the circuit in P be denoted by (weight (θ)⌞1 , . . . , weight (θ)⌞n). Set also\nweight (θ) = maxi⩽n weight (θ)⌞i. Now define\nδ = min\nθ∈C\nh\nweight (θ) −max\ni {weight (θ)⌞i | weight (θ)⌞i < weight (θ)}\ni\n.\nBy definition, we have δ > 0. By construction, for i ̸∈V , either z⌞i\n2 = z⌞i\n1 + x,\nor z⌞i\n2 ⩽z⌞i\n1 + x −δ. Furthermore, there is at least one index i and one index\nj such that z⌞i\n1 = 0 and z⌞j\n2 = 0. At last, for j ̸∈V , we have by construction\nz⌞j\n2 ⩾mini∈V z⌞i\n2 −NM, or z⌞j\n2 = −∞.\nAlltogether, it shows that there are\nfinitely many possible values for z2 = (z⌞1\n2 , . . . , z⌞n\n2 ).\nConsequently, any acyclic path in U is of finite length. Since the number of\ninitial states is finite, it follows easily from K ̈onig Lemma that the number of\nstates of U is finite.\n□\nLemma 3 The automaton U is unambiguous.\nproof. Define the surjective map\nΨ :\nU\n−→\nP\n(z, p)\n7−→\np .\n21\n\nBy construction of U, the following properties hold.\ni) The map Ψ restricted to the initial states of U defines a bijection between\nthe initial states of U and P.\nii) Consider\nh\np\na−→q\ni\nP. Then ∀(z, p) ∈Ψ−1(p), ∃!(z′, q) ∈Ψ−1(q) such\nthat\nh\n(z, p) a−→(z′, q)\ni\nU.\niii) A state (z, q) is a final state of U if and only if q is a final state of P.\np\nq\na\na\na\na\nΨ−1(p)\nΨ−1(q)\nU\nP\nFigure 12: The properties of the map Ψ.\nThese three properties together imply that there is a bijection between suc-\ncessful paths in P and successful paths in U. As P is unambiguous, so is U.\n□\nLemma 4 The automaton U recognizes the series L\ni∈I S(Ai).\nproof. Let lbe an integer and u = a0a1 · · · al−1 be a word in the common\nsupport of the series S(Ai).\nBy Lemma 3, there exists exactly one successful path labelled by u in the\nautomaton U:\nπ =\nh\n→(z0, q0)\na0\n−→(z1, q1)\na1\n−→· · ·\nal−2\n−→(zl−1, ql−1)\nal−1\n−→(zl, ql) →\ni\nU\n• Fix i ∈{1, . . . , n}. Assume that z⌞i\nl= −∞.\nThen i is not a victorious coordinate of π. Let j be a victorious coordinate, we\nshow that ⟨S(Ai), u⟩< ⟨S(Aj), u⟩. Hence the coefficient of u in L\ni∈I S(Ai) is\nnot realized by the coordinate i, which means that there is no damage in having\nz⌞i\nl= −∞.\nIn the path π, there exists a minimal state qh such that the coordinate z⌞i\nh\nis equal to −∞. That means that the difference between z⌞i\nh and z⌞j\nh would have\nbeen larger than NM. Let π′ in P be the path that corresponds to π (by the\nproof of Lemma 3, there is a canonical bijection between successful paths of U\nand P) and let q′\nh be the state of π′ that corresponds to qh. Let π′\nh be the end of\n22\n\nπ′ from q′\nh onwards (including the final arrow). Let us prove that the difference\nof weights on π′\nh between the coordinates i and j is smaller than NM, that is:\nweight (π′\nh)⌞i −weight (π′\nh)⌞j ⩽NM.\n(7)\nActually, on every circuit, the weight with respect to i is smaller than or equal\nto the weight with respect to j (which is victorious), and, if we delete all the\ncircuits in π′\nh, we obtain an acyclic path that is necessarily shorter than N −1.\nOn every transition, the difference between the weights of the coordinates i and\nj is at most M. Likewise, the difference between terminal functions is smaller\nthan M. Hence we proved (7). It means that the weight of coordinate i cannot\ncatch up with the one of coordinate j. In particular, we have: ⟨S(Ai), u⟩<\n⟨S(Aj), u⟩⩽⟨L\ni∈I S(Ai), u⟩.\n• Assume that z⌞i\nl̸= −∞. Set α = (α⌞i\nq⌞i\n0 )i∈I and β = (β⌞i\nq⌞i\nl)i∈I. Let π′ be the\npath in P that corresponds to π:\nπ′ =\nh α→q0\na0|x0\n−−−→q1\na1|x1\n−−−→· · ·\nal−1|xl−1\n−−−−−−→ql\nβ→\ni\nP.\nWe have, by construction of the automaton U:\n⟨S(Ai), u⟩=α⌞i\nq⌞i\n0 +\nl−1\nX\nk=0\nx⌞i\nk + β⌞i\nq⌞i\nl\n=ˇα + z⌞i\n0 +\nl−1\nX\nk=0\n(yk + z⌞i\nk+1 −z⌞i\nk) + β⌞i\nq⌞i\nl\n=ˇα +\nl−1\nX\nk=0\nyk + z⌞i\nl + β⌞i\nq⌞i\nl\nTherefore, ⟨S(Ai), u⟩= ⟨L\nj∈I S(Aj), u⟩if and only if z⌞i\nl+β⌞i\nq⌞i\nl= maxj[z⌞j\nl+β⌞j\nq⌞j\nl].\nNow observe that by construction,\n⟨U, u⟩= ˇα +\nl−1\nX\nk=0\nyk + max\ni [z⌞i\nl+ β⌞i\nq⌞i\nl].\nThe equality ⟨L\ni∈I S(Ai), u⟩= ⟨U, u⟩follows easily.\n□\nWe now have all the ingredients to prove the proposition below.\nProposition 5 Consider a finite family (Ai)i∈I of trim and unambiguous max-\nplus automata having the same support. Let P be the corresponding product\nautomaton. The series L\ni∈I S(Ai) is unambiguous if and only if P satisfies the\nproperty (P). In this case, the automaton U defined above is finite, unambiguous,\nand realizes the series L\ni∈I S(Ai).\nproof.\nLemmas 2, 3 and 4 show that (P) is a sufficient condition for\nL\ni∈I S(Ai) to be unambiguous.\nLet us prove that (P) is also a necessary\ncondition.\n23\n\nBy way of contradiction, assume that S = L\ni∈I S(Ai) is recognized by an\nunambiguous automaton U and that (P) does not hold. There exists a path\nπ of P that can be decomposed into π0, θ1, π1, θ2, . . . , πr, where every θi is a\ncircuit and T\ni Vict (θi) = ∅.\nLet ui be the label of πi and vi the label of\nθi.\nLet s be the maximal integer such that V = T\ni⩽s Vict (θi) ̸= ∅.\nLet\nwk,l = u0vk\n1u1 · · · vk\ns usvl\ns+1us+1us+2 · · · ur. For every k, l, wk,l is accepted by P\nand thus by U (with an unique successful path). Let k0, l0 be greater than the\nnumber of states d of U. By the pigeon-hole principle, every path in U labelled\nby vk0\ni\n(for i ∈{1, . . . , s}) has a sub-circuit labelled by vki\ni\n(with ki < d).\nLikewise, the path labelled by vl0\ns+1 has a sub-circuit labelled by vl1\ns+1. It means\nthat there exist (gi, ki, di)i∈[1,s] and (gs+1, l1, ds+1) such that the successful path\nlabelled by wk0,l0 in U has the following shape:\nu0vg1\n1\nvd1\n1 u1vg2\n2\nvds+1\ns+1 us+1us+2 · · · ur\nvk1\n1\nvk2\n2\nvl1\ns+1\nLet K = Q\ni⩽s ki. Since U is unambiguous, for every pair of integers (α, β),\nthe word wk0+αK,l0+βl1 is accepted by a path that has the same shape; hence,\nthere exist x = ⟨S, wk0,l0⟩, ρ and λ such that, for every (α, β) ∈\nN ×\nN,\n⟨S, wk0+αK,l0+βl1⟩= x + αρ + βλ.\nThe word wk0+αK,l0+βl1 labels in P a successful path that is the concatena-\ntion of π0, (k0+αK) times θ0, π1,. . . ,πs, (l0+βl1) times θs+1,. . . . Therefore, for\nevery β, there exists Nβ such that, for every α > Nβ, the successful coordinates\nof the path labelled by wk0+αK,l0+βl1 belong to V and the weight is equal to\ny + αρ1 + βλ1, where y is a constant, ρ1 is the sum of the maximal weights of\nthe circuits θ1 to θs, and λ1 = maxi∈V weight (θs+i)⌞i.\nLikewise, for every α, there exists Mα such that, for every β > Mα, the\nsuccessful coordinate of the path labelled by wk0+αK,l0+βl1 is a victorious coor-\ndinate of θs+1 and the weight of this path is equal to z + αρ2 + βλ2, where z\nis a constant, ρ2 is the maximum over the victorious coordinates of θs+1 of the\nsums of the weights of the circuits θ1 to θs, and λ2 is the maximal weight of\nθs+1.\nTo summerize, the following equalities hold:\n∀α, β, ⟨S, wk0+αK,l0+βl1⟩= x + αρ + βλ\n∀β, ∀α > Nβ, ⟨S, wk0+αK,l0+βl1⟩= y + αρ1 + βλ1\n∀α, ∀β > Mα, ⟨S, wk0+αK,l0+βl1⟩= z + αρ2 + βλ2\nTherefore, ρ1 = ρ = ρ2 and λ1 = λ = λ2. Thus, there exists a coordinate that\nbelongs to V and that is victorious on θs+1; this contradicts the maximality of\ns.\n24\n\nIt would be possible to use an argument similar to the one in §3.6, to prove\nthe above.\n□\nThe main result is now a corollary of Proposition 5:\nTheorem 2 One can decide in an effective way, whether the series recognized\nby a finitely ambiguous max-plus automaton is unambiguous, and whether it is\nsequential.\nMore precisely, turn first the finitely ambiguous automaton into an equiva-\nlent finite union of unambiguous automata, all having the same support (Corol-\nlary 1). Then check the property (P) on the new family of automata. If (P)\nis satisfied the series is unambiguous; build the unambiguous automaton U\n(Proposition 5), then decide the sequentiality of U (Theorem 1).\nReferences\n[1] C. Allauzen and M. Mohri. Efficient algorithms for testing the twins prop-\nerty. Journal of Automata, Languages and Combinatorics, 8(2):117–144,\n2003.\n[2] M.-P. 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Automata, languages and machines, vol. A. Academic Press,\n1974.\n[10] S. Gaubert. Performance evaluation of (max,+) automata. IEEE Trans.\nAut. Cont., 40(12):2014–2025, 1995.\n25\n\n[11] S. Gaubert and J. Mairesse. Task resource models and (max,+) automata.\nIn J. Gunawardena, editor, Idempotency, volume 11, pages 133–144. Cam-\nbridge University Press, 1998.\n[12] S. Gaubert and J. Mairesse. Modeling and analysis of timed Petri nets\nusing heaps of pieces. IEEE Trans. Aut. Cont., 44(4):683–698, 1999.\n[13] K. Hashigushi. Algorithms for determining relative star height and star\nheight. Inf. Comput., 78(2):124–169, 1988.\n[14] K. Hashigushi, K. Ishiguro, and S. Jimbo. Decidability of the equivalence\nproblem for finitely ambiguous finance automata. Int. J. Algebra Comput.,\n12(3):445–461, 2002.\n[15] D. Krob. The equality problem for rational series with multiplicities in the\ntropical semiring is undecidable. Int. J. Algebra Comput., 4(3):405–425,\n1994.\n[16] D. Krob and A. Bonnier-Rigny.\nA complete system of identities for\none-letter rational expressions with multiplicities in the tropical semiring.\nTheor. Comput. Sci., 134:27–50, 1994.\n[17] W. Kuich and A. Salomaa. Semirings, Automata, Languages, volume 5 of\nEATCS. Springer-Verlag, 1986.\n[18] M. Mohri.\nFinite-state transducers in language and speech processing.\nComput. Linguist., 23(2):269–311, 1997.\n[19] P. Moller. Th ́eorie alg ́ebrique des syst`emes `a ́ev ́enements discrets. PhD\nthesis, ́Ecole des Mines de Paris, 1988.\n[20] J. Sakarovitch. A construction in finite automata that has remained hidden.\nTheor. Comput. Sci., 204:205–231, 1998.\n[21] M.-P. Sch ̈utzenberger. On the definition of a family of automata. Informa-\ntion and Control, 4(2–3):245–270, 1961.\n[22] M.-P. Sch ̈utzenberger. Sur les relations rationnelles entre mono ̈ıdes libres.\nTheor. Comput. Sci., 3:243–259, 1976.\n[23] I. Simon. Recognizable sets with multiplicities in the tropical semiring. In\nMathematical Foundations of Computer Science, Proc. 13th Symp., number\n324 in LNCS, pages 107–120, 1988.\n[24] G.X. Viennot. Heaps of pieces, I: Basic definitions and combinatorial lem-\nmas. In Labelle and Leroux, editors, Combinatoire ́Enum ́erative, number\n1234 in Lect. Notes in Math., pages 321–350. Springer, 1986.\n[25] A. Weber. Finite-valued distance automata. Theor. Comput. Sci., 134:225–\n251, 1994.\n26\n\n[26] A. Weber and R. Klemm. Economy of description for single-valued trans-\nducers. Information and Computation, 118(2):327–340, 1995.\n[27] A. Weber and H. Seidl. On the degree of ambiguity of finite automata.\nTheor. Comput. Sci., 88(2):325–349, 1991.\n27","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0709.4117v1 [cs.CC] 26 Sep 2007\nDeciding Unambiguity and Sequentiality starting\nfrom a Finitely Ambiguous Max-Plus Automaton\nInes Klimann, Sylvain Lombardy, Jean Mairesse,\nand Christophe Prieur∗\nJuly 4, 2004\nAbstract\nFinite automata with weights in the max-plus semiring are considered.\nThe main result is: it is decidable in an effective way whether a series that\nis recognized by a finitely ambiguous max-plus automaton is unambiguous,\nor is sequential. A collection of examples is given to illustrate the hierarchy\nof max-plus series with respect to ambiguity.\n1\nIntroduction\nA max-plus automaton is a finite automaton with multiplicities in the max-plus\nsemiring\nRmax = (R ∪{−∞}, max, +). Roughly speaking, it is an automaton\nwith two tapes: an input tape labelled by a finite alphabet Σ, and an output\ntape weighted in\nRmax. The weight of a word in Σ∗is the maximum over all\nsuccessful paths of the sum of the weights along the path.\nMax-plus automata, and their min-plus counterparts, are studied under var-\nious names in the literature: distance automata, finance automata, cost au-\ntomata. They have also appeared in various contexts: to study logical problems\nin formal language theory (star height, finite power property) [13, 23], to model\nthe dynamic of some Discrete Event Systems (DES) [10, 12], or in the context\nof automatic speech recognition [18].\nTwo automata are equivalent if they recognize the same series, i.e. if they\nhave the same input/output behavior. The problem of equivalence of two max-\nplus automata is undecidable [15]. The same problem for finitely ambiguous\nmax-plus automata is decidable [14, 25].\nThe sequentiality problem is defined as follows: given a max-plus automaton,\nis there an equivalent max-plus automaton which is sequential (i.e. determinis-\ntic in input). Let us give some motivations on why the sequentiality problem\nis important. In the case of a sequential automaton, the time complexity of\n∗LIAFA, CNRS (umr 7089) - Universit ́e Paris 7, 2, place Jussieu - 75251 Paris Cedex 5 -\nFrance. email: {klimann,lombardy,mairesse,prieur}@liafa.jussieu.fr\n1"},{"paragraph_id":"p2","order":2,"text":"computing the output is roughly linear in the length of the input. This time\nefficiency is central in speech processing, see [18]. Consider now a DES mod-\nelled by a max-plus automaton. If the automaton is unambiguous, or a fortiori\nsequential, then one can compute the optimal, as well as the average behavior,\nof the DES, see [10, 11].\nSequentiality is decidable for unambiguous max-plus automata [18].\nIn the\npresent paper, we prove that sequentiality is decidable for finitely ambiguous\nmax-plus automata. To the best of our knowledge, it is not known if the finite\nambiguity of a max-plus series (defined via an infinitely ambiguous automaton)\nis a decidable problem. In particular, the status of the sequentiality problem\nis still open for a general max-plus automaton (even if the multiplicities are\nrestricted to be in\nZmax,\nNmax or\nZ−\nmax). To be complete, it is necessary to\nmention that in [18, §3.5], it is claimed that any max-plus automaton admits\nan effectively computable equivalent unambiguous one.\nIf that was true, it\nwould imply the decidability of the sequentiality for general max-plus automata.\nHowever, the statement is erroneous and counter-examples are provided in §3\nof the present paper1.\nThe sequentiality problem can be asked for automata over any semiring\nK.\nFor transducers, i.e. when\nK is the set of rational subsets of a free monoid\n(with union and concatenation as the two laws), the problem is completely\nsolved in the functional case (when, for every input, the output is a language of\ncardinality at most one) [3, 7, 8]. For a general transducer, the problem is wide\nopen. Observe that the semiring {a⩾n, n ∈\nN} = {ana∗, n ∈\nN} is isomorphic\nto\nNmin: ana∗+ ama∗= amin(n,m)a∗and ana∗· ama∗= an+ma∗. Similarly,\nthe semiring {a⩽n, n ∈\nN} is isomorphic to\nNmax (where a⩽n = {ε, a, . . ., an}).\nHence automata over\nNmax or\nNmin translate into transducers, but not functional\nones. Also the translation does not work for automata over\nRmax. Hence, the\nvast literature on transducers is of limited use in our context.\nIn the present paper, we work with\nRmax. Decidability and complexity should\nbe interpreted under the assumption that two real numbers can be added or\ncompared in constant time.\n2\nPreliminaries\n2.1\nMax-plus semiring and series\nThe free monoid over a finite set (alphabet) Σ is denoted by Σ∗and the empty\nword is denoted by ε. The structure\nRmax = (R ∪{−∞}, max, +) is a semiring,\nwhich is called the max-plus semiring.\nIt is convenient to use the notations\n⊕= max and ⊗= +. The neutral elements of ⊕and ⊗are denoted respectively\nby\n0 = −∞and\n1 = 0.\nThe subsemirings\nNmax,\nZmax, . . . , are defined in\nthe natural way.\nThe min-plus semiring\nRmin is obtained by replacing max\nby min and −∞by +∞in the definition of\nRmax. The results of this paper\ncan be easily adapted to the min-plus setting. Observe that the subsemiring\n1The version of [18] available on the author’s website has been correctly modified.\n2"},{"paragraph_id":"p3","order":3,"text":"B = ({ 0,\n1}, ⊕, ⊗) is isomorphic to the Boolean semiring. For matrices A, B of\nappropriate sizes with entries in\nRmax, we set (A⊕B)ij = Aij ⊕Bij, (A⊗B)ij =\nL\nk Aik ⊗Bkj, and for a ∈\nRmax, (a ⊗A)ij = a ⊗Aij. We usually omit the ⊗\nsign, writing for instance AB instead of A ⊗B.\nConsider the set\nRmax⟨⟨Σ∗⟩⟩of (formal power) series (over Σ∗with coeffi-\ncients in\nRmax), that is the set of maps from Σ∗to\nRmax. We denote by ⟨S, u⟩\nthe coefficient of the word u in the series S. The support of a series S is the\nset Supp S = {u ∈Σ∗|\n⟨S, u⟩̸=\n0}. It is convenient to use the notation\nS = L\nu∈Σ∗⟨S, u⟩u = L\nu∈Supp (S)⟨S, u⟩u. Equipped with the addition (⊕) and\nthe Cauchy product (⊗), the set\nRmax⟨⟨Σ∗⟩⟩forms a semiring. The image of\nλ ∈\nRmax by the canonical injection into\nRmax⟨⟨Σ∗⟩⟩is still denoted by λ. In\nparticular, the neutral elements of\nRmax⟨⟨Σ∗⟩⟩are\n0 and\n1. The characteristic\nseries of a language L is the series\n1L such that ⟨1L, w⟩=\n1 if w ∈L, and\n⟨1L, w⟩=\n0 otherwise.\n2.2\nMax-plus automaton\nLet Q and Σ be two finite sets. A max-plus automaton of set of states (di-\nmension) Q over the alphabet Σ, is a triple A = (α, μ, β), where α ∈\nR1×Q\nmax ,\nβ ∈\nRQ×1\nmax , and where μ : Σ∗→\nRQ×Q\nmax\nis a morphism of monoids. The mor-\nphism μ is uniquely determined by the family of matrices {μ(a), a ∈Σ}, and\nfor w = a1 · · · an, we have μ(w) = μ(a1) ⊗· · · ⊗μ(an). The series recognized\n(or realized) by A is by definition S(A) = L\nu∈Σ∗(αμ(u)β)u.\nThis is just a\nspecialization to the max-plus semiring of the classical notion of an automaton\nwith multiplicities over a semiring [4, 9, 17].\nBy the Kleene-Sch ̈utzenberger\nTheorem [21], the set of series recognized by a max-plus automaton is equal to\nthe set of rational series over\nRmax. We denote it by Rat.\nA state i ∈Q is initial, resp. final, if αi ̸=\n0, resp. βi ̸=\n0. As usual a\nmax-plus automaton is represented graphically by a labelled weighted digraph\nwith ingoing and outgoing arcs for initial and final states, see e.g. Figure 6\n(the input or output weights equal to\n1 are omitted). The terminology of graph\ntheory is used accordingly (e.g. (simple) path or circuit of an automaton, union\nof automata, . . . ).\nA path which is both starting with an ingoing arc and\nending with an outgoing arc is called a successful path. The label of a path is\nthe concatenation of the labels of the successive arcs (so called transitions), the\nweight of a path is the product (⊗) of the weights of the successive arcs (including\nthe ingoing and the outgoing arc, need it be). We denote by weight (π) the\nweight of the path π. We use the following notations for paths in an automaton\nA = (α, μ, β):\np →q,\n→p →q,\np →q →,\np\nu|x\n−−→q,\nh\np\nu|x\n−−→q\ni\nA , if μ(u)pq = x in A .\nThe first example is a path (of any length) from p to q, the second also\nincludes an ingoing arc, the third an outgoing arc, in the fourth the weight and\nthe label are added and in the fifth the underlying automaton is recalled.\nAn automaton is trim if any state belongs to at least one successful path.\n3"},{"paragraph_id":"p4","order":4,"text":"Let I be a finite set. The tensor product automaton of (Ai = (α⌞i, μ⌞i, β⌞i))i∈I,\ndenoted by ⊙i∈IAi, is defined as follows. It is the max-plus automaton (A, M, B)\nof dimension Q = Q\ni Qi, where Qi is the dimension of Ai, and such that\n∀p, q ∈Q,\nAp =\nO\ni∈I\nα⌞i\npi,\n∀a ∈Σ, M(a)p,q =\nO\ni∈I\nμ⌞i(a)pi,qi,\nBp =\nO\ni∈I\nβ⌞i\npi .\n2.3\nHeap model\nA heap or Tetris model [24], consists of a finite set of slots R, and a finite\nset of rectangular pieces Σ. Each piece a ∈Σ is of height 1 and occupies a\ndetermined subset R(a) of the slots. To a word u = u1 · · · uk ∈Σ∗is associated\nthe heap obtained by piling up in order the pieces u1, . . . , uk, starting with\na horizontal ground and according to the Tetris game mechanism (pieces are\nsubject to gravity and fall down vertically until they meet either a previously\npiled up piece or the ground). Consider the morphism generated by the matrices\nM(a) ∈\nRR×R\nmax , a ∈Σ, defined by\nM(a)ij ="},{"paragraph_id":"p5","order":5,"text":"1\nif i, j ∈R(a),\n0\nif i = j ̸∈R(a),\n−∞\notherwise .\nLet x(u)i be the height of the heap u on slot i ∈R. We have ([5, 11, 12]):\nx(u)i =\n1M(u)δi, where\n1 = (1, . . . ,\n1) ∈\nR1×R\nmax and δi ∈\nRR×1\nmax is defined by\n(δi)j =\n1 if j = i and\n0 otherwise. In other words, the application x(·)i : Σ∗→\nRmax is recognized by the max-plus automaton (1, M, δi). We call (1, M, δ), δ =\nL\ni∈I δi, I ⊆R, a heap automaton (associated with the heap model). Among\nmax-plus automata, heap automata are particularly convenient and playful, due\nto the underlying geometric interpretation. Here, they are used as a source of\nexamples and counter-examples, e.g. Figures 3, 4 and 7.\nWe represent a heap automaton graphically as in Figure 1.\nb\na\nR(a) = {1, 2}\nR(b) = {2, 3}\nR = {1, 2, 3}\n(1, M, δ2)\nFigure 1: A heap automaton\n2.4\nAmbiguity and Sequentiality\nConsider a max-plus automaton A = (α, μ, β) of dimension Q over Σ.\nThe\nautomaton is sequential if there is a unique initial state and if for all i ∈Q, and\n4"},{"paragraph_id":"p6","order":6,"text":"for all a ∈Σ, there is at most one j ∈Q such that μ(a)ij ̸=\n0. In the case of\na Boolean automaton, we also say deterministic for sequential. The automaton\nA is unambiguous if for any word u ∈Σ∗, there is at most one successful path\nof label u.\nThe automaton is finitely ambiguous if there exists some k ∈\nN\nsuch that for any word u ∈Σ∗, there are at most k successful paths of label\nu.\nThe minimal such k is called the degree of ambiguity of the automaton.\nClearly, ‘sequential’ implies ‘unambiguous’ which implies ‘finitely ambiguous’.\nThe automaton is infinitely ambiguous if it is not finitely ambiguous.\nConsider a series S ∈Rat. The series is sequential (resp. unambiguous,\nfinitely ambiguous) if there exists a sequential (resp. unambiguous, finitely am-\nbiguous) max-plus automaton recognizing it. The series is infinitely ambiguous\nif there exists no finitely ambiguous max-plus automaton recognizing it. The\ndegree of ambiguity of a finitely ambiguous series is the minimal degree of ambi-\nguity of an automaton recognizing it. The sets of sequential, unambiguous, and\nfinitely ambiguous series are denoted respectively by Seq, NAmb, and FAmb.\nDefine FSeq = {S | ∃k, ∃S1, . . . , Sk ∈Seq, S = S1 ⊕· · · ⊕Sk}.\nConsider a total order on Σ∗. Given a series S ̸=\n0, define the normalized\nseries φ(S) by φ(S) = L\nu∈Σ∗(⟨S, u⟩−⟨S, u0⟩)u, where u0 is the smallest word\nof Supp S. The (left) quotient of a series S by a word w is the series w−1S\ndefined by w−1S = L\nu∈Σ∗⟨S, wu⟩u.\nA series S is rational if and only if the semi-module of series ⟨w−1S, w ∈Σ∗⟩\nis finitely generated, i.e. if there exists S1, . . . , Sk, such that:\n∀w ∈Σ∗, ∃λ1, . . . , λk ∈\nRmax, w−1S =\nM\ni\nλiSi.\nA series S is sequential if and only if the set of series {φ(w−1S), w ∈Σ∗} is\nfinite.\nProposition 1 A trim automaton A of dimension Q is infinitely ambiguous if\nand only if there exist p, q ∈Q, p ̸= q, and v ∈Σ∗, such that p\nv\n−→p, p\nv\n−→q,\nq\nv\n−→q. This can be checked in polynomial time.\nFor a proof, see [27] and the references therein. Observe that the (in)finite\nambiguity is independent of the underlying semiring.\nNext result is due to\nMohri [18] and is an adaptation of a classical result of Choffrut on functional\ntransducers, see [3, 7, 8] (for the decidability) and [2, 26] (for the polynomial\ncomplexity).\nTheorem 1 Let A be an unambiguous max-plus automaton.\nThere exists a\npolynomial time algorithm to decide whether S(A) is a sequential series.\nIf A is unambiguous and S(A) is sequential, a sequential automaton recog-\nnizing the series can be effectively constructed from A using an adaptation of\nthe subset construction of Boolean automata [1, 6, 18].\nIt is useful to detail Theorem 1. We need to introduce several definitions.\nGiven two words u, v ∈Σ∗, let u ∧v be the longest common prefix of u and\n5"},{"paragraph_id":"p7","order":7,"text":"v, and define d(u, v) = |u| + |v| −2|u ∧v|. It is easy to check that d(., .) is a\ndistance on Σ∗. A series S is M-Lipschitz (M ∈\nR+) if:\n∀u, v ∈Supp S, |⟨S, u⟩−⟨S, v⟩| ⩽Md(u, v) ;\nand S is Lipschitz if it is M-Lipschitz for some M. The set of Lipschitz series is\ndenoted by Lip. Consider a trim max-plus automaton A of dimension Q. Two\nstates p, q ∈Q are twins if:\nh x0\n−→i\nu1|x1\n−−−→p\nu2|x2\n−−−→p,\ny0\n−→j\nu1|y1\n−−−→q\nu2|y2\n−−−→q\ni\n=⇒[x2 = y2] .\nIf all the states are twins, the automaton A is said to satisfy the twin property.\nWe denote the set of all such automata by Twin. The following implications\nhold:\nh\nA ∈Twin\ni\n=⇒\nh\nS(A) ∈Seq\ni\n=⇒\nh\nS(A) ∈Lip\ni\n.\n(1)\nFurthermore,\nh\nA ∈NAmb,\nS(A) ∈Lip\ni\n=⇒\nh\nA ∈Twin\ni\n.\n(2)\nThe twin property can be checked in polynomial time, hence Theorem 1 follows\nfrom the above implications.\n3\nHierarchy of Series\nThe examples in this section illustrate the classes of series on which we work.\nSeq ⊊(NAmb ∩FSeq)\n(§3.1)\n⊊\n⊊\nFSeq\n(§3.2)\nNAmb\n(§3.3)\n⊊\n⊊\nFAmb\n(§3.4)\n⊊\nRat\n(§3.5) ⊊Series\n(§3.6)\n3.1\nA Series in Seq ∩NAmb ∩FSeq\nAn example over a one-letter alphabet is provided in Figure 2. The recognized\nseries is\n⟨S, an⟩=\n(\n0\nif n is odd,\nn\nif n is even.\na|0\na|0\na|1\na|1\nFigure 2: Seq ∩NAmb ∩FSeq\nThe series is not Lipschitz, since |⟨S, an+1⟩−⟨S, an⟩| ⩾n, and consequently\nthe series cannot be sequential (see (1)). It is clear that it is an unambiguous\n6"},{"paragraph_id":"p8","order":8,"text":"series (the only successful path of label an is the right or left one depending\non the parity of n) and a sum of sequential series. In fact, any max-plus ra-\ntional series over a one-letter alphabet is unambiguous and a sum of sequential\nseries [16, 19].\na|1, b|0\na|0, b|1\na\nb\nFigure 3: FSeq ∩NAmb\n3.2\nA Series in FSeq ∩NAmb\nThe series ⟨S, u⟩= |u|a ⊕|u|b over the alphabet {a, b} is a sum of two sequential\nseries: the heap automaton of Figure 3 recognizes this series.\nAssume that S is unambiguous. The series S is 1-Lipschitz. So it has to be\nsequential, see (2) and (1). Consequently, there exist series S1,. . . Sk such that:\n∀u ∈Σ∗, ∃i, ∃λu ∈\nRmax\nu−1S = λu ⊗Si.\nBy the pigeon-hole principle, there must exist i ∈{1, . . . k} and two integers\nm < n such that\n∃λn, λm\n(an)−1S = λn ⊗Si,\n(am)−1S = λm ⊗Si.\nConsequently, we have\n⟨(an)−1S, bm+1⟩−⟨(an)−1S, ε⟩= ⟨(am)−1S, bm+1⟩−⟨(am)−1S, ε⟩.\nHowever\n⟨(an)−1S, bm+1⟩−⟨(an)−1S, ε⟩= ⟨S, anbm+1⟩−⟨S, an⟩= n −n = 0\n⟨(am)−1S, bm+1⟩−⟨(am)−1S, ε⟩= ⟨S, ambm+1⟩−⟨S, am⟩= m + 1 −m = 1.\nThis is a contradiction, consequently S is not sequential and thus cannot be\nan unambiguous series.\n3.3\nSeries in NAmb ∩FSeq\na) The first example is the series S given by the heap automaton of Figure 4 (a),\nor equivalently by the automaton of Figure 4 (b).\nConsider the series ̃S defined by ⟨ ̃S, w⟩= ⟨S, w⟩−|w|.\nAn automaton\nrecognizing ̃S can clearly be obtained from an automaton recognizing S by\nremoving 1 from each output weight. Hence S and ̃S are both sum of sequential\nseries or none of them is.\n7"},{"paragraph_id":"p9","order":9,"text":"a|1, b|0\na|1, b|1\na|1\na|1\nb\na\n(a)\nb|0\na|1, b|1\na|1\n(b)\nFigure 4: NAmb ∩FSeq\na|0\na|0, b|0\nb| −1\nFigure 5: NAmb ∩FSeq\nThe series ̃S is recognized by the automaton of Figure 5.\nSuppose that\n ̃S = S1 ⊕S2 ⊕· · · ⊕Sk, where k ∈\nN and the Si are sequential series.\nSince the Si are sequential series, they are Lipschitz. Let N be the maximal\nLipschitz coefficient of the Si. Let (Ni)i⩾0 be a sequence of integers such that\nN0 > N,\nN(Nk−1 + 1) < Nk −Nk−1 for all k ⩾1.\nThe coefficient of abNk in ̃S is −Nk, and it comes, for instance, from S1.\nThe coefficient of abNkabNk−1 is −Nk−1. We have:\nd(abNk, abNkabNk−1) = Nk−1+1 and |⟨ ̃S, abNk⟩−⟨ ̃S, abNkabNk−1⟩| = Nk−Nk−1.\nThe coefficient of abNkabNk−1 in ̃S does not come from S1, since\n|⟨S1, abNk⟩−⟨S1, abNkabNk−1⟩| ⩽N(Nk−1 + 1)\n< Nk −Nk−1 = |⟨S1, abNk⟩−⟨ ̃S, abNkabNk−1⟩|.\nIn the same way, we prove that any two words of the set\n{abNk, abNkabNk−1, . . . , abNkabNk−1 · · · abN0}\ncannot be recognized by the same Si. But this set has cardinality k + 1 and\nthus there is a contradiction.\nb) The second example is the series given by the automaton of Figure 6. The\nseries recognized by this automaton is:\n⟨S, am1bn1 · · · ampbnp⟩=\nX\nmieven\nmi,\n8"},{"paragraph_id":"p10","order":10,"text":"where m1 ∈\nN, mk+1 ∈\nN −{0}, nk ∈\nN −{0} for 1 ⩽k ⩽p −1, and np ∈\nN.\nThe automaton is clearly unambiguous. Furthermore, it is not a finite sum of\nsequential series. To simplify notations, let us prove that S is not the sum of\ntwo sequential series. Suppose that S = S1 ⊕S2, with S1, S2 ∈Seq.\nThe series Si, i ∈{1, 2}, are sequential, so they are Lipschitz by (1). Let\nN be such that Si, i ∈{1, 2}, are N-Lipschitz. Let us consider words of the\nform arbnas, with n > 0. We discuss on the parity of r and s. The coefficient\nof the word a2p+1bna2q+1 in S, which is equal to 0, comes from one of the Si.\nFor instance\n⟨S, a2p+1bna2q+1⟩= 0 = ⟨S1, a2p+1bna2q+1⟩.\n(3)\nSet q > N. Since S1 is N-Lipschitz and d(a2p+1bna2q+1, a2p+1bna2q) = 1,\nwe have\n⟨S, a2p+1bna2q⟩= 2q = ⟨S2, a2p+1bna2q⟩.\n(4)\nFix q and n. Since S1 and S2 are Lipschitz, there exists an integer M such\nthat:\n∀u, v ∈Supp Si, d(u, v) ⩽2n + 4q + 2 ⇒|⟨Si, u⟩−⟨Si, v⟩| ⩽M.\n(5)\nWe have:\nd(a2pbna2q, a2p+1bna2q+1) = 2n+4q+2\nand\nd(a2pbna2q, a2p+1bna2q) = 2n+4q+1.\nSo, by Equation (5), we know that:\n– If a2pbna2q ∈Supp S1, then\n2p + 2q = |⟨S1, a2pbna2q⟩−⟨S1, a2p+1bna2q+1⟩| ⩽M,\nwhich is wrong for p large enough.\n– If a2pbna2q ∈Supp S2, then\n2p = |⟨S2, a2pbna2q⟩−⟨S2, a2p+1bna2q⟩| ⩽M,\nwhich is also wrong for p large enough.\nConsequently, S is not the sum of two sequential series.\nTo extend the re-\nsult to the sum of m sequential series, one has to consider words of the form\nar1bn1ar2 · · · arm−1bnm−1arm.\n3.4\nSeries in NAmb ∩FSeq ∩FAmb\na) Consider the heap automaton given in Figure 7 (a).\nThe corresponding\nseries is at most two-ambiguous since it is also recognized by the two-ambiguous\nautomaton of Figure 7 (b).\nIt cannot be unambiguous: on {a, b}∗, since it\ncoincides with the series of Figure 3 which is in NAmb. It cannot be a finite\nsum of sequential series: on {b, c}∗, it coincides with the series of Figure 4 which\nis in FSeq.\n9"},{"paragraph_id":"p11","order":11,"text":"i\nj\na|0\na|0\na|1\na|1\nb|0\nb|0\nb|0\nb|0\nFigure 6: NAmb ∩FSeq\na|0, b|1, c|0\na|0, b|1, c|1\nb|1\nb|1\na|1, b|0, c|0\nb\na\nc\n(a)\na|0, c|0\na|0, b|1, c|1\nb|1\na|1, b|0, c|0\n(b)\nFigure 7: NAmb ∩FSeq ∩FAmb\nb) Another example is provided by the automaton A of Figure 8.\nDenote by S the series recognized by this automaton, by S1 the series recog-\nnized by the left part, say A1, of the automaton, and by S2 the series recognized\nby the right part, say A2.\nThe automaton A1 is the one introduced in Section 3.3 and the automaton\nA2 is the same one after permutation of the a’s and b’s in the labels. Recall\nthat A1 and A2 are unambiguous, so S is at most two-ambiguous.\nLet us prove that S is not a finite sum of sequential series. Denote by L the\nlanguage of words whose blocks of b’s have odd length. Let u be a word of L:\na|0\na|0\na|1\na|1\nb|0\nb|0\nb|0\nb|0\nb|0\nb|0\nb|1\nb|1\na|0\na|0\na|0\na|0\nFigure 8: NAmb ∩FSeq ∩FAmb\n10"},{"paragraph_id":"p12","order":12,"text":"c|1\nc|1\na|1, b|0, c|1\na|0, b|1, c|1\n(a)\na\nb\nc\n(b)\nFigure 9: FAmb ∩Rat\nin A2, the b-blocks of u are always read in the upper part of the automaton, so\n⟨S2, u⟩= 0. Since the coefficient of u in A1 is at least 0, we have S⊙\n1L = S1⊙\n1L.\nSuppose that S is a finite sum of sequential series. Then so is S⊙\n1L and S1⊙\n1L.\nAnd this is false since one can choose an odd n in the proof of Section 3.3 for\nthe automaton of Figure 6.\nLet us prove that S is not unambiguous. Let M be the rational language\nof words whose a-blocks and b-blocks have even lengths and let u be a word\nof M.\nIn A1, the a-blocks have to be read in the lower part of A1 and so\n⟨S1, u⟩= |u|a. In the same way: ⟨S2, u⟩= |u|b. So we have S ⊙\n1M = S′ ⊙\n1M,\nwhere S′ is the series recognized by the automaton of Figure 3. Consequently,\nif S is unambiguous, so is S′ ⊙\n1M. We now apply the arguments of Section 3.2\nto show that S′ ⊙\n1M is not unambiguous.\nc) Besides, Weber has given examples of series which are k-ambiguous and\nnot (k −1)-ambiguous [25, Theorem 4.2].\n3.5\nSeries in FAmb ∩Rat\nConsider the series S recognized by the automaton of Figure 9. Assume that S\nis finitely ambiguous. Using the result of Corollary 1 below, S is recognized by\na finite union of unambiguous automata with the same support, say A1, . . . , Ak.\nDenote by Si the series recognized by Ai, for 1 ⩽i ⩽k, and by n the\nmaximal dimension of an automaton Ai. Observe that Supp S = Σ∗. Since all\nthe Si have the same support, we have Supp Si = Σ∗.\nNow, consider the word w0 =\n anbnc\n k. For any i, there is a single successful\npath labelled by w0 in Ai. Note that a path of length n contains necessarily a\ncircuit.\nSo, each automaton Ai contains a path of the form:\nπi :\na · · · a\na · · · a\nb · · · b\nb · · · b\nc\na · · · a\na · · · a\na · · · a\nb · · · b\na · · · a\nFor every j ∈{1, . . ., k}, we choose in the subpath labelled by the j-th factor\nan (resp. bn) a circuit that is called the j-th a-loop (resp. the j-th b-loop).\n11"},{"paragraph_id":"p13","order":13,"text":"The coefficient of a word in S is less than or equal to its lentgh, it is thus\nthe same for its coefficients in the Si. Consequently, the mean weights of the\nloops of πi are less than or equal to 1. Denote by av (πi, a, j) the mean weight\nof the j-th a-loop in the path πi, and define av (πi, b, j) similarly.\nSet j ∈{1, . . ., k}. For λ ∈\nN −{0}, consider the word\nwλ = (anbnc) · · · (anbnc) (an+λn!bn+λn!c)\n|\n{z\n}\nj-th block\n(anbnc) · · · (anbnc)\nThis word can be read on each path πi by turning into the j-th a- and\nb-loops, whose lengths are less than or equal to n and so divide n!.\nLet i ∈{1, . . . , k} be such that ⟨S, w0⟩= ⟨Si, w0⟩.\nWe have ⟨S, wλ⟩−\n⟨S, w0⟩= λn!, and so ⟨Si, wλ⟩−⟨Si, w0⟩⩽λn!.\nBut ⟨Si, wλ⟩−⟨Si, w0⟩=\n(av (πi, a, j) + av (πi, b, j))λn!, consequently\nav (πi, a, j) + av (πi, b, j) ⩽1.\n(6)\nConsider any u in {01, 10}k. For all p ∈\nN −{0}, let us define the word\nvp(u) = (an+λ1n!bn+μ1n!c) · · · (an+λjn!bn+μjn!c) · · · (an+λkn!bn+μkn!c),\nwhere (λj, μj) = (p, 0) if (u2j−1, u2j) = (1, 0) (we say then that the dominant\nj-th loop is the j-th a-loop) and (λj, μj) = (0, p) otherwise (the dominant j-th\nloop is the j-th b-loop), for any j ∈{1, . . . , k}.\nBy the pigeon-hole principle, for some i, there are infinitely many words of\nthe form vp(u) such that ⟨S, vp(u)⟩= ⟨Si, vp(u)⟩= k + nk + kpn!. Such words\nare read on the path πi. The dominant j-th loop in πi has then necessarily\nmean weight 1, and by Equation (6), the non-dominant j-th loop in πi has\nmean weight less than or equal to 0.\nConsequently, we have built an injection from the language {01, 10}k into\nthe set of paths {πi}. But the language has cardinality 2k and the set of paths\nhas cardinality k. So we have a contradiction.\n3.6\nRational and Non-Rational Series (Rat)\nA max-plus series is non-rational as soon as its support is a non-rational lan-\nguage. Here, we present a less trivial example of non-rational max-plus series.\nIn this paragraph, it is necessary to distinguish between\nRmin and\nRmax:\nfor R = Rat or NAmb, we use the respective notations\nRminR,\nRmaxR.\nIf\nS ∈\nRmax⟨⟨Σ∗⟩⟩, we identify S with ̃S ∈\nRmin⟨⟨Σ∗⟩⟩such that ⟨ ̃S, w⟩= ⟨S, w⟩if\nw ∈Supp S and ⟨ ̃S, w⟩= +∞if ⟨S, w⟩= −∞.\nClearly, we have\nRmaxNAmb =\nRminNAmb = NAmb.\nOn the other hand, it is easy to find S ∈\nRminFSeq ∩\nRminNAmb such that\nS ̸∈\nRmaxRat.\n12"},{"paragraph_id":"p14","order":14,"text":"Consider for instance the series S = min(|w|a, |w|b) (recognized by the au-\ntomaton of Figure 3 seen as a min-plus automaton). Let us prove that S does\nnot belong to\nRmaxRat. If it does: let S1, . . . Sn be a minimal generating fam-\nily of ⟨u−1S, u ∈Σ∗⟩(see §2.4), we have: ∀u ∈Σ∗, ∃λ(u)\n1 , . . . λ(u)\nn , u−1S =\nL\ni λ(u)\ni\n⊗Si. The restrictions of the quotients of S to b∗are bounded, hence\nso are the restrictions of the Si. Let ki be such that: ⟨Si, bki⟩= maxk⟨Si, bk⟩.\nIt follows that for any word u: maxk⟨u−1S, bk⟩= maxki⟨u−1S, bki⟩. Consider\nk > maxi ki. Then arises a contradiction:\nmax\nl\n⟨(ak)−1S, bl⟩= k > max\nki ⟨(ak)−1S, bki⟩= max\ni\nki.\n3.7\nAmbiguity vs. sequentiality and Ambiguity vs. Lips-\nchitz\nHere are some examples of series that are in several classes described in Section 3:\nNAmb\nFAmb ∩NAmb\nFAmb\nSeq\na\nimpossible\nimpossible\nFSeq\n∩\nSeq\na|0\na|0\na|1\na|1\na\nb\nSec 3.2\nimpossible\nFSeq\nb\na\nSec 3.3\nb\na\nc\nSec 3.4\na\nb\nc\nSec 3.5\n13"},{"paragraph_id":"p15","order":15,"text":"NAmb\nFAmb ∩NAmb\nFAmb\nLip\na\na\nb\nSec 3.2\na\nb\nc\nSec 3.5\nLip\nb\na\nSec 3.3\nb\na\nc\nSec 3.4\na\nb\nc\n4\nFrom Finitely Ambiguous to Union of Unam-\nbiguous\nWeber [25] has proved that a finitely ambiguous\nNmax-automaton can be turned\ninto an union of unambiguous ones.\nWe present a completely different and\nsimpler proof that holds in any semiring, in particular\nRmax.\nIn this section, we work on the structure of the automata. So we consider\nsimply Boolean automata.\nBelow, given a set S, we identify the vectors of\nBS with the subsets of S,\ni.e. x ∈\nBS is identified with {i ∈S | xi =\n1}.\nLet A = (α, μ, β) be a trim automaton. The past of a state p is the set of\nwords that label a path from some initial state to p. The future of p is the set\nof words that label a path from p to some final state. We write:\nPastA(p) = {w ∈Σ∗| (αμ(w))p =\n1},\nFutA(p) = {w ∈Σ∗| (μ(w)β)p =\n1}.\nLet A = (α, μ : Σ∗→\nBQ×Q, β) be an automaton. Let us recall the usual\ndeterminization procedure of A via the subset construction. Let R be the least\nsubset of\nBQ inductively defined by:\nα ∈R,\nX ∈R ⇒∀a ∈Σ, Xμ(a) ∈R.\nLet D = D(A) = (J, ν : Σ∗→\nBR×R, U) be the determinized automaton\nof A defined by:\nJ = {α},\nU = {P ∈R | Pβ =\n1},\nν(a)P,P ′ =\n1 ⇐⇒P ′ = Pμ(a).\nLemma 1 i) Let A be an automaton and D its determinized automaton. Then\nfor each state P of D,\nPastD(P) ⊆\n\\\np∈P\nPastA(p),\nand\nFutD(P) =\n[\np∈P\nFutA(p).\n14"},{"paragraph_id":"p16","order":16,"text":"p\nq\nr\npq\nqr\npr\nr\nq\np\np, {p, q}\nq, {p, q}\nq, {q, r}\nr, {q, r}\np, {p, r}\nr, {p, r}\nr, {r}\nq, {q}\np, {p}\na b\na\nb\nb\na\nb\na\nb\nb\na\nb\na\nb\na\nb\na\nb\na\nb\nb\na\na\nb\nb\na\nb\nb\na\nb\nb\nFigure 10: A (left), the determinized automaton (top) and the Sch ̈utzenberger cov-\nering\nii) Let A and B be two automata and A ⊙B their tensor product (cf. §2.4),\nthen, for all state (p, q) of A ⊙B,\nPastA⊙B(p, q) = PastA(p) ∩PastB(q),\nFutA⊙B(p, q) = FutA(p) ∩FutB(q).\nThe constructions and results given in Propositions 2 and 3 are inspired by\nSch ̈utzenberger [22]. They have been explicitely stated by Sakarovitch in [20].\nLet A be an automaton and D its determinized automaton. The trim part\nof the product A ⊙D is called the Sch ̈utzenberger covering S of A.\nProposition 2\nLet A = (α, μ, β) be a trim automaton, D its determinized\nautomaton and S its Sch ̈utzenberger covering.\ni) The states of S are exactly the pairs (p, P), where P is a state of D and\np ∈P. We call the set {(p, P) | p ∈P} of states of S a column (in gray on\nFigure 10).\nii) The canonical surjection ψ from the transitions of S onto the transitions\nof A induces a one-to-one mapping between the successful paths of S and A.\niii) Let P be a state of D. Then, for every p in P,\nPastS(p, P) = PastD(P),\nFutS(p, P) = FutA(p).\nThus, all the states of a given column have the same past.\nproof. i) A state (p, P) of S is initial if and only if p is initial in A (i.e.\np ∈α) and P is initial in D (i.e. P = {α}). Now, let (p, P) be a state of S\nsuch that p ∈P and (q, Q) a successor of (p, P) by a. Then, there exist two\ntransitions:\nh\np\na\n−→q\ni\nA\nand\nh\nP\na\n−→Q\ni\nD.\nBy definition of D, q belongs thus to Q.\n15"},{"paragraph_id":"p17","order":17,"text":"Conversely, let P be a state of D and p an element of P. For every w in\nPastD(P), w belongs to PastA(p) (Lemma 1). Therefore there is a path in S\nfrom an initial state to (p, P).\nii) Let π be a successful path of A, with label w = w1w2 · · · wn. Let θ be\nthe (unique) successful path with label w in D:\nπ =\nh\n→p0\nw1\n−→p1\nw2\n−→. . .\nwn\n−→pn →\ni\nA,\nθ =\nh\n→P0\nw1\n−→P1\nw2\n−→. . .\nwn\n−→Pn →\ni\nD.\nThere is a path in S: π′ =→(p0, P0)\nw1\n−→(p1, P1)\nw2\n−→. . .\nwn\n−→(pn, Pn) →.\nThe function π 7→π′ is obviously one-to-one.\niii) By results from Lemma 1:\nPastS(p, P) = PastA(p) ∩PastD(P)\n∀p ∈P, PastD(P) ⊆PastA(p)"},{"paragraph_id":"p18","order":18,"text":"⇒∀p ∈P, PastS(p, P) = PastD(P).\nFutS(p, P) = FutA(p) ∩FutD(P)\n∀p ∈P, FutA(p) ⊆FutD(P)"},{"paragraph_id":"p19","order":19,"text":"⇒∀p ∈P, FutS(p, P) = FutA(p).\n□\nDefinition 1 In S, different transitions with the same label, the same desti-\nnation and whose origins belong to the same column are said to be competing.\nLikewise, different final states of the same column are competing. A competing\nset is a maximal set of competing transitions or competing final states.\nLet U be an automaton obtained from S by removing all transitions except one\nin every competing set and by turning all final states of a column, except one,\ninto non-final states. The choice of the transition (or the final state) to keep in\na competing set is arbitrary.\nFor instance, the covering S of Figure 10 has two competing sets (drawn\nwith double lines); the first one contains two transitions with label b that arrive\nin (r, {r}), the second one contains the states (p, {p, r}) and (r, {p, r}) which are\nboth final. The above selection principle gives rise to four possible automata,\nthe automaton of Fig. 11 being one of them.\nProposition 3 Let S and U be two automata defined as above. Then,\ni) ∀P, ∀p ∈P, PastU(p, P) = PastS(p, P).\nii) Futures of states in a column of U are disjoint and\n∀P, ∀p ∈P,\n[\np∈P\nFutU(p, P) =\n[\np∈P\nFutS(p, P) .\nConsequently, the automaton U is unambiguous and equivalent to A.\nproof. i) The proof is by induction on the length of words. If (p, P) is\ninitial in S, it is still initial in U. Let wa be a word of PastS(p, P) and π a\n16"},{"paragraph_id":"p20","order":20,"text":"path labelled by this word from an initial state to (p, P). We consider the last\ntransition of π:\nh\n(q, P ′)\na\n−→(p, P)\ni\nS.\nIf this transition does not belong to a competing set, it still appears in U and, by\ninduction, w ∈PastU(q, P ′), thus wa ∈PastU(p, P). If this transition belongs\nto a competing set, there exist q′ ∈P ′ and a transition\nh\n(q′, P ′)\na\n−→(p, P)\ni\nS\nwhich still appears in U, and by induction, since PastS(q, P ′) = PastS(q′, P ′),\nw ∈PastU(q′, P ′), so wa ∈PastU(p, P).\nii) We prove this by induction on the length of words. If there are several\nfinal states in a column of S, exactly one remains in U, so there is at most\none state whose future contains the empty word. Now let (p, P) and (p′, P ′) be\ntwo states in the same column such that the word au belongs to FutA(p) and\nFutA(p′):\nh\np\na\n−→q\nu\n−→t →\ni\nA\nh\np′\na\n−→q′\nu\n−→t′ →\ni\nA\nBoth transitions p\na\n−→q and p′\na\n−→q′ correspond to the same transition in D.\nThus q and q′ belong to the same column and, by induction, q = q′. Since there\nis no competing set in U, p = p′.\nObviously FutU(p, P) ⊆FutS(p, P). If au is in the future of a state (p0, P0) of S,\nthere exist a state (p1, P1) and a transition (p0, P0)\na\n−→(p1, P1), such that u is in\nFutS(p1, P1). By induction, there exists p′\n1 in P1 such that u is in FutU(p′\n1, P1),\nand there exists a transition (p′\n0, P0)\na\n−→(p′\n1, P1), thus au is in FutU(p′\n0, P0).\nLet w be a word accepted by A. For any factorization uv of w, there is\nexactly one column P of U such that, for every p in P, u is in PastU(p, P) and\nthere is exactly one state (p, P) in this column such that v is in FutU(p, P). This\ncharacterizes the only successful path with label w in U.\n□\nWe show now how the Sch ̈utzenberger covering can be used to convert a\nfinitely ambiguous automaton A into a finite union of unambiguous automata,\neach of them recognizing the same language as A.\nProposition 4 Let S be the Sch ̈utzenberger covering of a finitely ambiguous\nautomaton. Then, competing transitions of S do not belong to any circuit of S.\nThus a path of S contains at most one transition of each competing set.\nproof. Assume that a competing transition τ belongs to a circuit:\n→i\nu\n−→(p, P)\na\n−→\nτ\n(q, Q)\nw\n−→(p, P)\na\n−→\nτ\n(q, Q)\nv\n−→t →.\n17"},{"paragraph_id":"p21","order":21,"text":"a\nb\na\nb\nb\na\na\nb\nb\na\nb\nb\na\nb\nFigure 11: An unambiguous automaton equivalent to S\nHence, u(aw)∗is a subset of PastA(p). Let τ ′ be another transition that belongs\nto the same competing set: (p′, P)\na\n−→\nτ ′ (q, Q). From Lemma 1, u(aw)∗is a subset\nof PastA(p′). Thus, for every n, for every k in {0, . . . , n}, there exists a path:\n→i\nu(aw)k\n−→(p′, P)\na\n−→\nτ ′\nh\n(q, Q)\nw\n−→(p, P)\na\n−→\nτ\n(q, Q)\nin−k\nv\n−→t →.\nTherefore, there are at least n + 1 successful paths with label u(wa)nv in S,\nwhich is in contradiction with the finite ambiguity of S and A.\nIf there exists a path of S that contains two competing transitions τ and τ ′:\n(p, P)\na\n−→\nτ\n(q, Q)\nw\n−→(p′, P)\na\n−→\nτ ′ (q, Q),\nthen τ ′ belongs to a circuit, which is impossible.\n□\nAssume that A is finitely ambiguous. As a consequence of Proposition 4, for\nevery path in S (and thus for every path in A), one can compute an unambiguous\nautomaton U that contains this path. Consider the following algorithm.\nAs they do not belong to any circuit, competing sets of S are partially\nordered.\n– Compute C, the set of maximal competing sets of S (there is no path from\nany element of C to another competing set).\n– Let S1 and S2 be two copies of S. For every competing set X in C, let x\nbe an element of X;\n– if x is a transition, remove every transition of X\\{x} in S1 and remove x\nin S2;\n– if x is a final state, make every state of X\\{x} in S1 non-final and make\nx in S2 non-final.\n– Apply inductively this algorithm to S1 and S2.\n18"},{"paragraph_id":"p22","order":22,"text":"The result is a finite set of unambiguous automata. Each of them recognizes the\nlanguage of A and every path of S appears in at least one of these automata.\nNotice that the cardinality of this set may be larger than the degree of ambiguity\nof A. Denote by F the automaton obtained by taking the union of the automata\nin this set.\nAssume now that A is any automaton with multiplicities over an idempotent\nsemiring. Since there is a canonical mapping from the transitions (resp. initial\nstates, resp. final states) of the Sch ̈utzenberger covering S onto the transitions\n(resp. initial states, resp. final states) of A, one can decorate every transition\n(resp. initial state, resp. final state) of S with the corresponding multiplicity\nin A. This decoration can be carried out in the same way on the automaton F.\nObviously, since there is a one-to-one mapping between the successful paths\nof A and those of S, the series realized by S is equal to the one realized by A.\nFurthermore, as every path of S appears in F, the automaton F realizes the\nsame series as A. Notice that a path of S may appear several times in F, with\nno consequence since the semiring is idempotent.\nThe construction of F could be modified in order to get a one-to-one relation\nbetween paths of A and paths of F, but then the automata in the union would\nnot have the same support, which would be less convenient in the sequel.\nCorollary 1 A finitely ambiguous max-plus automaton can be effectively turned\ninto an equivalent finite union of unambiguous max-plus automata, all with the\nsame support.\n5\nThe Decidability Result\nIn this section, we show that a series, realized by a finite union of unambiguous\nautomata having the same support, is unambiguous if and only if a certain\nproperty denoted by (P) holds. Associated with Theorem 1 and Corollary 1,\nthis enables to prove Theorem 2, stated at the end of the paper.\nConsider a finite family of max-plus automata (Ai)i∈I with respective dimen-\nsions (Qi)i∈I. Set Ai = (α⌞i, μ⌞i, β⌞i). The corresponding product automaton P\nis an automaton with multiplicities in the product semiring\nRI\nmax, defined as\nfollows.\nSet Q =\nY\ni∈I\nQi and consider A, B ∈(RI\nmax)Q, M : Σ∗→(RI\nmax)Q×Q with\n∀p, q ∈Q, Ap = (α⌞i\npi)i∈I,\n∀a ∈Σ, M(a)p,q =\n(\n(μ⌞i(a)pi,qi)i∈I\nif ∀i, μ⌞i(a)pi,qi ̸=\n0\n(0, . . . ,\n0)\notherwise\nBp = (β⌞i\npi)i∈I .\nA state q ∈Q is initial if ∀i, (Aq)i ̸=\n0. A state q ∈Q is final if ∀i, (Bq)i ̸=\n0.\nThe trim part of (A, M, B) with respect to the above definition of initial and\nfinal states is the product automaton P.\n19"},{"paragraph_id":"p23","order":23,"text":"Clearly, if the automata (Ai)i∈I are unambiguous and all have the same\nsupport, then the product automaton P is also unambiguous and satisfies\n∀u ∈Σ∗, ∀i ∈I, ⟨S(P), u⟩i = α⌞iμ⌞i(u)β⌞i\n⇒\nM\ni∈I\n⟨S(P), u⟩i = ⟨\nM\ni∈I\nS(Ai), u⟩=\nM\ni∈I\nα⌞iμ⌞i(u)β⌞i.\nDefinition 2 Let θ be a simple circuit of P, whose weight is (x⌞i)i∈I. The set\nof victorious coordinates of θ, denoted by Vict (θ), is the set of coordinates on\nwhich the weight of θ is maximal, i.e. Vict (θ) ="},{"paragraph_id":"p24","order":24,"text":"i ∈I | x⌞i = max\nj∈I {x⌞j}"},{"paragraph_id":"p25","order":25,"text":".\nThis definition is extended in a natural way to a strongly connected subgraph\nC of P: the set of victorious coordinates of C is the intersection of the sets of\nvictorious coordinates of the simple circuits of C. We also extend the definition\nto a path π of P: the set of victorious coordinates of π is the intersection of the\nsets of victorious coordinates of the strongly connected subgraphs of P crossed\nby π.\nLet us define the ‘dominance’ property (P):\nFor each successful path π of the product automaton P, the set of victorious\ncoordinates of π is not empty.\nObviously, the number of simple circuits is finite. Hence (P) is a decidable\nproperty.\nLet (Ai = (α⌞i ∈\nRQi\nmax, μ⌞i : Σ∗→\nRQi×Qi\nmax\n, β⌞i ∈\nRQi\nmax))i∈I be a finite family\nof unambiguous trim automata, all with the same support, and let P be the\nproduct automaton with set of states Q ⊆Πi∈IQi. We assume that P satisfies\nthe dominance property (P).\nLet N = |Q| and M = max( max\ni,a,p,q μ⌞i(a)p,q, max\ni,p β⌞i\np)−min( min\ni,a,p,q μ⌞i(a)p,q, min\ni,p β⌞i\np),\nwhere the minima are taken over non- 0 terms. In words, M is the difference\nbetween the largest and the smallest non-initial weights appearing in the au-\ntomata.\nWe use the following notations as shortcuts. For x = (x⌞i)i∈I ∈\nRI\nmax, set\nˇx = mini∈I{x⌞i | x⌞i ̸= −∞} and x = x −(ˇx, . . . , ˇx).\nSet I = {1, . . ., n}. We now define an automaton U that is shown to be\nunambiguous and to realize the series L\ni∈I S(Ai).\nThe states of U belong to\nRn\nmax × Q.\nInitial states.\nAll the initial states are defined as follows. If q = (q⌞1, . . . , q⌞n)\nis a tuple such that q⌞i is an initial state of Ai, and if we set α = (α⌞1\nq⌞1, . . . , α⌞n\nq⌞n),\nthen (α, q) is an initial state of U and the weight of the ingoing arc is ˇα.\nStates and transitions.\nIf (z, p) is a state of U, then for each transition in P\nof type: p\na|x\n−−→q such that x⌞i ̸= −∞for all i, there is a transition in U leaving\n20"},{"paragraph_id":"p26","order":26,"text":"p, labelled by the letter a, and that we now describe. Set t = z + x. Let V be\nthe set of victorious coordinates of the maximal strongly connected subgraph of\nq in P. Since P satisfies (P), the set V ∩{t⌞k ̸= −∞} is non-empty. Let j ∈V\nbe such that t⌞j = mink∈V {t⌞k | t⌞k ̸= −∞}, and let y ∈\nRn\nmax be defined by:\n∀i,\ny⌞i =\n(\n−∞\nif t⌞i < t⌞j −NM,\nt⌞i\notherwise.\nNow (y, q) is a state of U and we have the following transition:\nh\n(z, p)\na|ˇy\n−−→(y, q)\ni\nU .\nFinal states.\nAll the final states are defined as follows. If (z, q) is a state of\nU, and if q⌞i is a final state of Ai for all i, then (z, q) is a final state of U and\nthe weight of the outgoing arc is maxi∈I{z⌞i + β⌞i\nq⌞i}.\nLemma 2 The set of states of U is finite.\nproof. First, given a state (z1, q) of U, we show that there are finitely many\nstates of the form (z2, q) that can be reached from (z1, q).\nObserve that a path leading from (z1, q) to (z2, q) in U corresponds to a\ncircuit leading from q to q in P that can be fully decomposed into simple\ncircuits belonging to the strongly connected component of q. Let V be the set of\nvictorious coordinates of the strongly connected component of q. By definition\nof victorious coordinates, for all i ∈V the value of z⌞i\n2 −z⌞i\n1 is a constant, that\nwe denote by x, and for all i ̸∈V one has z⌞i\n2 ⩽z⌞i\n1 + x.\nLet C be the (finite) set of simple circuits of P. For a circuit θ ∈C, let the\nweight of the circuit in P be denoted by (weight (θ)⌞1 , . . . , weight (θ)⌞n). Set also\nweight (θ) = maxi⩽n weight (θ)⌞i. Now define\nδ = min\nθ∈C\nh\nweight (θ) −max\ni {weight (θ)⌞i | weight (θ)⌞i < weight (θ)}\ni\n.\nBy definition, we have δ > 0. By construction, for i ̸∈V , either z⌞i\n2 = z⌞i\n1 + x,\nor z⌞i\n2 ⩽z⌞i\n1 + x −δ. Furthermore, there is at least one index i and one index\nj such that z⌞i\n1 = 0 and z⌞j\n2 = 0. At last, for j ̸∈V , we have by construction\nz⌞j\n2 ⩾mini∈V z⌞i\n2 −NM, or z⌞j\n2 = −∞.\nAlltogether, it shows that there are\nfinitely many possible values for z2 = (z⌞1\n2 , . . . , z⌞n\n2 ).\nConsequently, any acyclic path in U is of finite length. Since the number of\ninitial states is finite, it follows easily from K ̈onig Lemma that the number of\nstates of U is finite.\n□\nLemma 3 The automaton U is unambiguous.\nproof. Define the surjective map\nΨ :\nU\n−→\nP\n(z, p)\n7−→\np .\n21"},{"paragraph_id":"p27","order":27,"text":"By construction of U, the following properties hold.\ni) The map Ψ restricted to the initial states of U defines a bijection between\nthe initial states of U and P.\nii) Consider\nh\np\na−→q\ni\nP. Then ∀(z, p) ∈Ψ−1(p), ∃!(z′, q) ∈Ψ−1(q) such\nthat\nh\n(z, p) a−→(z′, q)\ni\nU.\niii) A state (z, q) is a final state of U if and only if q is a final state of P.\np\nq\na\na\na\na\nΨ−1(p)\nΨ−1(q)\nU\nP\nFigure 12: The properties of the map Ψ.\nThese three properties together imply that there is a bijection between suc-\ncessful paths in P and successful paths in U. As P is unambiguous, so is U.\n□\nLemma 4 The automaton U recognizes the series L\ni∈I S(Ai).\nproof. Let lbe an integer and u = a0a1 · · · al−1 be a word in the common\nsupport of the series S(Ai).\nBy Lemma 3, there exists exactly one successful path labelled by u in the\nautomaton U:\nπ =\nh\n→(z0, q0)\na0\n−→(z1, q1)\na1\n−→· · ·\nal−2\n−→(zl−1, ql−1)\nal−1\n−→(zl, ql) →\ni\nU\n• Fix i ∈{1, . . . , n}. Assume that z⌞i\nl= −∞.\nThen i is not a victorious coordinate of π. Let j be a victorious coordinate, we\nshow that ⟨S(Ai), u⟩< ⟨S(Aj), u⟩. Hence the coefficient of u in L\ni∈I S(Ai) is\nnot realized by the coordinate i, which means that there is no damage in having\nz⌞i\nl= −∞.\nIn the path π, there exists a minimal state qh such that the coordinate z⌞i\nh\nis equal to −∞. That means that the difference between z⌞i\nh and z⌞j\nh would have\nbeen larger than NM. Let π′ in P be the path that corresponds to π (by the\nproof of Lemma 3, there is a canonical bijection between successful paths of U\nand P) and let q′\nh be the state of π′ that corresponds to qh. Let π′\nh be the end of\n22"},{"paragraph_id":"p28","order":28,"text":"π′ from q′\nh onwards (including the final arrow). Let us prove that the difference\nof weights on π′\nh between the coordinates i and j is smaller than NM, that is:\nweight (π′\nh)⌞i −weight (π′\nh)⌞j ⩽NM.\n(7)\nActually, on every circuit, the weight with respect to i is smaller than or equal\nto the weight with respect to j (which is victorious), and, if we delete all the\ncircuits in π′\nh, we obtain an acyclic path that is necessarily shorter than N −1.\nOn every transition, the difference between the weights of the coordinates i and\nj is at most M. Likewise, the difference between terminal functions is smaller\nthan M. Hence we proved (7). It means that the weight of coordinate i cannot\ncatch up with the one of coordinate j. In particular, we have: ⟨S(Ai), u⟩<\n⟨S(Aj), u⟩⩽⟨L\ni∈I S(Ai), u⟩.\n• Assume that z⌞i\nl̸= −∞. Set α = (α⌞i\nq⌞i\n0 )i∈I and β = (β⌞i\nq⌞i\nl)i∈I. Let π′ be the\npath in P that corresponds to π:\nπ′ =\nh α→q0\na0|x0\n−−−→q1\na1|x1\n−−−→· · ·\nal−1|xl−1\n−−−−−−→ql\nβ→\ni\nP.\nWe have, by construction of the automaton U:\n⟨S(Ai), u⟩=α⌞i\nq⌞i\n0 +\nl−1\nX\nk=0\nx⌞i\nk + β⌞i\nq⌞i\nl\n=ˇα + z⌞i\n0 +\nl−1\nX\nk=0\n(yk + z⌞i\nk+1 −z⌞i\nk) + β⌞i\nq⌞i\nl\n=ˇα +\nl−1\nX\nk=0\nyk + z⌞i\nl + β⌞i\nq⌞i\nl\nTherefore, ⟨S(Ai), u⟩= ⟨L\nj∈I S(Aj), u⟩if and only if z⌞i\nl+β⌞i\nq⌞i\nl= maxj[z⌞j\nl+β⌞j\nq⌞j\nl].\nNow observe that by construction,\n⟨U, u⟩= ˇα +\nl−1\nX\nk=0\nyk + max\ni [z⌞i\nl+ β⌞i\nq⌞i\nl].\nThe equality ⟨L\ni∈I S(Ai), u⟩= ⟨U, u⟩follows easily.\n□\nWe now have all the ingredients to prove the proposition below.\nProposition 5 Consider a finite family (Ai)i∈I of trim and unambiguous max-\nplus automata having the same support. Let P be the corresponding product\nautomaton. The series L\ni∈I S(Ai) is unambiguous if and only if P satisfies the\nproperty (P). In this case, the automaton U defined above is finite, unambiguous,\nand realizes the series L\ni∈I S(Ai).\nproof.\nLemmas 2, 3 and 4 show that (P) is a sufficient condition for\nL\ni∈I S(Ai) to be unambiguous.\nLet us prove that (P) is also a necessary\ncondition.\n23"},{"paragraph_id":"p29","order":29,"text":"By way of contradiction, assume that S = L\ni∈I S(Ai) is recognized by an\nunambiguous automaton U and that (P) does not hold. There exists a path\nπ of P that can be decomposed into π0, θ1, π1, θ2, . . . , πr, where every θi is a\ncircuit and T\ni Vict (θi) = ∅.\nLet ui be the label of πi and vi the label of\nθi.\nLet s be the maximal integer such that V = T\ni⩽s Vict (θi) ̸= ∅.\nLet\nwk,l = u0vk\n1u1 · · · vk\ns usvl\ns+1us+1us+2 · · · ur. For every k, l, wk,l is accepted by P\nand thus by U (with an unique successful path). Let k0, l0 be greater than the\nnumber of states d of U. By the pigeon-hole principle, every path in U labelled\nby vk0\ni\n(for i ∈{1, . . . , s}) has a sub-circuit labelled by vki\ni\n(with ki < d).\nLikewise, the path labelled by vl0\ns+1 has a sub-circuit labelled by vl1\ns+1. It means\nthat there exist (gi, ki, di)i∈[1,s] and (gs+1, l1, ds+1) such that the successful path\nlabelled by wk0,l0 in U has the following shape:\nu0vg1\n1\nvd1\n1 u1vg2\n2\nvds+1\ns+1 us+1us+2 · · · ur\nvk1\n1\nvk2\n2\nvl1\ns+1\nLet K = Q\ni⩽s ki. Since U is unambiguous, for every pair of integers (α, β),\nthe word wk0+αK,l0+βl1 is accepted by a path that has the same shape; hence,\nthere exist x = ⟨S, wk0,l0⟩, ρ and λ such that, for every (α, β) ∈\nN ×\nN,\n⟨S, wk0+αK,l0+βl1⟩= x + αρ + βλ.\nThe word wk0+αK,l0+βl1 labels in P a successful path that is the concatena-\ntion of π0, (k0+αK) times θ0, π1,. . . ,πs, (l0+βl1) times θs+1,. . . . Therefore, for\nevery β, there exists Nβ such that, for every α > Nβ, the successful coordinates\nof the path labelled by wk0+αK,l0+βl1 belong to V and the weight is equal to\ny + αρ1 + βλ1, where y is a constant, ρ1 is the sum of the maximal weights of\nthe circuits θ1 to θs, and λ1 = maxi∈V weight (θs+i)⌞i.\nLikewise, for every α, there exists Mα such that, for every β > Mα, the\nsuccessful coordinate of the path labelled by wk0+αK,l0+βl1 is a victorious coor-\ndinate of θs+1 and the weight of this path is equal to z + αρ2 + βλ2, where z\nis a constant, ρ2 is the maximum over the victorious coordinates of θs+1 of the\nsums of the weights of the circuits θ1 to θs, and λ2 is the maximal weight of\nθs+1.\nTo summerize, the following equalities hold:\n∀α, β, ⟨S, wk0+αK,l0+βl1⟩= x + αρ + βλ\n∀β, ∀α > Nβ, ⟨S, wk0+αK,l0+βl1⟩= y + αρ1 + βλ1\n∀α, ∀β > Mα, ⟨S, wk0+αK,l0+βl1⟩= z + αρ2 + βλ2\nTherefore, ρ1 = ρ = ρ2 and λ1 = λ = λ2. Thus, there exists a coordinate that\nbelongs to V and that is victorious on θs+1; this contradicts the maximality of\ns.\n24"},{"paragraph_id":"p30","order":30,"text":"It would be possible to use an argument similar to the one in §3.6, to prove\nthe above.\n□\nThe main result is now a corollary of Proposition 5:\nTheorem 2 One can decide in an effective way, whether the series recognized\nby a finitely ambiguous max-plus automaton is unambiguous, and whether it is\nsequential.\nMore precisely, turn first the finitely ambiguous automaton into an equiva-\nlent finite union of unambiguous automata, all having the same support (Corol-\nlary 1). Then check the property (P) on the new family of automata. If (P)\nis satisfied the series is unambiguous; build the unambiguous automaton U\n(Proposition 5), then decide the sequentiality of U (Theorem 1).\nReferences\n[1] C. Allauzen and M. Mohri. Efficient algorithms for testing the twins prop-\nerty. Journal of Automata, Languages and Combinatorics, 8(2):117–144,\n2003.\n[2] M.-P. B ́eal, O. Carton, C. Prieur, and J. Sakarovitch. Squaring transducers:\nAn efficient procedure for deciding functionality and sequentiality. Theor.\nComput. Sci., 292:45–63, 2003.\n[3] J. Berstel. Transductions and context-free languages. B. G. Teubner, 1979.\n[4] J. Berstel and C. Reutenauer.\nRational Series and their Languages.\nSpringer Verlag, 1988.\n[5] M. Brilman and J.M. Vincent. Dynamics of synchronized parallel systems.\nStochastic Models, 13(3):605–619, 1997.\n[6] A.L. Buchsbaum, R. Giancarlo, and J.R. Westbrook. On the determiniza-\ntion of weighted finite automata. SIAM J. Comput., 30(5):1502–1531, 2000.\n[7] C. Choffrut. Une caract ́erisation des fonctions s ́equentielles et des fonctions\nsous-s ́equentielles en tant que relations rationnelles. Theor. Comput. Sci.,\n5:325–337, 1977.\n[8] C. Choffrut. Contribution `a l’ ́etude de quelques familles remarquables de\nfonctions rationnelles. Th`ese d’ ́etat, Univ. Paris VII, 1978.\n[9] S. Eilenberg. Automata, languages and machines, vol. A. Academic Press,\n1974.\n[10] S. Gaubert. Performance evaluation of (max,+) automata. IEEE Trans.\nAut. Cont., 40(12):2014–2025, 1995.\n25"},{"paragraph_id":"p31","order":31,"text":"[11] S. Gaubert and J. Mairesse. Task resource models and (max,+) automata.\nIn J. Gunawardena, editor, Idempotency, volume 11, pages 133–144. Cam-\nbridge University Press, 1998.\n[12] S. Gaubert and J. Mairesse. Modeling and analysis of timed Petri nets\nusing heaps of pieces. IEEE Trans. Aut. Cont., 44(4):683–698, 1999.\n[13] K. Hashigushi. Algorithms for determining relative star height and star\nheight. Inf. Comput., 78(2):124–169, 1988.\n[14] K. Hashigushi, K. Ishiguro, and S. Jimbo. Decidability of the equivalence\nproblem for finitely ambiguous finance automata. Int. J. Algebra Comput.,\n12(3):445–461, 2002.\n[15] D. Krob. The equality problem for rational series with multiplicities in the\ntropical semiring is undecidable. Int. J. Algebra Comput., 4(3):405–425,\n1994.\n[16] D. Krob and A. Bonnier-Rigny.\nA complete system of identities for\none-letter rational expressions with multiplicities in the tropical semiring.\nTheor. Comput. Sci., 134:27–50, 1994.\n[17] W. Kuich and A. Salomaa. Semirings, Automata, Languages, volume 5 of\nEATCS. Springer-Verlag, 1986.\n[18] M. Mohri.\nFinite-state transducers in language and speech processing.\nComput. Linguist., 23(2):269–311, 1997.\n[19] P. Moller. Th ́eorie alg ́ebrique des syst`emes `a ́ev ́enements discrets. PhD\nthesis, ́Ecole des Mines de Paris, 1988.\n[20] J. Sakarovitch. A construction in finite automata that has remained hidden.\nTheor. Comput. Sci., 204:205–231, 1998.\n[21] M.-P. Sch ̈utzenberger. On the definition of a family of automata. Informa-\ntion and Control, 4(2–3):245–270, 1961.\n[22] M.-P. Sch ̈utzenberger. Sur les relations rationnelles entre mono ̈ıdes libres.\nTheor. Comput. Sci., 3:243–259, 1976.\n[23] I. Simon. Recognizable sets with multiplicities in the tropical semiring. In\nMathematical Foundations of Computer Science, Proc. 13th Symp., number\n324 in LNCS, pages 107–120, 1988.\n[24] G.X. Viennot. Heaps of pieces, I: Basic definitions and combinatorial lem-\nmas. In Labelle and Leroux, editors, Combinatoire ́Enum ́erative, number\n1234 in Lect. Notes in Math., pages 321–350. Springer, 1986.\n[25] A. Weber. Finite-valued distance automata. Theor. Comput. Sci., 134:225–\n251, 1994.\n26"},{"paragraph_id":"p32","order":32,"text":"[26] A. Weber and R. Klemm. Economy of description for single-valued trans-\nducers. Information and Computation, 118(2):327–340, 1995.\n[27] A. Weber and H. Seidl. On the degree of ambiguity of finite automata.\nTheor. Comput. Sci., 88(2):325–349, 1991.\n27"}],"pages":[{"page":1,"text":"arXiv:0709.4117v1 [cs.CC] 26 Sep 2007\nDeciding Unambiguity and Sequentiality starting\nfrom a Finitely Ambiguous Max-Plus Automaton\nInes Klimann, Sylvain Lombardy, Jean Mairesse,\nand Christophe Prieur∗\nJuly 4, 2004\nAbstract\nFinite automata with weights in the max-plus semiring are considered.\nThe main result is: it is decidable in an effective way whether a series that\nis recognized by a finitely ambiguous max-plus automaton is unambiguous,\nor is sequential. A collection of examples is given to illustrate the hierarchy\nof max-plus series with respect to ambiguity.\n1\nIntroduction\nA max-plus automaton is a finite automaton with multiplicities in the max-plus\nsemiring\nRmax = (R ∪{−∞}, max, +). Roughly speaking, it is an automaton\nwith two tapes: an input tape labelled by a finite alphabet Σ, and an output\ntape weighted in\nRmax. The weight of a word in Σ∗is the maximum over all\nsuccessful paths of the sum of the weights along the path.\nMax-plus automata, and their min-plus counterparts, are studied under var-\nious names in the literature: distance automata, finance automata, cost au-\ntomata. They have also appeared in various contexts: to study logical problems\nin formal language theory (star height, finite power property) [13, 23], to model\nthe dynamic of some Discrete Event Systems (DES) [10, 12], or in the context\nof automatic speech recognition [18].\nTwo automata are equivalent if they recognize the same series, i.e. if they\nhave the same input/output behavior. The problem of equivalence of two max-\nplus automata is undecidable [15]. The same problem for finitely ambiguous\nmax-plus automata is decidable [14, 25].\nThe sequentiality problem is defined as follows: given a max-plus automaton,\nis there an equivalent max-plus automaton which is sequential (i.e. determinis-\ntic in input). Let us give some motivations on why the sequentiality problem\nis important. In the case of a sequential automaton, the time complexity of\n∗LIAFA, CNRS (umr 7089) - Universit ́e Paris 7, 2, place Jussieu - 75251 Paris Cedex 5 -\nFrance. email: {klimann,lombardy,mairesse,prieur}@liafa.jussieu.fr\n1"},{"page":2,"text":"computing the output is roughly linear in the length of the input. This time\nefficiency is central in speech processing, see [18]. Consider now a DES mod-\nelled by a max-plus automaton. If the automaton is unambiguous, or a fortiori\nsequential, then one can compute the optimal, as well as the average behavior,\nof the DES, see [10, 11].\nSequentiality is decidable for unambiguous max-plus automata [18].\nIn the\npresent paper, we prove that sequentiality is decidable for finitely ambiguous\nmax-plus automata. To the best of our knowledge, it is not known if the finite\nambiguity of a max-plus series (defined via an infinitely ambiguous automaton)\nis a decidable problem. In particular, the status of the sequentiality problem\nis still open for a general max-plus automaton (even if the multiplicities are\nrestricted to be in\nZmax,\nNmax or\nZ−\nmax). To be complete, it is necessary to\nmention that in [18, §3.5], it is claimed that any max-plus automaton admits\nan effectively computable equivalent unambiguous one.\nIf that was true, it\nwould imply the decidability of the sequentiality for general max-plus automata.\nHowever, the statement is erroneous and counter-examples are provided in §3\nof the present paper1.\nThe sequentiality problem can be asked for automata over any semiring\nK.\nFor transducers, i.e. when\nK is the set of rational subsets of a free monoid\n(with union and concatenation as the two laws), the problem is completely\nsolved in the functional case (when, for every input, the output is a language of\ncardinality at most one) [3, 7, 8]. For a general transducer, the problem is wide\nopen. Observe that the semiring {a⩾n, n ∈\nN} = {ana∗, n ∈\nN} is isomorphic\nto\nNmin: ana∗+ ama∗= amin(n,m)a∗and ana∗· ama∗= an+ma∗. Similarly,\nthe semiring {a⩽n, n ∈\nN} is isomorphic to\nNmax (where a⩽n = {ε, a, . . ., an}).\nHence automata over\nNmax or\nNmin translate into transducers, but not functional\nones. Also the translation does not work for automata over\nRmax. Hence, the\nvast literature on transducers is of limited use in our context.\nIn the present paper, we work with\nRmax. Decidability and complexity should\nbe interpreted under the assumption that two real numbers can be added or\ncompared in constant time.\n2\nPreliminaries\n2.1\nMax-plus semiring and series\nThe free monoid over a finite set (alphabet) Σ is denoted by Σ∗and the empty\nword is denoted by ε. The structure\nRmax = (R ∪{−∞}, max, +) is a semiring,\nwhich is called the max-plus semiring.\nIt is convenient to use the notations\n⊕= max and ⊗= +. The neutral elements of ⊕and ⊗are denoted respectively\nby\n0 = −∞and\n1 = 0.\nThe subsemirings\nNmax,\nZmax, . . . , are defined in\nthe natural way.\nThe min-plus semiring\nRmin is obtained by replacing max\nby min and −∞by +∞in the definition of\nRmax. The results of this paper\ncan be easily adapted to the min-plus setting. Observe that the subsemiring\n1The version of [18] available on the author’s website has been correctly modified.\n2"},{"page":3,"text":"B = ({ 0,\n1}, ⊕, ⊗) is isomorphic to the Boolean semiring. For matrices A, B of\nappropriate sizes with entries in\nRmax, we set (A⊕B)ij = Aij ⊕Bij, (A⊗B)ij =\nL\nk Aik ⊗Bkj, and for a ∈\nRmax, (a ⊗A)ij = a ⊗Aij. We usually omit the ⊗\nsign, writing for instance AB instead of A ⊗B.\nConsider the set\nRmax⟨⟨Σ∗⟩⟩of (formal power) series (over Σ∗with coeffi-\ncients in\nRmax), that is the set of maps from Σ∗to\nRmax. We denote by ⟨S, u⟩\nthe coefficient of the word u in the series S. The support of a series S is the\nset Supp S = {u ∈Σ∗|\n⟨S, u⟩̸=\n0}. It is convenient to use the notation\nS = L\nu∈Σ∗⟨S, u⟩u = L\nu∈Supp (S)⟨S, u⟩u. Equipped with the addition (⊕) and\nthe Cauchy product (⊗), the set\nRmax⟨⟨Σ∗⟩⟩forms a semiring. The image of\nλ ∈\nRmax by the canonical injection into\nRmax⟨⟨Σ∗⟩⟩is still denoted by λ. In\nparticular, the neutral elements of\nRmax⟨⟨Σ∗⟩⟩are\n0 and\n1. The characteristic\nseries of a language L is the series\n1L such that ⟨1L, w⟩=\n1 if w ∈L, and\n⟨1L, w⟩=\n0 otherwise.\n2.2\nMax-plus automaton\nLet Q and Σ be two finite sets. A max-plus automaton of set of states (di-\nmension) Q over the alphabet Σ, is a triple A = (α, μ, β), where α ∈\nR1×Q\nmax ,\nβ ∈\nRQ×1\nmax , and where μ : Σ∗→\nRQ×Q\nmax\nis a morphism of monoids. The mor-\nphism μ is uniquely determined by the family of matrices {μ(a), a ∈Σ}, and\nfor w = a1 · · · an, we have μ(w) = μ(a1) ⊗· · · ⊗μ(an). The series recognized\n(or realized) by A is by definition S(A) = L\nu∈Σ∗(αμ(u)β)u.\nThis is just a\nspecialization to the max-plus semiring of the classical notion of an automaton\nwith multiplicities over a semiring [4, 9, 17].\nBy the Kleene-Sch ̈utzenberger\nTheorem [21], the set of series recognized by a max-plus automaton is equal to\nthe set of rational series over\nRmax. We denote it by Rat.\nA state i ∈Q is initial, resp. final, if αi ̸=\n0, resp. βi ̸=\n0. As usual a\nmax-plus automaton is represented graphically by a labelled weighted digraph\nwith ingoing and outgoing arcs for initial and final states, see e.g. Figure 6\n(the input or output weights equal to\n1 are omitted). The terminology of graph\ntheory is used accordingly (e.g. (simple) path or circuit of an automaton, union\nof automata, . . . ).\nA path which is both starting with an ingoing arc and\nending with an outgoing arc is called a successful path. The label of a path is\nthe concatenation of the labels of the successive arcs (so called transitions), the\nweight of a path is the product (⊗) of the weights of the successive arcs (including\nthe ingoing and the outgoing arc, need it be). We denote by weight (π) the\nweight of the path π. We use the following notations for paths in an automaton\nA = (α, μ, β):\np →q,\n→p →q,\np →q →,\np\nu|x\n−−→q,\nh\np\nu|x\n−−→q\ni\nA , if μ(u)pq = x in A .\nThe first example is a path (of any length) from p to q, the second also\nincludes an ingoing arc, the third an outgoing arc, in the fourth the weight and\nthe label are added and in the fifth the underlying automaton is recalled.\nAn automaton is trim if any state belongs to at least one successful path.\n3"},{"page":4,"text":"Let I be a finite set. The tensor product automaton of (Ai = (α⌞i, μ⌞i, β⌞i))i∈I,\ndenoted by ⊙i∈IAi, is defined as follows. It is the max-plus automaton (A, M, B)\nof dimension Q = Q\ni Qi, where Qi is the dimension of Ai, and such that\n∀p, q ∈Q,\nAp =\nO\ni∈I\nα⌞i\npi,\n∀a ∈Σ, M(a)p,q =\nO\ni∈I\nμ⌞i(a)pi,qi,\nBp =\nO\ni∈I\nβ⌞i\npi .\n2.3\nHeap model\nA heap or Tetris model [24], consists of a finite set of slots R, and a finite\nset of rectangular pieces Σ. Each piece a ∈Σ is of height 1 and occupies a\ndetermined subset R(a) of the slots. To a word u = u1 · · · uk ∈Σ∗is associated\nthe heap obtained by piling up in order the pieces u1, . . . , uk, starting with\na horizontal ground and according to the Tetris game mechanism (pieces are\nsubject to gravity and fall down vertically until they meet either a previously\npiled up piece or the ground). Consider the morphism generated by the matrices\nM(a) ∈\nRR×R\nmax , a ∈Σ, defined by\nM(a)ij =\n \n \n \n1\nif i, j ∈R(a),\n0\nif i = j ̸∈R(a),\n−∞\notherwise .\nLet x(u)i be the height of the heap u on slot i ∈R. We have ([5, 11, 12]):\nx(u)i =\n1M(u)δi, where\n1 = (1, . . . ,\n1) ∈\nR1×R\nmax and δi ∈\nRR×1\nmax is defined by\n(δi)j =\n1 if j = i and\n0 otherwise. In other words, the application x(·)i : Σ∗→\nRmax is recognized by the max-plus automaton (1, M, δi). We call (1, M, δ), δ =\nL\ni∈I δi, I ⊆R, a heap automaton (associated with the heap model). Among\nmax-plus automata, heap automata are particularly convenient and playful, due\nto the underlying geometric interpretation. Here, they are used as a source of\nexamples and counter-examples, e.g. Figures 3, 4 and 7.\nWe represent a heap automaton graphically as in Figure 1.\nb\na\nR(a) = {1, 2}\nR(b) = {2, 3}\nR = {1, 2, 3}\n(1, M, δ2)\nFigure 1: A heap automaton\n2.4\nAmbiguity and Sequentiality\nConsider a max-plus automaton A = (α, μ, β) of dimension Q over Σ.\nThe\nautomaton is sequential if there is a unique initial state and if for all i ∈Q, and\n4"},{"page":5,"text":"for all a ∈Σ, there is at most one j ∈Q such that μ(a)ij ̸=\n0. In the case of\na Boolean automaton, we also say deterministic for sequential. The automaton\nA is unambiguous if for any word u ∈Σ∗, there is at most one successful path\nof label u.\nThe automaton is finitely ambiguous if there exists some k ∈\nN\nsuch that for any word u ∈Σ∗, there are at most k successful paths of label\nu.\nThe minimal such k is called the degree of ambiguity of the automaton.\nClearly, ‘sequential’ implies ‘unambiguous’ which implies ‘finitely ambiguous’.\nThe automaton is infinitely ambiguous if it is not finitely ambiguous.\nConsider a series S ∈Rat. The series is sequential (resp. unambiguous,\nfinitely ambiguous) if there exists a sequential (resp. unambiguous, finitely am-\nbiguous) max-plus automaton recognizing it. The series is infinitely ambiguous\nif there exists no finitely ambiguous max-plus automaton recognizing it. The\ndegree of ambiguity of a finitely ambiguous series is the minimal degree of ambi-\nguity of an automaton recognizing it. The sets of sequential, unambiguous, and\nfinitely ambiguous series are denoted respectively by Seq, NAmb, and FAmb.\nDefine FSeq = {S | ∃k, ∃S1, . . . , Sk ∈Seq, S = S1 ⊕· · · ⊕Sk}.\nConsider a total order on Σ∗. Given a series S ̸=\n0, define the normalized\nseries φ(S) by φ(S) = L\nu∈Σ∗(⟨S, u⟩−⟨S, u0⟩)u, where u0 is the smallest word\nof Supp S. The (left) quotient of a series S by a word w is the series w−1S\ndefined by w−1S = L\nu∈Σ∗⟨S, wu⟩u.\nA series S is rational if and only if the semi-module of series ⟨w−1S, w ∈Σ∗⟩\nis finitely generated, i.e. if there exists S1, . . . , Sk, such that:\n∀w ∈Σ∗, ∃λ1, . . . , λk ∈\nRmax, w−1S =\nM\ni\nλiSi.\nA series S is sequential if and only if the set of series {φ(w−1S), w ∈Σ∗} is\nfinite.\nProposition 1 A trim automaton A of dimension Q is infinitely ambiguous if\nand only if there exist p, q ∈Q, p ̸= q, and v ∈Σ∗, such that p\nv\n−→p, p\nv\n−→q,\nq\nv\n−→q. This can be checked in polynomial time.\nFor a proof, see [27] and the references therein. Observe that the (in)finite\nambiguity is independent of the underlying semiring.\nNext result is due to\nMohri [18] and is an adaptation of a classical result of Choffrut on functional\ntransducers, see [3, 7, 8] (for the decidability) and [2, 26] (for the polynomial\ncomplexity).\nTheorem 1 Let A be an unambiguous max-plus automaton.\nThere exists a\npolynomial time algorithm to decide whether S(A) is a sequential series.\nIf A is unambiguous and S(A) is sequential, a sequential automaton recog-\nnizing the series can be effectively constructed from A using an adaptation of\nthe subset construction of Boolean automata [1, 6, 18].\nIt is useful to detail Theorem 1. We need to introduce several definitions.\nGiven two words u, v ∈Σ∗, let u ∧v be the longest common prefix of u and\n5"},{"page":6,"text":"v, and define d(u, v) = |u| + |v| −2|u ∧v|. It is easy to check that d(., .) is a\ndistance on Σ∗. A series S is M-Lipschitz (M ∈\nR+) if:\n∀u, v ∈Supp S, |⟨S, u⟩−⟨S, v⟩| ⩽Md(u, v) ;\nand S is Lipschitz if it is M-Lipschitz for some M. The set of Lipschitz series is\ndenoted by Lip. Consider a trim max-plus automaton A of dimension Q. Two\nstates p, q ∈Q are twins if:\nh x0\n−→i\nu1|x1\n−−−→p\nu2|x2\n−−−→p,\ny0\n−→j\nu1|y1\n−−−→q\nu2|y2\n−−−→q\ni\n=⇒[x2 = y2] .\nIf all the states are twins, the automaton A is said to satisfy the twin property.\nWe denote the set of all such automata by Twin. The following implications\nhold:\nh\nA ∈Twin\ni\n=⇒\nh\nS(A) ∈Seq\ni\n=⇒\nh\nS(A) ∈Lip\ni\n.\n(1)\nFurthermore,\nh\nA ∈NAmb,\nS(A) ∈Lip\ni\n=⇒\nh\nA ∈Twin\ni\n.\n(2)\nThe twin property can be checked in polynomial time, hence Theorem 1 follows\nfrom the above implications.\n3\nHierarchy of Series\nThe examples in this section illustrate the classes of series on which we work.\nSeq ⊊(NAmb ∩FSeq)\n(§3.1)\n⊊\n⊊\nFSeq\n(§3.2)\nNAmb\n(§3.3)\n⊊\n⊊\nFAmb\n(§3.4)\n⊊\nRat\n(§3.5) ⊊Series\n(§3.6)\n3.1\nA Series in Seq ∩NAmb ∩FSeq\nAn example over a one-letter alphabet is provided in Figure 2. The recognized\nseries is\n⟨S, an⟩=\n(\n0\nif n is odd,\nn\nif n is even.\na|0\na|0\na|1\na|1\nFigure 2: Seq ∩NAmb ∩FSeq\nThe series is not Lipschitz, since |⟨S, an+1⟩−⟨S, an⟩| ⩾n, and consequently\nthe series cannot be sequential (see (1)). It is clear that it is an unambiguous\n6"},{"page":7,"text":"series (the only successful path of label an is the right or left one depending\non the parity of n) and a sum of sequential series. In fact, any max-plus ra-\ntional series over a one-letter alphabet is unambiguous and a sum of sequential\nseries [16, 19].\na|1, b|0\na|0, b|1\na\nb\nFigure 3: FSeq ∩NAmb\n3.2\nA Series in FSeq ∩NAmb\nThe series ⟨S, u⟩= |u|a ⊕|u|b over the alphabet {a, b} is a sum of two sequential\nseries: the heap automaton of Figure 3 recognizes this series.\nAssume that S is unambiguous. The series S is 1-Lipschitz. So it has to be\nsequential, see (2) and (1). Consequently, there exist series S1,. . . Sk such that:\n∀u ∈Σ∗, ∃i, ∃λu ∈\nRmax\nu−1S = λu ⊗Si.\nBy the pigeon-hole principle, there must exist i ∈{1, . . . k} and two integers\nm < n such that\n∃λn, λm\n(an)−1S = λn ⊗Si,\n(am)−1S = λm ⊗Si.\nConsequently, we have\n⟨(an)−1S, bm+1⟩−⟨(an)−1S, ε⟩= ⟨(am)−1S, bm+1⟩−⟨(am)−1S, ε⟩.\nHowever\n⟨(an)−1S, bm+1⟩−⟨(an)−1S, ε⟩= ⟨S, anbm+1⟩−⟨S, an⟩= n −n = 0\n⟨(am)−1S, bm+1⟩−⟨(am)−1S, ε⟩= ⟨S, ambm+1⟩−⟨S, am⟩= m + 1 −m = 1.\nThis is a contradiction, consequently S is not sequential and thus cannot be\nan unambiguous series.\n3.3\nSeries in NAmb ∩FSeq\na) The first example is the series S given by the heap automaton of Figure 4 (a),\nor equivalently by the automaton of Figure 4 (b).\nConsider the series ̃S defined by ⟨ ̃S, w⟩= ⟨S, w⟩−|w|.\nAn automaton\nrecognizing ̃S can clearly be obtained from an automaton recognizing S by\nremoving 1 from each output weight. Hence S and ̃S are both sum of sequential\nseries or none of them is.\n7"},{"page":8,"text":"a|1, b|0\na|1, b|1\na|1\na|1\nb\na\n(a)\nb|0\na|1, b|1\na|1\n(b)\nFigure 4: NAmb ∩FSeq\na|0\na|0, b|0\nb| −1\nFigure 5: NAmb ∩FSeq\nThe series ̃S is recognized by the automaton of Figure 5.\nSuppose that\n ̃S = S1 ⊕S2 ⊕· · · ⊕Sk, where k ∈\nN and the Si are sequential series.\nSince the Si are sequential series, they are Lipschitz. Let N be the maximal\nLipschitz coefficient of the Si. Let (Ni)i⩾0 be a sequence of integers such that\nN0 > N,\nN(Nk−1 + 1) < Nk −Nk−1 for all k ⩾1.\nThe coefficient of abNk in ̃S is −Nk, and it comes, for instance, from S1.\nThe coefficient of abNkabNk−1 is −Nk−1. We have:\nd(abNk, abNkabNk−1) = Nk−1+1 and |⟨ ̃S, abNk⟩−⟨ ̃S, abNkabNk−1⟩| = Nk−Nk−1.\nThe coefficient of abNkabNk−1 in ̃S does not come from S1, since\n|⟨S1, abNk⟩−⟨S1, abNkabNk−1⟩| ⩽N(Nk−1 + 1)\n< Nk −Nk−1 = |⟨S1, abNk⟩−⟨ ̃S, abNkabNk−1⟩|.\nIn the same way, we prove that any two words of the set\n{abNk, abNkabNk−1, . . . , abNkabNk−1 · · · abN0}\ncannot be recognized by the same Si. But this set has cardinality k + 1 and\nthus there is a contradiction.\nb) The second example is the series given by the automaton of Figure 6. The\nseries recognized by this automaton is:\n⟨S, am1bn1 · · · ampbnp⟩=\nX\nmieven\nmi,\n8"},{"page":9,"text":"where m1 ∈\nN, mk+1 ∈\nN −{0}, nk ∈\nN −{0} for 1 ⩽k ⩽p −1, and np ∈\nN.\nThe automaton is clearly unambiguous. Furthermore, it is not a finite sum of\nsequential series. To simplify notations, let us prove that S is not the sum of\ntwo sequential series. Suppose that S = S1 ⊕S2, with S1, S2 ∈Seq.\nThe series Si, i ∈{1, 2}, are sequential, so they are Lipschitz by (1). Let\nN be such that Si, i ∈{1, 2}, are N-Lipschitz. Let us consider words of the\nform arbnas, with n > 0. We discuss on the parity of r and s. The coefficient\nof the word a2p+1bna2q+1 in S, which is equal to 0, comes from one of the Si.\nFor instance\n⟨S, a2p+1bna2q+1⟩= 0 = ⟨S1, a2p+1bna2q+1⟩.\n(3)\nSet q > N. Since S1 is N-Lipschitz and d(a2p+1bna2q+1, a2p+1bna2q) = 1,\nwe have\n⟨S, a2p+1bna2q⟩= 2q = ⟨S2, a2p+1bna2q⟩.\n(4)\nFix q and n. Since S1 and S2 are Lipschitz, there exists an integer M such\nthat:\n∀u, v ∈Supp Si, d(u, v) ⩽2n + 4q + 2 ⇒|⟨Si, u⟩−⟨Si, v⟩| ⩽M.\n(5)\nWe have:\nd(a2pbna2q, a2p+1bna2q+1) = 2n+4q+2\nand\nd(a2pbna2q, a2p+1bna2q) = 2n+4q+1.\nSo, by Equation (5), we know that:\n– If a2pbna2q ∈Supp S1, then\n2p + 2q = |⟨S1, a2pbna2q⟩−⟨S1, a2p+1bna2q+1⟩| ⩽M,\nwhich is wrong for p large enough.\n– If a2pbna2q ∈Supp S2, then\n2p = |⟨S2, a2pbna2q⟩−⟨S2, a2p+1bna2q⟩| ⩽M,\nwhich is also wrong for p large enough.\nConsequently, S is not the sum of two sequential series.\nTo extend the re-\nsult to the sum of m sequential series, one has to consider words of the form\nar1bn1ar2 · · · arm−1bnm−1arm.\n3.4\nSeries in NAmb ∩FSeq ∩FAmb\na) Consider the heap automaton given in Figure 7 (a).\nThe corresponding\nseries is at most two-ambiguous since it is also recognized by the two-ambiguous\nautomaton of Figure 7 (b).\nIt cannot be unambiguous: on {a, b}∗, since it\ncoincides with the series of Figure 3 which is in NAmb. It cannot be a finite\nsum of sequential series: on {b, c}∗, it coincides with the series of Figure 4 which\nis in FSeq.\n9"},{"page":10,"text":"i\nj\na|0\na|0\na|1\na|1\nb|0\nb|0\nb|0\nb|0\nFigure 6: NAmb ∩FSeq\na|0, b|1, c|0\na|0, b|1, c|1\nb|1\nb|1\na|1, b|0, c|0\nb\na\nc\n(a)\na|0, c|0\na|0, b|1, c|1\nb|1\na|1, b|0, c|0\n(b)\nFigure 7: NAmb ∩FSeq ∩FAmb\nb) Another example is provided by the automaton A of Figure 8.\nDenote by S the series recognized by this automaton, by S1 the series recog-\nnized by the left part, say A1, of the automaton, and by S2 the series recognized\nby the right part, say A2.\nThe automaton A1 is the one introduced in Section 3.3 and the automaton\nA2 is the same one after permutation of the a’s and b’s in the labels. Recall\nthat A1 and A2 are unambiguous, so S is at most two-ambiguous.\nLet us prove that S is not a finite sum of sequential series. Denote by L the\nlanguage of words whose blocks of b’s have odd length. Let u be a word of L:\na|0\na|0\na|1\na|1\nb|0\nb|0\nb|0\nb|0\nb|0\nb|0\nb|1\nb|1\na|0\na|0\na|0\na|0\nFigure 8: NAmb ∩FSeq ∩FAmb\n10"},{"page":11,"text":"c|1\nc|1\na|1, b|0, c|1\na|0, b|1, c|1\n(a)\na\nb\nc\n(b)\nFigure 9: FAmb ∩Rat\nin A2, the b-blocks of u are always read in the upper part of the automaton, so\n⟨S2, u⟩= 0. Since the coefficient of u in A1 is at least 0, we have S⊙\n1L = S1⊙\n1L.\nSuppose that S is a finite sum of sequential series. Then so is S⊙\n1L and S1⊙\n1L.\nAnd this is false since one can choose an odd n in the proof of Section 3.3 for\nthe automaton of Figure 6.\nLet us prove that S is not unambiguous. Let M be the rational language\nof words whose a-blocks and b-blocks have even lengths and let u be a word\nof M.\nIn A1, the a-blocks have to be read in the lower part of A1 and so\n⟨S1, u⟩= |u|a. In the same way: ⟨S2, u⟩= |u|b. So we have S ⊙\n1M = S′ ⊙\n1M,\nwhere S′ is the series recognized by the automaton of Figure 3. Consequently,\nif S is unambiguous, so is S′ ⊙\n1M. We now apply the arguments of Section 3.2\nto show that S′ ⊙\n1M is not unambiguous.\nc) Besides, Weber has given examples of series which are k-ambiguous and\nnot (k −1)-ambiguous [25, Theorem 4.2].\n3.5\nSeries in FAmb ∩Rat\nConsider the series S recognized by the automaton of Figure 9. Assume that S\nis finitely ambiguous. Using the result of Corollary 1 below, S is recognized by\na finite union of unambiguous automata with the same support, say A1, . . . , Ak.\nDenote by Si the series recognized by Ai, for 1 ⩽i ⩽k, and by n the\nmaximal dimension of an automaton Ai. Observe that Supp S = Σ∗. Since all\nthe Si have the same support, we have Supp Si = Σ∗.\nNow, consider the word w0 =\n anbnc\n k. For any i, there is a single successful\npath labelled by w0 in Ai. Note that a path of length n contains necessarily a\ncircuit.\nSo, each automaton Ai contains a path of the form:\nπi :\na · · · a\na · · · a\nb · · · b\nb · · · b\nc\na · · · a\na · · · a\na · · · a\nb · · · b\na · · · a\nFor every j ∈{1, . . ., k}, we choose in the subpath labelled by the j-th factor\nan (resp. bn) a circuit that is called the j-th a-loop (resp. the j-th b-loop).\n11"},{"page":12,"text":"The coefficient of a word in S is less than or equal to its lentgh, it is thus\nthe same for its coefficients in the Si. Consequently, the mean weights of the\nloops of πi are less than or equal to 1. Denote by av (πi, a, j) the mean weight\nof the j-th a-loop in the path πi, and define av (πi, b, j) similarly.\nSet j ∈{1, . . ., k}. For λ ∈\nN −{0}, consider the word\nwλ = (anbnc) · · · (anbnc) (an+λn!bn+λn!c)\n|\n{z\n}\nj-th block\n(anbnc) · · · (anbnc)\nThis word can be read on each path πi by turning into the j-th a- and\nb-loops, whose lengths are less than or equal to n and so divide n!.\nLet i ∈{1, . . . , k} be such that ⟨S, w0⟩= ⟨Si, w0⟩.\nWe have ⟨S, wλ⟩−\n⟨S, w0⟩= λn!, and so ⟨Si, wλ⟩−⟨Si, w0⟩⩽λn!.\nBut ⟨Si, wλ⟩−⟨Si, w0⟩=\n(av (πi, a, j) + av (πi, b, j))λn!, consequently\nav (πi, a, j) + av (πi, b, j) ⩽1.\n(6)\nConsider any u in {01, 10}k. For all p ∈\nN −{0}, let us define the word\nvp(u) = (an+λ1n!bn+μ1n!c) · · · (an+λjn!bn+μjn!c) · · · (an+λkn!bn+μkn!c),\nwhere (λj, μj) = (p, 0) if (u2j−1, u2j) = (1, 0) (we say then that the dominant\nj-th loop is the j-th a-loop) and (λj, μj) = (0, p) otherwise (the dominant j-th\nloop is the j-th b-loop), for any j ∈{1, . . . , k}.\nBy the pigeon-hole principle, for some i, there are infinitely many words of\nthe form vp(u) such that ⟨S, vp(u)⟩= ⟨Si, vp(u)⟩= k + nk + kpn!. Such words\nare read on the path πi. The dominant j-th loop in πi has then necessarily\nmean weight 1, and by Equation (6), the non-dominant j-th loop in πi has\nmean weight less than or equal to 0.\nConsequently, we have built an injection from the language {01, 10}k into\nthe set of paths {πi}. But the language has cardinality 2k and the set of paths\nhas cardinality k. So we have a contradiction.\n3.6\nRational and Non-Rational Series (Rat)\nA max-plus series is non-rational as soon as its support is a non-rational lan-\nguage. Here, we present a less trivial example of non-rational max-plus series.\nIn this paragraph, it is necessary to distinguish between\nRmin and\nRmax:\nfor R = Rat or NAmb, we use the respective notations\nRminR,\nRmaxR.\nIf\nS ∈\nRmax⟨⟨Σ∗⟩⟩, we identify S with ̃S ∈\nRmin⟨⟨Σ∗⟩⟩such that ⟨ ̃S, w⟩= ⟨S, w⟩if\nw ∈Supp S and ⟨ ̃S, w⟩= +∞if ⟨S, w⟩= −∞.\nClearly, we have\nRmaxNAmb =\nRminNAmb = NAmb.\nOn the other hand, it is easy to find S ∈\nRminFSeq ∩\nRminNAmb such that\nS ̸∈\nRmaxRat.\n12"},{"page":13,"text":"Consider for instance the series S = min(|w|a, |w|b) (recognized by the au-\ntomaton of Figure 3 seen as a min-plus automaton). Let us prove that S does\nnot belong to\nRmaxRat. If it does: let S1, . . . Sn be a minimal generating fam-\nily of ⟨u−1S, u ∈Σ∗⟩(see §2.4), we have: ∀u ∈Σ∗, ∃λ(u)\n1 , . . . λ(u)\nn , u−1S =\nL\ni λ(u)\ni\n⊗Si. The restrictions of the quotients of S to b∗are bounded, hence\nso are the restrictions of the Si. Let ki be such that: ⟨Si, bki⟩= maxk⟨Si, bk⟩.\nIt follows that for any word u: maxk⟨u−1S, bk⟩= maxki⟨u−1S, bki⟩. Consider\nk > maxi ki. Then arises a contradiction:\nmax\nl\n⟨(ak)−1S, bl⟩= k > max\nki ⟨(ak)−1S, bki⟩= max\ni\nki.\n3.7\nAmbiguity vs. sequentiality and Ambiguity vs. Lips-\nchitz\nHere are some examples of series that are in several classes described in Section 3:\nNAmb\nFAmb ∩NAmb\nFAmb\nSeq\na\nimpossible\nimpossible\nFSeq\n∩\nSeq\na|0\na|0\na|1\na|1\na\nb\nSec 3.2\nimpossible\nFSeq\nb\na\nSec 3.3\nb\na\nc\nSec 3.4\na\nb\nc\nSec 3.5\n13"},{"page":14,"text":"NAmb\nFAmb ∩NAmb\nFAmb\nLip\na\na\nb\nSec 3.2\na\nb\nc\nSec 3.5\nLip\nb\na\nSec 3.3\nb\na\nc\nSec 3.4\na\nb\nc\n4\nFrom Finitely Ambiguous to Union of Unam-\nbiguous\nWeber [25] has proved that a finitely ambiguous\nNmax-automaton can be turned\ninto an union of unambiguous ones.\nWe present a completely different and\nsimpler proof that holds in any semiring, in particular\nRmax.\nIn this section, we work on the structure of the automata. So we consider\nsimply Boolean automata.\nBelow, given a set S, we identify the vectors of\nBS with the subsets of S,\ni.e. x ∈\nBS is identified with {i ∈S | xi =\n1}.\nLet A = (α, μ, β) be a trim automaton. The past of a state p is the set of\nwords that label a path from some initial state to p. The future of p is the set\nof words that label a path from p to some final state. We write:\nPastA(p) = {w ∈Σ∗| (αμ(w))p =\n1},\nFutA(p) = {w ∈Σ∗| (μ(w)β)p =\n1}.\nLet A = (α, μ : Σ∗→\nBQ×Q, β) be an automaton. Let us recall the usual\ndeterminization procedure of A via the subset construction. Let R be the least\nsubset of\nBQ inductively defined by:\nα ∈R,\nX ∈R ⇒∀a ∈Σ, Xμ(a) ∈R.\nLet D = D(A) = (J, ν : Σ∗→\nBR×R, U) be the determinized automaton\nof A defined by:\nJ = {α},\nU = {P ∈R | Pβ =\n1},\nν(a)P,P ′ =\n1 ⇐⇒P ′ = Pμ(a).\nLemma 1 i) Let A be an automaton and D its determinized automaton. Then\nfor each state P of D,\nPastD(P) ⊆\n\\\np∈P\nPastA(p),\nand\nFutD(P) =\n[\np∈P\nFutA(p).\n14"},{"page":15,"text":"p\nq\nr\npq\nqr\npr\nr\nq\np\np, {p, q}\nq, {p, q}\nq, {q, r}\nr, {q, r}\np, {p, r}\nr, {p, r}\nr, {r}\nq, {q}\np, {p}\na b\na\nb\nb\na\nb\na\nb\nb\na\nb\na\nb\na\nb\na\nb\na\nb\nb\na\na\nb\nb\na\nb\nb\na\nb\nb\nFigure 10: A (left), the determinized automaton (top) and the Sch ̈utzenberger cov-\nering\nii) Let A and B be two automata and A ⊙B their tensor product (cf. §2.4),\nthen, for all state (p, q) of A ⊙B,\nPastA⊙B(p, q) = PastA(p) ∩PastB(q),\nFutA⊙B(p, q) = FutA(p) ∩FutB(q).\nThe constructions and results given in Propositions 2 and 3 are inspired by\nSch ̈utzenberger [22]. They have been explicitely stated by Sakarovitch in [20].\nLet A be an automaton and D its determinized automaton. The trim part\nof the product A ⊙D is called the Sch ̈utzenberger covering S of A.\nProposition 2\nLet A = (α, μ, β) be a trim automaton, D its determinized\nautomaton and S its Sch ̈utzenberger covering.\ni) The states of S are exactly the pairs (p, P), where P is a state of D and\np ∈P. We call the set {(p, P) | p ∈P} of states of S a column (in gray on\nFigure 10).\nii) The canonical surjection ψ from the transitions of S onto the transitions\nof A induces a one-to-one mapping between the successful paths of S and A.\niii) Let P be a state of D. Then, for every p in P,\nPastS(p, P) = PastD(P),\nFutS(p, P) = FutA(p).\nThus, all the states of a given column have the same past.\nproof. i) A state (p, P) of S is initial if and only if p is initial in A (i.e.\np ∈α) and P is initial in D (i.e. P = {α}). Now, let (p, P) be a state of S\nsuch that p ∈P and (q, Q) a successor of (p, P) by a. Then, there exist two\ntransitions:\nh\np\na\n−→q\ni\nA\nand\nh\nP\na\n−→Q\ni\nD.\nBy definition of D, q belongs thus to Q.\n15"},{"page":16,"text":"Conversely, let P be a state of D and p an element of P. For every w in\nPastD(P), w belongs to PastA(p) (Lemma 1). Therefore there is a path in S\nfrom an initial state to (p, P).\nii) Let π be a successful path of A, with label w = w1w2 · · · wn. Let θ be\nthe (unique) successful path with label w in D:\nπ =\nh\n→p0\nw1\n−→p1\nw2\n−→. . .\nwn\n−→pn →\ni\nA,\nθ =\nh\n→P0\nw1\n−→P1\nw2\n−→. . .\nwn\n−→Pn →\ni\nD.\nThere is a path in S: π′ =→(p0, P0)\nw1\n−→(p1, P1)\nw2\n−→. . .\nwn\n−→(pn, Pn) →.\nThe function π 7→π′ is obviously one-to-one.\niii) By results from Lemma 1:\nPastS(p, P) = PastA(p) ∩PastD(P)\n∀p ∈P, PastD(P) ⊆PastA(p)\n \n⇒∀p ∈P, PastS(p, P) = PastD(P).\nFutS(p, P) = FutA(p) ∩FutD(P)\n∀p ∈P, FutA(p) ⊆FutD(P)\n \n⇒∀p ∈P, FutS(p, P) = FutA(p).\n□\nDefinition 1 In S, different transitions with the same label, the same desti-\nnation and whose origins belong to the same column are said to be competing.\nLikewise, different final states of the same column are competing. A competing\nset is a maximal set of competing transitions or competing final states.\nLet U be an automaton obtained from S by removing all transitions except one\nin every competing set and by turning all final states of a column, except one,\ninto non-final states. The choice of the transition (or the final state) to keep in\na competing set is arbitrary.\nFor instance, the covering S of Figure 10 has two competing sets (drawn\nwith double lines); the first one contains two transitions with label b that arrive\nin (r, {r}), the second one contains the states (p, {p, r}) and (r, {p, r}) which are\nboth final. The above selection principle gives rise to four possible automata,\nthe automaton of Fig. 11 being one of them.\nProposition 3 Let S and U be two automata defined as above. Then,\ni) ∀P, ∀p ∈P, PastU(p, P) = PastS(p, P).\nii) Futures of states in a column of U are disjoint and\n∀P, ∀p ∈P,\n[\np∈P\nFutU(p, P) =\n[\np∈P\nFutS(p, P) .\nConsequently, the automaton U is unambiguous and equivalent to A.\nproof. i) The proof is by induction on the length of words. If (p, P) is\ninitial in S, it is still initial in U. Let wa be a word of PastS(p, P) and π a\n16"},{"page":17,"text":"path labelled by this word from an initial state to (p, P). We consider the last\ntransition of π:\nh\n(q, P ′)\na\n−→(p, P)\ni\nS.\nIf this transition does not belong to a competing set, it still appears in U and, by\ninduction, w ∈PastU(q, P ′), thus wa ∈PastU(p, P). If this transition belongs\nto a competing set, there exist q′ ∈P ′ and a transition\nh\n(q′, P ′)\na\n−→(p, P)\ni\nS\nwhich still appears in U, and by induction, since PastS(q, P ′) = PastS(q′, P ′),\nw ∈PastU(q′, P ′), so wa ∈PastU(p, P).\nii) We prove this by induction on the length of words. If there are several\nfinal states in a column of S, exactly one remains in U, so there is at most\none state whose future contains the empty word. Now let (p, P) and (p′, P ′) be\ntwo states in the same column such that the word au belongs to FutA(p) and\nFutA(p′):\nh\np\na\n−→q\nu\n−→t →\ni\nA\nh\np′\na\n−→q′\nu\n−→t′ →\ni\nA\nBoth transitions p\na\n−→q and p′\na\n−→q′ correspond to the same transition in D.\nThus q and q′ belong to the same column and, by induction, q = q′. Since there\nis no competing set in U, p = p′.\nObviously FutU(p, P) ⊆FutS(p, P). If au is in the future of a state (p0, P0) of S,\nthere exist a state (p1, P1) and a transition (p0, P0)\na\n−→(p1, P1), such that u is in\nFutS(p1, P1). By induction, there exists p′\n1 in P1 such that u is in FutU(p′\n1, P1),\nand there exists a transition (p′\n0, P0)\na\n−→(p′\n1, P1), thus au is in FutU(p′\n0, P0).\nLet w be a word accepted by A. For any factorization uv of w, there is\nexactly one column P of U such that, for every p in P, u is in PastU(p, P) and\nthere is exactly one state (p, P) in this column such that v is in FutU(p, P). This\ncharacterizes the only successful path with label w in U.\n□\nWe show now how the Sch ̈utzenberger covering can be used to convert a\nfinitely ambiguous automaton A into a finite union of unambiguous automata,\neach of them recognizing the same language as A.\nProposition 4 Let S be the Sch ̈utzenberger covering of a finitely ambiguous\nautomaton. Then, competing transitions of S do not belong to any circuit of S.\nThus a path of S contains at most one transition of each competing set.\nproof. Assume that a competing transition τ belongs to a circuit:\n→i\nu\n−→(p, P)\na\n−→\nτ\n(q, Q)\nw\n−→(p, P)\na\n−→\nτ\n(q, Q)\nv\n−→t →.\n17"},{"page":18,"text":"a\nb\na\nb\nb\na\na\nb\nb\na\nb\nb\na\nb\nFigure 11: An unambiguous automaton equivalent to S\nHence, u(aw)∗is a subset of PastA(p). Let τ ′ be another transition that belongs\nto the same competing set: (p′, P)\na\n−→\nτ ′ (q, Q). From Lemma 1, u(aw)∗is a subset\nof PastA(p′). Thus, for every n, for every k in {0, . . . , n}, there exists a path:\n→i\nu(aw)k\n−→(p′, P)\na\n−→\nτ ′\nh\n(q, Q)\nw\n−→(p, P)\na\n−→\nτ\n(q, Q)\nin−k\nv\n−→t →.\nTherefore, there are at least n + 1 successful paths with label u(wa)nv in S,\nwhich is in contradiction with the finite ambiguity of S and A.\nIf there exists a path of S that contains two competing transitions τ and τ ′:\n(p, P)\na\n−→\nτ\n(q, Q)\nw\n−→(p′, P)\na\n−→\nτ ′ (q, Q),\nthen τ ′ belongs to a circuit, which is impossible.\n□\nAssume that A is finitely ambiguous. As a consequence of Proposition 4, for\nevery path in S (and thus for every path in A), one can compute an unambiguous\nautomaton U that contains this path. Consider the following algorithm.\nAs they do not belong to any circuit, competing sets of S are partially\nordered.\n– Compute C, the set of maximal competing sets of S (there is no path from\nany element of C to another competing set).\n– Let S1 and S2 be two copies of S. For every competing set X in C, let x\nbe an element of X;\n– if x is a transition, remove every transition of X\\{x} in S1 and remove x\nin S2;\n– if x is a final state, make every state of X\\{x} in S1 non-final and make\nx in S2 non-final.\n– Apply inductively this algorithm to S1 and S2.\n18"},{"page":19,"text":"The result is a finite set of unambiguous automata. Each of them recognizes the\nlanguage of A and every path of S appears in at least one of these automata.\nNotice that the cardinality of this set may be larger than the degree of ambiguity\nof A. Denote by F the automaton obtained by taking the union of the automata\nin this set.\nAssume now that A is any automaton with multiplicities over an idempotent\nsemiring. Since there is a canonical mapping from the transitions (resp. initial\nstates, resp. final states) of the Sch ̈utzenberger covering S onto the transitions\n(resp. initial states, resp. final states) of A, one can decorate every transition\n(resp. initial state, resp. final state) of S with the corresponding multiplicity\nin A. This decoration can be carried out in the same way on the automaton F.\nObviously, since there is a one-to-one mapping between the successful paths\nof A and those of S, the series realized by S is equal to the one realized by A.\nFurthermore, as every path of S appears in F, the automaton F realizes the\nsame series as A. Notice that a path of S may appear several times in F, with\nno consequence since the semiring is idempotent.\nThe construction of F could be modified in order to get a one-to-one relation\nbetween paths of A and paths of F, but then the automata in the union would\nnot have the same support, which would be less convenient in the sequel.\nCorollary 1 A finitely ambiguous max-plus automaton can be effectively turned\ninto an equivalent finite union of unambiguous max-plus automata, all with the\nsame support.\n5\nThe Decidability Result\nIn this section, we show that a series, realized by a finite union of unambiguous\nautomata having the same support, is unambiguous if and only if a certain\nproperty denoted by (P) holds. Associated with Theorem 1 and Corollary 1,\nthis enables to prove Theorem 2, stated at the end of the paper.\nConsider a finite family of max-plus automata (Ai)i∈I with respective dimen-\nsions (Qi)i∈I. Set Ai = (α⌞i, μ⌞i, β⌞i). The corresponding product automaton P\nis an automaton with multiplicities in the product semiring\nRI\nmax, defined as\nfollows.\nSet Q =\nY\ni∈I\nQi and consider A, B ∈(RI\nmax)Q, M : Σ∗→(RI\nmax)Q×Q with\n∀p, q ∈Q, Ap = (α⌞i\npi)i∈I,\n∀a ∈Σ, M(a)p,q =\n(\n(μ⌞i(a)pi,qi)i∈I\nif ∀i, μ⌞i(a)pi,qi ̸=\n0\n(0, . . . ,\n0)\notherwise\nBp = (β⌞i\npi)i∈I .\nA state q ∈Q is initial if ∀i, (Aq)i ̸=\n0. A state q ∈Q is final if ∀i, (Bq)i ̸=\n0.\nThe trim part of (A, M, B) with respect to the above definition of initial and\nfinal states is the product automaton P.\n19"},{"page":20,"text":"Clearly, if the automata (Ai)i∈I are unambiguous and all have the same\nsupport, then the product automaton P is also unambiguous and satisfies\n∀u ∈Σ∗, ∀i ∈I, ⟨S(P), u⟩i = α⌞iμ⌞i(u)β⌞i\n⇒\nM\ni∈I\n⟨S(P), u⟩i = ⟨\nM\ni∈I\nS(Ai), u⟩=\nM\ni∈I\nα⌞iμ⌞i(u)β⌞i.\nDefinition 2 Let θ be a simple circuit of P, whose weight is (x⌞i)i∈I. The set\nof victorious coordinates of θ, denoted by Vict (θ), is the set of coordinates on\nwhich the weight of θ is maximal, i.e. Vict (θ) =\n \ni ∈I | x⌞i = max\nj∈I {x⌞j}\n \n.\nThis definition is extended in a natural way to a strongly connected subgraph\nC of P: the set of victorious coordinates of C is the intersection of the sets of\nvictorious coordinates of the simple circuits of C. We also extend the definition\nto a path π of P: the set of victorious coordinates of π is the intersection of the\nsets of victorious coordinates of the strongly connected subgraphs of P crossed\nby π.\nLet us define the ‘dominance’ property (P):\nFor each successful path π of the product automaton P, the set of victorious\ncoordinates of π is not empty.\nObviously, the number of simple circuits is finite. Hence (P) is a decidable\nproperty.\nLet (Ai = (α⌞i ∈\nRQi\nmax, μ⌞i : Σ∗→\nRQi×Qi\nmax\n, β⌞i ∈\nRQi\nmax))i∈I be a finite family\nof unambiguous trim automata, all with the same support, and let P be the\nproduct automaton with set of states Q ⊆Πi∈IQi. We assume that P satisfies\nthe dominance property (P).\nLet N = |Q| and M = max( max\ni,a,p,q μ⌞i(a)p,q, max\ni,p β⌞i\np)−min( min\ni,a,p,q μ⌞i(a)p,q, min\ni,p β⌞i\np),\nwhere the minima are taken over non- 0 terms. In words, M is the difference\nbetween the largest and the smallest non-initial weights appearing in the au-\ntomata.\nWe use the following notations as shortcuts. For x = (x⌞i)i∈I ∈\nRI\nmax, set\nˇx = mini∈I{x⌞i | x⌞i ̸= −∞} and x = x −(ˇx, . . . , ˇx).\nSet I = {1, . . ., n}. We now define an automaton U that is shown to be\nunambiguous and to realize the series L\ni∈I S(Ai).\nThe states of U belong to\nRn\nmax × Q.\nInitial states.\nAll the initial states are defined as follows. If q = (q⌞1, . . . , q⌞n)\nis a tuple such that q⌞i is an initial state of Ai, and if we set α = (α⌞1\nq⌞1, . . . , α⌞n\nq⌞n),\nthen (α, q) is an initial state of U and the weight of the ingoing arc is ˇα.\nStates and transitions.\nIf (z, p) is a state of U, then for each transition in P\nof type: p\na|x\n−−→q such that x⌞i ̸= −∞for all i, there is a transition in U leaving\n20"},{"page":21,"text":"p, labelled by the letter a, and that we now describe. Set t = z + x. Let V be\nthe set of victorious coordinates of the maximal strongly connected subgraph of\nq in P. Since P satisfies (P), the set V ∩{t⌞k ̸= −∞} is non-empty. Let j ∈V\nbe such that t⌞j = mink∈V {t⌞k | t⌞k ̸= −∞}, and let y ∈\nRn\nmax be defined by:\n∀i,\ny⌞i =\n(\n−∞\nif t⌞i < t⌞j −NM,\nt⌞i\notherwise.\nNow (y, q) is a state of U and we have the following transition:\nh\n(z, p)\na|ˇy\n−−→(y, q)\ni\nU .\nFinal states.\nAll the final states are defined as follows. If (z, q) is a state of\nU, and if q⌞i is a final state of Ai for all i, then (z, q) is a final state of U and\nthe weight of the outgoing arc is maxi∈I{z⌞i + β⌞i\nq⌞i}.\nLemma 2 The set of states of U is finite.\nproof. First, given a state (z1, q) of U, we show that there are finitely many\nstates of the form (z2, q) that can be reached from (z1, q).\nObserve that a path leading from (z1, q) to (z2, q) in U corresponds to a\ncircuit leading from q to q in P that can be fully decomposed into simple\ncircuits belonging to the strongly connected component of q. Let V be the set of\nvictorious coordinates of the strongly connected component of q. By definition\nof victorious coordinates, for all i ∈V the value of z⌞i\n2 −z⌞i\n1 is a constant, that\nwe denote by x, and for all i ̸∈V one has z⌞i\n2 ⩽z⌞i\n1 + x.\nLet C be the (finite) set of simple circuits of P. For a circuit θ ∈C, let the\nweight of the circuit in P be denoted by (weight (θ)⌞1 , . . . , weight (θ)⌞n). Set also\nweight (θ) = maxi⩽n weight (θ)⌞i. Now define\nδ = min\nθ∈C\nh\nweight (θ) −max\ni {weight (θ)⌞i | weight (θ)⌞i < weight (θ)}\ni\n.\nBy definition, we have δ > 0. By construction, for i ̸∈V , either z⌞i\n2 = z⌞i\n1 + x,\nor z⌞i\n2 ⩽z⌞i\n1 + x −δ. Furthermore, there is at least one index i and one index\nj such that z⌞i\n1 = 0 and z⌞j\n2 = 0. At last, for j ̸∈V , we have by construction\nz⌞j\n2 ⩾mini∈V z⌞i\n2 −NM, or z⌞j\n2 = −∞.\nAlltogether, it shows that there are\nfinitely many possible values for z2 = (z⌞1\n2 , . . . , z⌞n\n2 ).\nConsequently, any acyclic path in U is of finite length. Since the number of\ninitial states is finite, it follows easily from K ̈onig Lemma that the number of\nstates of U is finite.\n□\nLemma 3 The automaton U is unambiguous.\nproof. Define the surjective map\nΨ :\nU\n−→\nP\n(z, p)\n7−→\np .\n21"},{"page":22,"text":"By construction of U, the following properties hold.\ni) The map Ψ restricted to the initial states of U defines a bijection between\nthe initial states of U and P.\nii) Consider\nh\np\na−→q\ni\nP. Then ∀(z, p) ∈Ψ−1(p), ∃!(z′, q) ∈Ψ−1(q) such\nthat\nh\n(z, p) a−→(z′, q)\ni\nU.\niii) A state (z, q) is a final state of U if and only if q is a final state of P.\np\nq\na\na\na\na\nΨ−1(p)\nΨ−1(q)\nU\nP\nFigure 12: The properties of the map Ψ.\nThese three properties together imply that there is a bijection between suc-\ncessful paths in P and successful paths in U. As P is unambiguous, so is U.\n□\nLemma 4 The automaton U recognizes the series L\ni∈I S(Ai).\nproof. Let lbe an integer and u = a0a1 · · · al−1 be a word in the common\nsupport of the series S(Ai).\nBy Lemma 3, there exists exactly one successful path labelled by u in the\nautomaton U:\nπ =\nh\n→(z0, q0)\na0\n−→(z1, q1)\na1\n−→· · ·\nal−2\n−→(zl−1, ql−1)\nal−1\n−→(zl, ql) →\ni\nU\n• Fix i ∈{1, . . . , n}. Assume that z⌞i\nl= −∞.\nThen i is not a victorious coordinate of π. Let j be a victorious coordinate, we\nshow that ⟨S(Ai), u⟩< ⟨S(Aj), u⟩. Hence the coefficient of u in L\ni∈I S(Ai) is\nnot realized by the coordinate i, which means that there is no damage in having\nz⌞i\nl= −∞.\nIn the path π, there exists a minimal state qh such that the coordinate z⌞i\nh\nis equal to −∞. That means that the difference between z⌞i\nh and z⌞j\nh would have\nbeen larger than NM. Let π′ in P be the path that corresponds to π (by the\nproof of Lemma 3, there is a canonical bijection between successful paths of U\nand P) and let q′\nh be the state of π′ that corresponds to qh. Let π′\nh be the end of\n22"},{"page":23,"text":"π′ from q′\nh onwards (including the final arrow). Let us prove that the difference\nof weights on π′\nh between the coordinates i and j is smaller than NM, that is:\nweight (π′\nh)⌞i −weight (π′\nh)⌞j ⩽NM.\n(7)\nActually, on every circuit, the weight with respect to i is smaller than or equal\nto the weight with respect to j (which is victorious), and, if we delete all the\ncircuits in π′\nh, we obtain an acyclic path that is necessarily shorter than N −1.\nOn every transition, the difference between the weights of the coordinates i and\nj is at most M. Likewise, the difference between terminal functions is smaller\nthan M. Hence we proved (7). It means that the weight of coordinate i cannot\ncatch up with the one of coordinate j. In particular, we have: ⟨S(Ai), u⟩<\n⟨S(Aj), u⟩⩽⟨L\ni∈I S(Ai), u⟩.\n• Assume that z⌞i\nl̸= −∞. Set α = (α⌞i\nq⌞i\n0 )i∈I and β = (β⌞i\nq⌞i\nl)i∈I. Let π′ be the\npath in P that corresponds to π:\nπ′ =\nh α→q0\na0|x0\n−−−→q1\na1|x1\n−−−→· · ·\nal−1|xl−1\n−−−−−−→ql\nβ→\ni\nP.\nWe have, by construction of the automaton U:\n⟨S(Ai), u⟩=α⌞i\nq⌞i\n0 +\nl−1\nX\nk=0\nx⌞i\nk + β⌞i\nq⌞i\nl\n=ˇα + z⌞i\n0 +\nl−1\nX\nk=0\n(yk + z⌞i\nk+1 −z⌞i\nk) + β⌞i\nq⌞i\nl\n=ˇα +\nl−1\nX\nk=0\nyk + z⌞i\nl + β⌞i\nq⌞i\nl\nTherefore, ⟨S(Ai), u⟩= ⟨L\nj∈I S(Aj), u⟩if and only if z⌞i\nl+β⌞i\nq⌞i\nl= maxj[z⌞j\nl+β⌞j\nq⌞j\nl].\nNow observe that by construction,\n⟨U, u⟩= ˇα +\nl−1\nX\nk=0\nyk + max\ni [z⌞i\nl+ β⌞i\nq⌞i\nl].\nThe equality ⟨L\ni∈I S(Ai), u⟩= ⟨U, u⟩follows easily.\n□\nWe now have all the ingredients to prove the proposition below.\nProposition 5 Consider a finite family (Ai)i∈I of trim and unambiguous max-\nplus automata having the same support. Let P be the corresponding product\nautomaton. The series L\ni∈I S(Ai) is unambiguous if and only if P satisfies the\nproperty (P). In this case, the automaton U defined above is finite, unambiguous,\nand realizes the series L\ni∈I S(Ai).\nproof.\nLemmas 2, 3 and 4 show that (P) is a sufficient condition for\nL\ni∈I S(Ai) to be unambiguous.\nLet us prove that (P) is also a necessary\ncondition.\n23"},{"page":24,"text":"By way of contradiction, assume that S = L\ni∈I S(Ai) is recognized by an\nunambiguous automaton U and that (P) does not hold. There exists a path\nπ of P that can be decomposed into π0, θ1, π1, θ2, . . . , πr, where every θi is a\ncircuit and T\ni Vict (θi) = ∅.\nLet ui be the label of πi and vi the label of\nθi.\nLet s be the maximal integer such that V = T\ni⩽s Vict (θi) ̸= ∅.\nLet\nwk,l = u0vk\n1u1 · · · vk\ns usvl\ns+1us+1us+2 · · · ur. For every k, l, wk,l is accepted by P\nand thus by U (with an unique successful path). Let k0, l0 be greater than the\nnumber of states d of U. By the pigeon-hole principle, every path in U labelled\nby vk0\ni\n(for i ∈{1, . . . , s}) has a sub-circuit labelled by vki\ni\n(with ki < d).\nLikewise, the path labelled by vl0\ns+1 has a sub-circuit labelled by vl1\ns+1. It means\nthat there exist (gi, ki, di)i∈[1,s] and (gs+1, l1, ds+1) such that the successful path\nlabelled by wk0,l0 in U has the following shape:\nu0vg1\n1\nvd1\n1 u1vg2\n2\nvds+1\ns+1 us+1us+2 · · · ur\nvk1\n1\nvk2\n2\nvl1\ns+1\nLet K = Q\ni⩽s ki. Since U is unambiguous, for every pair of integers (α, β),\nthe word wk0+αK,l0+βl1 is accepted by a path that has the same shape; hence,\nthere exist x = ⟨S, wk0,l0⟩, ρ and λ such that, for every (α, β) ∈\nN ×\nN,\n⟨S, wk0+αK,l0+βl1⟩= x + αρ + βλ.\nThe word wk0+αK,l0+βl1 labels in P a successful path that is the concatena-\ntion of π0, (k0+αK) times θ0, π1,. . . ,πs, (l0+βl1) times θs+1,. . . . Therefore, for\nevery β, there exists Nβ such that, for every α > Nβ, the successful coordinates\nof the path labelled by wk0+αK,l0+βl1 belong to V and the weight is equal to\ny + αρ1 + βλ1, where y is a constant, ρ1 is the sum of the maximal weights of\nthe circuits θ1 to θs, and λ1 = maxi∈V weight (θs+i)⌞i.\nLikewise, for every α, there exists Mα such that, for every β > Mα, the\nsuccessful coordinate of the path labelled by wk0+αK,l0+βl1 is a victorious coor-\ndinate of θs+1 and the weight of this path is equal to z + αρ2 + βλ2, where z\nis a constant, ρ2 is the maximum over the victorious coordinates of θs+1 of the\nsums of the weights of the circuits θ1 to θs, and λ2 is the maximal weight of\nθs+1.\nTo summerize, the following equalities hold:\n∀α, β, ⟨S, wk0+αK,l0+βl1⟩= x + αρ + βλ\n∀β, ∀α > Nβ, ⟨S, wk0+αK,l0+βl1⟩= y + αρ1 + βλ1\n∀α, ∀β > Mα, ⟨S, wk0+αK,l0+βl1⟩= z + αρ2 + βλ2\nTherefore, ρ1 = ρ = ρ2 and λ1 = λ = λ2. Thus, there exists a coordinate that\nbelongs to V and that is victorious on θs+1; this contradicts the maximality of\ns.\n24"},{"page":25,"text":"It would be possible to use an argument similar to the one in §3.6, to prove\nthe above.\n□\nThe main result is now a corollary of Proposition 5:\nTheorem 2 One can decide in an effective way, whether the series recognized\nby a finitely ambiguous max-plus automaton is unambiguous, and whether it is\nsequential.\nMore precisely, turn first the finitely ambiguous automaton into an equiva-\nlent finite union of unambiguous automata, all having the same support (Corol-\nlary 1). Then check the property (P) on the new family of automata. If (P)\nis satisfied the series is unambiguous; build the unambiguous automaton U\n(Proposition 5), then decide the sequentiality of U (Theorem 1).\nReferences\n[1] C. Allauzen and M. Mohri. Efficient algorithms for testing the twins prop-\nerty. Journal of Automata, Languages and Combinatorics, 8(2):117–144,\n2003.\n[2] M.-P. B ́eal, O. Carton, C. Prieur, and J. Sakarovitch. Squaring transducers:\nAn efficient procedure for deciding functionality and sequentiality. Theor.\nComput. Sci., 292:45–63, 2003.\n[3] J. Berstel. Transductions and context-free languages. B. G. Teubner, 1979.\n[4] J. Berstel and C. Reutenauer.\nRational Series and their Languages.\nSpringer Verlag, 1988.\n[5] M. Brilman and J.M. Vincent. Dynamics of synchronized parallel systems.\nStochastic Models, 13(3):605–619, 1997.\n[6] A.L. Buchsbaum, R. Giancarlo, and J.R. Westbrook. On the determiniza-\ntion of weighted finite automata. SIAM J. Comput., 30(5):1502–1531, 2000.\n[7] C. Choffrut. Une caract ́erisation des fonctions s ́equentielles et des fonctions\nsous-s ́equentielles en tant que relations rationnelles. Theor. Comput. Sci.,\n5:325–337, 1977.\n[8] C. Choffrut. Contribution `a l’ ́etude de quelques familles remarquables de\nfonctions rationnelles. Th`ese d’ ́etat, Univ. Paris VII, 1978.\n[9] S. Eilenberg. Automata, languages and machines, vol. A. Academic Press,\n1974.\n[10] S. Gaubert. Performance evaluation of (max,+) automata. IEEE Trans.\nAut. Cont., 40(12):2014–2025, 1995.\n25"},{"page":26,"text":"[11] S. Gaubert and J. Mairesse. Task resource models and (max,+) automata.\nIn J. Gunawardena, editor, Idempotency, volume 11, pages 133–144. Cam-\nbridge University Press, 1998.\n[12] S. Gaubert and J. Mairesse. Modeling and analysis of timed Petri nets\nusing heaps of pieces. IEEE Trans. Aut. Cont., 44(4):683–698, 1999.\n[13] K. Hashigushi. Algorithms for determining relative star height and star\nheight. Inf. Comput., 78(2):124–169, 1988.\n[14] K. Hashigushi, K. Ishiguro, and S. Jimbo. Decidability of the equivalence\nproblem for finitely ambiguous finance automata. Int. J. Algebra Comput.,\n12(3):445–461, 2002.\n[15] D. Krob. The equality problem for rational series with multiplicities in the\ntropical semiring is undecidable. Int. J. Algebra Comput., 4(3):405–425,\n1994.\n[16] D. Krob and A. Bonnier-Rigny.\nA complete system of identities for\none-letter rational expressions with multiplicities in the tropical semiring.\nTheor. Comput. Sci., 134:27–50, 1994.\n[17] W. Kuich and A. Salomaa. Semirings, Automata, Languages, volume 5 of\nEATCS. Springer-Verlag, 1986.\n[18] M. Mohri.\nFinite-state transducers in language and speech processing.\nComput. Linguist., 23(2):269–311, 1997.\n[19] P. Moller. Th ́eorie alg ́ebrique des syst`emes `a ́ev ́enements discrets. PhD\nthesis, ́Ecole des Mines de Paris, 1988.\n[20] J. Sakarovitch. A construction in finite automata that has remained hidden.\nTheor. Comput. Sci., 204:205–231, 1998.\n[21] M.-P. Sch ̈utzenberger. On the definition of a family of automata. Informa-\ntion and Control, 4(2–3):245–270, 1961.\n[22] M.-P. Sch ̈utzenberger. Sur les relations rationnelles entre mono ̈ıdes libres.\nTheor. Comput. Sci., 3:243–259, 1976.\n[23] I. Simon. Recognizable sets with multiplicities in the tropical semiring. In\nMathematical Foundations of Computer Science, Proc. 13th Symp., number\n324 in LNCS, pages 107–120, 1988.\n[24] G.X. Viennot. Heaps of pieces, I: Basic definitions and combinatorial lem-\nmas. In Labelle and Leroux, editors, Combinatoire ́Enum ́erative, number\n1234 in Lect. Notes in Math., pages 321–350. Springer, 1986.\n[25] A. Weber. Finite-valued distance automata. Theor. Comput. Sci., 134:225–\n251, 1994.\n26"},{"page":27,"text":"[26] A. Weber and R. Klemm. Economy of description for single-valued trans-\nducers. Information and Computation, 118(2):327–340, 1995.\n[27] A. Weber and H. Seidl. On the degree of ambiguity of finite automata.\nTheor. Comput. Sci., 88(2):325–349, 1991.\n27"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"Rmax = (R ∪{−∞}, max, +). Roughly speaking, it is an automaton","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"N} = {ana∗, n ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"Nmin: ana∗+ ama∗= amin(n,m)a∗and ana∗· ama∗= an+ma∗. Similarly,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"Nmax (where a⩽n = {ε, a, . . ., an}).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"Rmax = (R ∪{−∞}, max, +) is a semiring,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"⊕= max and ⊗= +. The neutral elements of ⊕and ⊗are denoted respectively","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"0 = −∞and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"B = ({ 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"Rmax, we set (A⊕B)ij = Aij ⊕Bij, (A⊗B)ij =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"Rmax, (a ⊗A)ij = a ⊗Aij. We usually omit the ⊗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"set Supp S = {u ∈Σ∗|","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"⟨S, u⟩̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"S = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"u∈Σ∗⟨S, u⟩u = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"1L such that ⟨1L, w⟩=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"⟨1L, w⟩=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"mension) Q over the alphabet Σ, is a triple A = (α, μ, β), where α ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"for w = a1 · · · an, we have μ(w) = μ(a1) ⊗· · · ⊗μ(an). The series recognized","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"(or realized) by A is by definition S(A) = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"A state i ∈Q is initial, resp. final, if αi ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"0, resp. βi ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"A = (α, μ, β):","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"A , if μ(u)pq = x in A .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"Let I be a finite set. The tensor product automaton of (Ai = (α⌞i, μ⌞i, β⌞i))i∈I,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"of dimension Q = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"Ap =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"∀a ∈Σ, M(a)p,q =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"Bp =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"determined subset R(a) of the slots. To a word u = u1 · · · uk ∈Σ∗is associated","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"M(a)ij =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"if i = j ̸∈R(a),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"x(u)i =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"(δi)j =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"1 if j = i and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"Rmax is recognized by the max-plus automaton (1, M, δi). We call (1, M, δ), δ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"R(a) = {1, 2}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"R(b) = {2, 3}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"R = {1, 2, 3}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"Consider a max-plus automaton A = (α, μ, β) of dimension Q over Σ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"for all a ∈Σ, there is at most one j ∈Q such that μ(a)ij ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"Define FSeq = {S | ∃k, ∃S1, . . . , Sk ∈Seq, S = S1 ⊕· · · ⊕Sk}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"Consider a total order on Σ∗. Given a series S ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"series φ(S) by φ(S) = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"defined by w−1S = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"Rmax, w−1S =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"and only if there exist p, q ∈Q, p ̸= q, and v ∈Σ∗, such that p","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"v, and define d(u, v) = |u| + |v| −2|u ∧v|. It is easy to check that d(., .) is a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"=⇒[x2 = y2] .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"⟨S, an⟩=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"The series ⟨S, u⟩= |u|a ⊕|u|b over the alphabet {a, b} is a sum of two sequential","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"u−1S = λu ⊗Si.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"(an)−1S = λn ⊗Si,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"(am)−1S = λm ⊗Si.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"⟨(an)−1S, bm+1⟩−⟨(an)−1S, ε⟩= ⟨(am)−1S, bm+1⟩−⟨(am)−1S, ε⟩.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"⟨(an)−1S, bm+1⟩−⟨(an)−1S, ε⟩= ⟨S, anbm+1⟩−⟨S, an⟩= n −n = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"⟨(am)−1S, bm+1⟩−⟨(am)−1S, ε⟩= ⟨S, ambm+1⟩−⟨S, am⟩= m + 1 −m = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"Consider the series ̃S defined by ⟨ ̃S, w⟩= ⟨S, w⟩−|w|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"̃S = S1 ⊕S2 ⊕· · · ⊕Sk, where k ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"d(abNk, abNkabNk−1) = Nk−1+1 and |⟨ ̃S, abNk⟩−⟨ ̃S, abNkabNk−1⟩| = Nk−Nk−1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"< Nk −Nk−1 = |⟨S1, abNk⟩−⟨ ̃S, abNkabNk−1⟩|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"⟨S, am1bn1 · · · ampbnp⟩=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"two sequential series. Suppose that S = S1 ⊕S2, with S1, S2 ∈Seq.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"⟨S, a2p+1bna2q+1⟩= 0 = ⟨S1, a2p+1bna2q+1⟩.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"Set q > N. Since S1 is N-Lipschitz and d(a2p+1bna2q+1, a2p+1bna2q) = 1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"⟨S, a2p+1bna2q⟩= 2q = ⟨S2, a2p+1bna2q⟩.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"d(a2pbna2q, a2p+1bna2q+1) = 2n+4q+2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"d(a2pbna2q, a2p+1bna2q) = 2n+4q+1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"2p + 2q = |⟨S1, a2pbna2q⟩−⟨S1, a2p+1bna2q+1⟩| ⩽M,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"2p = |⟨S2, a2pbna2q⟩−⟨S2, a2p+1bna2q⟩| ⩽M,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"⟨S2, u⟩= 0. Since the coefficient of u in A1 is at least 0, we have S⊙","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"1L = S1⊙","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"⟨S1, u⟩= |u|a. In the same way: ⟨S2, u⟩= |u|b. So we have S ⊙","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"1M = S′ ⊙","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"maximal dimension of an automaton Ai. Observe that Supp S = Σ∗. Since all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"the Si have the same support, we have Supp Si = Σ∗.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"Now, consider the word w0 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"wλ = (anbnc) · · · (anbnc) (an+λn!bn+λn!c)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"Let i ∈{1, . . . , k} be such that ⟨S, w0⟩= ⟨Si, w0⟩.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"⟨S, w0⟩= λn!, and so ⟨Si, wλ⟩−⟨Si, w0⟩⩽λn!.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"But ⟨Si, wλ⟩−⟨Si, w0⟩=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"vp(u) = (an+λ1n!bn+μ1n!c) · · · (an+λjn!bn+μjn!c) · · · (an+λkn!bn+μkn!c),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"where (λj, μj) = (p, 0) if (u2j−1, u2j) = (1, 0) (we say then that the dominant","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"j-th loop is the j-th a-loop) and (λj, μj) = (0, p) otherwise (the dominant j-th","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"the form vp(u) such that ⟨S, vp(u)⟩= ⟨Si, vp(u)⟩= k + nk + kpn!. Such words","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"for R = Rat or NAmb, we use the respective notations","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"Rmin⟨⟨Σ∗⟩⟩such that ⟨ ̃S, w⟩= ⟨S, w⟩if","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"w ∈Supp S and ⟨ ̃S, w⟩= +∞if ⟨S, w⟩= −∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"RmaxNAmb =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"RminNAmb = NAmb.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"Consider for instance the series S = min(|w|a, |w|b) (recognized by the au-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"n , u−1S =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"so are the restrictions of the Si. Let ki be such that: ⟨Si, bki⟩= maxk⟨Si, bk⟩.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"It follows that for any word u: maxk⟨u−1S, bk⟩= maxki⟨u−1S, bki⟩. Consider","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"⟨(ak)−1S, bl⟩= k > max","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"ki ⟨(ak)−1S, bki⟩= max","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"BS is identified with {i ∈S | xi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"Let A = (α, μ, β) be a trim automaton. The past of a state p is the set of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"PastA(p) = {w ∈Σ∗| (αμ(w))p =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"FutA(p) = {w ∈Σ∗| (μ(w)β)p =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"Let A = (α, μ : Σ∗→","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"Let D = D(A) = (J, ν : Σ∗→","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"J = {α},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"U = {P ∈R | Pβ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"ν(a)P,P ′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"1 ⇐⇒P ′ = Pμ(a).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"FutD(P) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"PastA⊙B(p, q) = PastA(p) ∩PastB(q),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"FutA⊙B(p, q) = FutA(p) ∩FutB(q).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"Let A = (α, μ, β) be a trim automaton, D its determinized","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"PastS(p, P) = PastD(P),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"FutS(p, P) = FutA(p).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"p ∈α) and P is initial in D (i.e. P = {α}). Now, let (p, P) be a state of S","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"ii) Let π be a successful path of A, with label w = w1w2 · · · wn. Let θ be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"There is a path in S: π′ =→(p0, P0)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"PastS(p, P) = PastA(p) ∩PastD(P)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"⇒∀p ∈P, PastS(p, P) = PastD(P).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"FutS(p, P) = FutA(p) ∩FutD(P)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"⇒∀p ∈P, FutS(p, P) = FutA(p).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"i) ∀P, ∀p ∈P, PastU(p, P) = PastS(p, P).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq120","equation_number":null,"raw_text":"FutU(p, P) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq121","equation_number":null,"raw_text":"which still appears in U, and by induction, since PastS(q, P ′) = PastS(q′, P ′),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq122","equation_number":null,"raw_text":"Thus q and q′ belong to the same column and, by induction, q = q′. Since there","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq123","equation_number":null,"raw_text":"is no competing set in U, p = p′.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq124","equation_number":null,"raw_text":"sions (Qi)i∈I. Set Ai = (α⌞i, μ⌞i, β⌞i). The corresponding product automaton P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq125","equation_number":null,"raw_text":"Set Q =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq126","equation_number":null,"raw_text":"∀p, q ∈Q, Ap = (α⌞i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq127","equation_number":null,"raw_text":"∀a ∈Σ, M(a)p,q =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq128","equation_number":null,"raw_text":"if ∀i, μ⌞i(a)pi,qi ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq129","equation_number":null,"raw_text":"Bp = (β⌞i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq130","equation_number":null,"raw_text":"A state q ∈Q is initial if ∀i, (Aq)i ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq131","equation_number":null,"raw_text":"0. A state q ∈Q is final if ∀i, (Bq)i ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq132","equation_number":null,"raw_text":"∀u ∈Σ∗, ∀i ∈I, ⟨S(P), u⟩i = α⌞iμ⌞i(u)β⌞i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq133","equation_number":null,"raw_text":"⟨S(P), u⟩i = ⟨","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq134","equation_number":null,"raw_text":"S(Ai), u⟩=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq135","equation_number":null,"raw_text":"which the weight of θ is maximal, i.e. Vict (θ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq136","equation_number":null,"raw_text":"i ∈I | x⌞i = max","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq137","equation_number":null,"raw_text":"Let (Ai = (α⌞i ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq138","equation_number":null,"raw_text":"Let N = |Q| and M = max( max","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq139","equation_number":null,"raw_text":"We use the following notations as shortcuts. For x = (x⌞i)i∈I ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq140","equation_number":null,"raw_text":"ˇx = mini∈I{x⌞i | x⌞i ̸= −∞} and x = x −(ˇx, . . . , ˇx).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq141","equation_number":null,"raw_text":"Set I = {1, . . ., n}. We now define an automaton U that is shown to be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq142","equation_number":null,"raw_text":"All the initial states are defined as follows. If q = (q⌞1, . . . , q⌞n)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq143","equation_number":null,"raw_text":"is a tuple such that q⌞i is an initial state of Ai, and if we set α = (α⌞1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq144","equation_number":null,"raw_text":"−−→q such that x⌞i ̸= −∞for all i, there is a transition in U leaving","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq145","equation_number":null,"raw_text":"p, labelled by the letter a, and that we now describe. Set t = z + x. Let V be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq146","equation_number":null,"raw_text":"q in P. Since P satisfies (P), the set V ∩{t⌞k ̸= −∞} is non-empty. Let j ∈V","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq147","equation_number":null,"raw_text":"be such that t⌞j = mink∈V {t⌞k | t⌞k ̸= −∞}, and let y ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq148","equation_number":null,"raw_text":"y⌞i =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq149","equation_number":null,"raw_text":"weight (θ) = maxi⩽n weight (θ)⌞i. Now define","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq150","equation_number":null,"raw_text":"δ = min","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq151","equation_number":null,"raw_text":"2 = z⌞i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq152","equation_number":null,"raw_text":"1 = 0 and z⌞j","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq153","equation_number":null,"raw_text":"2 = 0. At last, for j ̸∈V , we have by construction","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq154","equation_number":null,"raw_text":"finitely many possible values for z2 = (z⌞1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq155","equation_number":null,"raw_text":"proof. Let lbe an integer and u = a0a1 · · · al−1 be a word in the common","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq156","equation_number":null,"raw_text":"l= −∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq157","equation_number":null,"raw_text":"l= −∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq158","equation_number":null,"raw_text":"l̸= −∞. Set α = (α⌞i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq159","equation_number":null,"raw_text":"0 )i∈I and β = (β⌞i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq160","equation_number":null,"raw_text":"⟨S(Ai), u⟩=α⌞i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq161","equation_number":null,"raw_text":"k=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq162","equation_number":null,"raw_text":"=ˇα + z⌞i","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq163","equation_number":null,"raw_text":"k=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq164","equation_number":null,"raw_text":"k=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq165","equation_number":null,"raw_text":"Therefore, ⟨S(Ai), u⟩= ⟨L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq166","equation_number":null,"raw_text":"l= maxj[z⌞j","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq167","equation_number":null,"raw_text":"⟨U, u⟩= ˇα +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq168","equation_number":null,"raw_text":"k=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq169","equation_number":null,"raw_text":"i∈I S(Ai), u⟩= ⟨U, u⟩follows easily.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq170","equation_number":null,"raw_text":"By way of contradiction, assume that S = L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq171","equation_number":null,"raw_text":"i Vict (θi) = ∅.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq172","equation_number":null,"raw_text":"Let s be the maximal integer such that V = T","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq173","equation_number":null,"raw_text":"i⩽s Vict (θi) ̸= ∅.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq174","equation_number":null,"raw_text":"wk,l = u0vk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq175","equation_number":null,"raw_text":"Let K = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq176","equation_number":null,"raw_text":"there exist x = ⟨S, wk0,l0⟩, ρ and λ such that, for every (α, β) ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq177","equation_number":null,"raw_text":"⟨S, wk0+αK,l0+βl1⟩= x + αρ + βλ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq178","equation_number":null,"raw_text":"the circuits θ1 to θs, and λ1 = maxi∈V weight (θs+i)⌞i.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq179","equation_number":null,"raw_text":"∀α, β, ⟨S, wk0+αK,l0+βl1⟩= x + αρ + βλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq180","equation_number":null,"raw_text":"∀β, ∀α > Nβ, ⟨S, wk0+αK,l0+βl1⟩= y + αρ1 + βλ1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq181","equation_number":null,"raw_text":"∀α, ∀β > Mα, ⟨S, wk0+αK,l0+βl1⟩= z + αρ2 + βλ2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq182","equation_number":null,"raw_text":"Therefore, ρ1 = ρ = ρ2 and λ1 = λ = λ2. Thus, there exists a coordinate that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":51477,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}