Methods and Applications of (max,+) Linear Algebra
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Methods and Applications of (max,+) Linear Algebra St´ephane Gaubert , Max Plus. N ˚ 3088 Janvier 1997 ` THEME 4. ISSN 0249-6399. apport de recherche.
(3)
(4) Methods and Applications of (max,+) Linear Algebra St´ephane Gaubert , Max Plus Th`eme 4 — Simulation et optimisation de syst`emes complexes Projet Meta2 Rapport de recherche n ˚ 3088 — Janvier 1997 — 24 pages
(5).
(6). Abstract: Exotic semirings such as the “ semiring” , or the “tropical !" #$%'&(
(7) semiring” , have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities). Despite this apparent profusion, there) is a(small set of common, non-naive, basic results and prob
(8) lems, in general not known outside the community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core *
(9) of results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of discrete event systems, analysis of Markov decision processes with average cost). Basic techniques include: solving all kinds of systems)of (linear equations, sometimes with exotic
(10) symmetrization and determinant techniques; using the Perron-Frobenius theory to study (
(11) the dynamics of linear maps. We point out some open problems and current developments. Key-words: Max-algebra, tropical semiring, dioid, idempotent semiring, linear equations, semimodule, Perron-Frobenius theorem, linear dynamical systems, discrete event systems, Markov decision processes, dynamic programming, asymptotic calculus.. (R´esum´e : tsvp) This survey paper has been prepared for an invited talk at STACS’97 +. Max Plus is a collective name for a working group on ,- .0/(1243 algebra, at INRIA Rocquencourt, comprising currently: Marianne Akian, Guy Cohen, S.G., Jean-Pierre Quadrat and Michel Viot.. Unit´e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T´el´ephone : (33) 01 39 63 55 11 – T´el´ecopie : (33) 01 39 63 53 30.
(12) M´ethodes et applications de l’alg`ebre lin´eaire (max,+) R´esum´e : Il s’agit d’une br`eve introduction au semi-anneau (max,+) et autres semi-anneaux tropicaux, pr´epar´ee pour une lecture invit´ee au STACS’97. Ces alg`ebres exotiques ont e´t´e e´ tudi´ees depuis la fin des ann´ees 50, avec des motivations tr`es vari´ees: (i) recherche op´erationnelle, d´ecision Markovienne, EDP de Hamilton-Jacobi, (ii) asymptotiques (asymptotiques a` temp´erature nulle en physique statistique, grandes d´eviations, m´ethode WKB), (iii) th´eorie des langages (emploi des automates a` multiplicit´e pour r´esoudre des probl`emes de d´ecision, comme la “propri´et´e de la puissance finie”); (iv) e´valuation de performance d’une bonne classe de syst`emes a` e´ v´enements discrets, qui sont lin´eaires dans l’alg`ebre (max,+). Ces applications ont conduit a` des d´eveloppements th´eoriques trop vari´es pour eˆ tre survol´es ici. On s’est content´e de motiver par sept “applications” typiques le petit noyau de r´esultats (simples, en g´en´eral peu connus en dehors de la communaut´e) qui semblent eˆ tre utiles dans la plupart des cas. Il s’agit essentiellement de “l’alg`ebre lin´eaire (max,+)”: th´eor`eme de la base pour les semimodules de type fini, th´eorie spectrale a` la Perron-Frobenius, m´ethodes effectives: point fixe, e´ limination, r´esiduation, d´eterminants de Cramer et sym´etrisation. Ces r´esultats sont pr´esent´es et illustr´es. On mentionne quelques probl`emes d’actualit´e. Mots-cl´e : Alg`ebre (max,+), semi-anneau tropical, dio¨ıde, semi-anneau idempotent, e´ quations lin´eaires, semi-module, th´eor`eme de Perron-Frobenius, syst`emes dynamiques lin´eaires, syst`emes a` e´v´enements discrets, processus de d´ecision Markoviens, programmation dynamique, calcul asymptotique..
(13) Methods and Applications of (max,+) Linear Algebra. 1. Introduction: the.
(14) . ) *
(15). 3. and tropical semirings . . . . . . The “max-algebra” or “ semiring” , is the set , equipped with as ad.
(16) for dition, and as multiplication. It is traditional to use the notation for ( ), and.
(17)
(18) . We denote1 by the zero element for (such that
(19) , here ) and by ! ! ! ! $ , here &% ). This structure satisfies all the the unit element for (such that " # semiring axioms, i.e. is associative, commutative, with zero element, is associative, has a unit, distributes over , and zero is absorbing (all the ring axioms
(20) are satisfied, except that
(21) need not be a group law). This semiring is commutative '"()*(+, , idempotent '"
(22) , and non zero elements have an inverse for (we call semifields the semirings that satisfy this property). The term dioid is sometimes used for an idempotent semiring. . Using the new symbols and instead of the familiar and notation is the price to pay (
(23) to easily handle all the familiar algebraic constructions. For instance, we will write, in the semiring: 76
(24) -( . ,( 0/1. 2*33345 98'
(25) : times @ @ @ @ A% % F% %0 % %0B B B DC % EC = % " %=C C
(26) IH
(27)
(28) H H "G 5 G "G *: "=G"G * : ,G . ; . *. :. . . KJLG. =<?> .
(29) $
(30) L<. We will systematically use the standard algebraic notions (matrices, vectors, linear operators, semimodules — i.e. modules over a semiring—, formal polynomials and polynomial functions, for
(31) mal series) in the context of the semiring, often without explicit mention. Essentially all the standard notions of algebra have obvious semiring analogues, provided they do not appeal to the invertibility of addition. *
(32) There are several useful variants of the semiring, displayed in Table 1. In the sequel, ! with a we will have to consider various semirings, and will universally use the notation NI % & +O P context dependent meaning (e.g. in but in , in but M M L . +O P in ). RQ T % ), is the The fact that is idempotent instead of being invertible ( is an exception, for S"M main original feature of these “exotic” algebras, which makes them so different from the more familiar ring and field structures. In fact VU U W U
(33) the idempotence and cancellativity axioms are exclusive: if for all ( , '"()* 2 () and 2" * , we get X
(34) , for all (simplify 2F K* 2F ). This paper is not a survey in the usual sense. There exist several comprehensive books and excellent survey articles on the subject, each one having its own bias and motivations. Applications of ) *
(35) algebras are too vast (they range from asymptotic methods to decidability problems), techniques are too various (from graph theory to measure theory and large deviations) to be surveyed in 1. The notation for the zero and unit is one of the disputed questions of the community. The symbols Y for zero, and Z for the unit, often used in the literature, are very distinctive and well suited to handwritten computations. But it is difficult to renounce to the traditional use of Y in Analysis. The notation [ 1]\ used by the Idempotent Analysis school has the advantage of making formulæ closer to their usual analogues.. RR n ˚ 3088.
(36) 4. St´ephane Gaubert , Max Plus. 0/ 1 70/ 1 9/ 1 : =.
(37) ,. ,. 1 -. .0/(1243. 1 -. .0/(1243. semiring idempotent semifield max algebra completed , 2 , 2 3 ,- . /1243 semiring [ for [ ( ) ,- . /1 3 semiring isomorphic to ,1243 semiring isomorphic to ( ) tropical semiring (famous in Language Theory) bottleneck algebra not dealt with here Boolean semiring isomorphic to , [1 \ 1 1 43 , for any of the above semirings Maslov semirings isomorphic to , 12 1 3 ,- . /1243. . . 2 2 1 - . 1/ -41"! 365 3 1243 782 2 1-4365 1243 , & 1- . / 1 -4365 3 ,. , false 1 true 1 or 1 and3 ,. 365. ,. &
(38) 1>; = , = = . ; ? A@.(+*,(, ZBDC. 1243. -E<C Z. 2. #!. % (+*- ,.$ 4 $&$&') '% $. =. <; . F "! H K (63'- =GIH J and tropical semirings LMLML = . 3. Table 1. The family of. ,- . /1 243. a paper of this format. But there is a small common set of useful basic results, applications and problems, that we try to spotlight here. We aim neither at completeness, nor at originality. But we wish
(39) to give an honest idea of the services that one should expect from techniques. The interested reader is referred to the books [15,44,10,2,31], to the survey papers listed in the bibliography, and to the recent collection of articles [24] for an up-to-date account of the maxplusian results and motivations. Bibliographical and historical comments are at the end of the paper.. 2 2.1. Seven good reasons to use the.
(40) . semiring. An Algebra for Optimal Control. A standard problem of calculus of variations, which appears in Mechanics (least action principle) and Optimal Control, is the following. Given a Lagrangian N and suitable boundary conditions (e.g. O %
(41) O QP
(42) fixed), compute. X Y . \
(43) ^]
(44) _ SR Z O O TDUQV W [ N % &. ). % &
(45). This problem is intrinsically % &SR variant, with ` acb rather than ,. d. 76
(46). <. linear. To see this, consider the (slightly more general) discrete. d fe(
(47) hg d ie!.
(48) >j ie
(49)
(50) .e 6 < << Mk m l m G M j
(51) n U d fe!F0
(52) Mj fe(
(53)
(54) Zs d ik
(55) $
(56) / o"p t m G
(57) u` acb /Sql r m G >j*
(58) v / /.
(59) G. . (1). (2a) (2b) (2c). INRIA.
(60) Methods and Applications of (max,+) Linear Algebra. 5. j f e(
(61) Me. >k. 6 . . where the ` a b isfe(taken of controls selected in a finite set 333 e all6 sequences < <<(Mk d
(62) , forover of controls , , belongs to a finite set of states, is a distinguished initial G U state, is the dynamics, is the instantaneous reward, and g J J s is the final reward (the value can be used to code forbidden final states or transitions). These data a deterministic Markov Decision Process (MDP) with 6 k additive reward. t m 3
(63) ,form The function which represents the optimal reward from time to time , as a function of / the starting point, is called the value function. It satisfies the backward dynamic programming equation. . . . t mm. . s. t om. . Introducing the transition matrix. .
(64). G. v . . .
(65) U. . >j. (the supremum over an empty set is. . t om. *
(66) . G. I , v ` acU b v W p . . . . qr g . . . G. Mj. .
(67) $
(68). <. (3). .
(69). U . Mj. *
(70) . G. (4). ), we obtain:. t m. o of a fiFACT 1 (D ETERMINISTIC MDP - LINEAR DYNAMICS ). The value function (
(71) nite deterministic Markov decision process with additive reward is given by the linear dynamics: *
(72). t mm. . s . t om. t om. . qr. <. . !. s. (5). ! " . The interpretation in terms of paths is elementary. If we must end at node , we take t m
(73) m
(74) is(the ! vector with all entries except the -th equal tok ), Then, the value function [ the maximal (additive) weight of a path of length , from to , in the graph canonically associated 2 with .. . . . Example 1 (Taxicab). Consider a taxicab which operates between 3 cities and one airport, as shown in Fig. 1. At each state, the taxi driver has to choose his next destination, with deterministic fares shown on the graph (for simplicity, we assume that the demand is deterministic, and k that the driver can choose the destination). The taxi driver considers maximizing his reward over journeys. The ) *
(75) matrix associated with this MDP is displayed in Fig. 1. Let us consider the optimization of the average reward:. # . G.
(76). . u` acv%b $ m'&` ( a b k l [ m . % . j. . G. Mj. <.
(77). Mj. I
(78) . (6) <<<. Here, the ` a e b is taken over infinite sequences of controls and the trajectory (2a) is < << l m % defined for . . We expect [ to grow (or to decrease) linearly, as a function of the horizon 2. .
(79) . ) !*) matrix + we associate the weighted (directed) graph, with set of nodes ) +/. 0 whenever +/. 01 [ .. With a with weight. RR n ˚ 3088. , and an arc. Q$. -,. , *1 3.
(80) 6. St´ephane Gaubert , Max Plus U. 4$ 5$. U. . city 1 7$. 6$. airport 2$. city 3. U . 3$. U . r . . U H. B. 8 . . : B. . 8. 1$. city 2. B. H U. 4$. . r. 3$. > . . . . . 4$. Fig. 1. Taxicab Deterministic MDP and its matrix. #. k. . . Thus, G the `^acb and. $.
(81) % . represents the optimal average reward (per time unit), starting from G . Assuming that ` acb commute in (6), we get:. . #. . G.
(82). m s. . $ 'm & `^( acb k % . . m s. . J. K
(83) . . (7). k. .
(84). J 3 is in the conventional semiring , algebra). (
(85) admits an eigenvector in the semiring:. . " . . i.e.. . . (
(86). (this is an hybrid formula, is in the To evaluate (7), let us assume that the matrix.
(87). . s(the eigenvector. . . . ".
(88). ". (8). NI. must be nonidentically ,
(89) is the eigenvalue). Let us assumes that. and have only finite entries. Then, there exist two finite constants such that . . *
(90) s m m m s m A
(91) m orwith In notation, . Then
(92). the conventional notation: . k.
(93). . . . m s. . k. .
(94). <. (9). We easily deduce from (9) the following.. s. #. FACT 2 (“E IGENELEMENTS = O PTIMAL R EWARD AND P OLICY ”). If the final reward is finite,
(95) and if has a finite eigenvector with eigenvalue
(96) , the optimal average reward G is a constant
(97) . An optimal control is obtained by (independent of the starting pointU G ),Mj*equal to the eigenvalue j
(98) $Mj
(99) playing in state any such that and g , where is in the arg max of (8) at state .. . !. " . . . The existence of a finite eigenvector is characterized in Theorems 11 and 15 below. We will not discuss here the extension of these results to the infinite dimensional case (e.g. (1)), which is one of the major themes of Idempotent Analysis [31]. Let us just mention that all the results presented here admit or should admit infinite dimensional generalizations, presumably up to important technical difficulties. There is another much simpler extension, to the (discrete) semi-Markov case, which is worth be ing mentioned. Let us equip the above MDP with an additional map J q % ;. . . INRIA.
(100) Methods and Applications of (max,+) Linear Algebra . . fe . G. . 7. 0
(101) Mj f e(
(102)
(103) j i e(
(104). e. e4 ,. represents the physical time elapsed between decision and decision , when control is chosen. This is very natural in most applications (for the taxicab example, the times of the different possible journeys in general differ). The optimal average reward per time unit now writes:. # . G.
(105). . % . " . " . . Of $course, specialization >j*
(106) the _ , and for , g . . mo#p. `^acv b $ m &` ( acb . fe. U . >j i e(
(107) $
(108). "
(109) . . r m#o G p. . r. ie. . .. G. . s f k G
(110) >j i e(
(111) $
(112).
(113)
(114). <. gives the original problem (6). Let us define. v . U ! Mj. U " v ` Wa p b U " v W p . <.
(115). " . (10) . . >j. . $ *
(116). (11). Arguing as in the Markov case, it is not too difficult to show the following. FACT 3 (G ENERALIZED S PECTRAL P ROBLEM ized spectral problem.
(117) " . . s. . S EMI -M ARKOV P ROCESSES ). If the general-. FOR .
(118). _. . . ". . (12). #. .
(119). has a finite solution , and if is finite, thenj the optimal average reward is_ G
(120) , for all G . An optimal control is obtained by playing any in the arg max of (11), with in the arg max of (12), when in state .. . . Algebraically, (12) is nothing but a generalized spectral problem. Indeed, with an obvious definition of the matrices , we can write:. .
(121). 2.2. " " . . . where &. <. (13). An Algebra for Asymptotics. In Statistical Physics, one looks at the asymptotics when the temperature S tends to zero of the spectrum of transfer matrices, which have the form. b r " r " Q. " . $ . . . . . S. <.
(122) $
(123). /. . The real parameters represent potential terms plus interaction energy Q(
(124) terms (when two adjacent 3 sites are in states and , respectively). The Perron eigenvalue determines the free energy Q Q
(125) Q 1Q
(126) * S
(127) per site . Clearly, is an eigenvalue of in the semiring , defined in Table 1. I(
(128) )(
(129) % Q [ J Q [ X I , the Let denote the maximal eigenvalue of . Since following result is natural.. . 3. . !. . . $ &. The Perron eigenvalue , 3 of a matrix with nonnegative entries is the maximal eigenvalue associated with a nonnegative eigenvector, which is equal to the spectral radius of .. RR n ˚ 3088.
(130) 8. St´ephane Gaubert , Max Plus. . FACT 4 (PQ ERRON F ROBENIUS A SYMPTOTICS ). The asymptotic growth rate of the Perron eigen *
(131) value of is equal to the maximal eigenvalue of the matrix : Q. . $&. % . [J S. (
(132). This follows easily from the Q Q eigenvector of also satisfies. ) (
(133). Q. .
(134). . I. . . <
(135). (14). spectral inequalities (24),(25) below. The normalized Perron. $ eigenvector of $&. Q. j. $ . [J. % . S. . Q. "
(136). " j. . . . where is a special which has been Q(
(137) characterized recently by Akian, as sum of exponentials have been Bapat, and Gaubert [1]. Precise asymptotic expansions of given, some of the terms having combinatorial interpretations. ) (
(138) More generally, algebra arises almost everywhere in asymptotic phenomena. Often, (
(139) the algebra is involved in an elementary way (e.g. when computing exponents of Puiseux expansions using the Newton Polygon). Less elementary applications are WKB type asymptotics (see [31]), which are related to Large Deviations (see e.g. [17]). 2.3. An Algebra for Discrete Event Systems
(140). )(
(141). The algebra is popular in the Discrete Event Systems community, since linear dynamics correspond to a well identified subclass of Discrete Event Systems, with only synchronization 6 phenomena, called Timed Event Graphs. Indeed, consider a system with repetitive tasks. We
(142) ase ie* sume that the -th execution of task (firing of transition ) has to wait time units for the -th execution of task . E.g. tasks represent the processing of parts in a manufacturing system, represents an initially available stock, and represents a production or transportation time.. . " . " . " ie. (
(143). FACT 5 (T IMED E VENT G RAPHS ARE rence of an event in a Timed Event Graph, G G. " fe. (
(144). . (
(145). " . . " " . L INEAR S YSTEMS ). The earliest date of occur, satisfies. G. ie . . . " . <.
(146). . (15). Eqn 15 coincides with the value iteration of the deterministic semi-Markov Decision Process in 2.1, that we only wrote in the Markov version (3). Therefore, the asymptotic behavior of (15) can be dealt
(147) with as in 2.1, using spectral theory. In particular, if the generalized spectral problem .
(148) % e ie(
(149) o.
(150). has a finite solution
(151) , then
(152) 1 r J G , for all (
(153) is the cycle time, or inverse of the asymptotic throughput). The study of the dynamics (15), and of its stochastic) [2], and non-linear extensions [11,23] (fluid Petri Nets, minmax functions), is the major
(154) theme of discrete event systems theory. Another-linear model is that of heaps of pieces. Let denote a set of positions or resources < <<(]6 (say ). A piece (or task) is a rigid (possibly non connected) block, represented
(155) geometrically by a set of occupied positions (or requested resources) , a lower con
(156)
(157)
(158)
(159) tour (starting time) , an upper contour (release time) S , . " . ". . " . . . $. . . & (. . ". . . INRIA.
(160) Methods and Applications of (max,+) Linear Algebra. 9. , S . The piece
(161) corresponds to the region of the J plane: . , which means that task requires the set of re J operators) #S . sources (machines, processors, , and that resource is used from time . to time S . A piece can be translated vertically of any , which gives the new region defined. S We can execute a task earlier or later, but we cannot change by . whichS are . invariants S the differences of the task. A ground is< a<row A
(162) e < o. or initial condition , vector . Resource becomes initially available at time . If we drop pieces r in this order, on the ground (letting the pieces fall down according to the gravity, forbidding horizontal translations, and rotations, as in the famous Tetris NI call game,< <see < o Fig 2), we obtain what we
(163) , a heap of pieces. The upper contour G of the heap # is the row vector in r whose -th component is equal to the position of the top of the highest piece occupying resource . .
(164). G . Physically, gives the makespan The height of the heap is by definition (= completion time) of the sequence of tasks , and G is the release time of resource I.
(165) , With each piece a set of pieces , we associate the matrix within .
(166). . ! #S if , and for diagonal entries not in (other. . such that . .
(167). .
(168). .
(169). .
(170).
(171). . . .
(172).
(173). . . . . .
(174) .
(175). .
(176).
(177). .
(178) . . .
(179). .
(180). . .
(181).
(182). .
(183). .
(184).
(185).
(186). .
(187). . (.
(188). . . .
(189). (.
(190). .
(191). . .
(192). . .
(193). .
(194).
(195).
(196). . .
(197).
(198). .
(199).
(200). .
(201). entries are ). The following result was found independently by Gaubert and Mairesse (in [24]), and Brilman and Vincent [6].. . (
(202). . .
(203). (.
(204). FACT 6 (T ETRIS GAME<< IS< LINEAR ). The upper contour G and the height of the (
(205) heap of pieces . o , piled up on the ground , are given by the products:. (. !. . r. G. .
(206). . . r. . denotes the column vector indexed by. #". . << <
(207). . . . o . .
(208) . .
(209). ! . .G.
(210). . !. with entries ).. ! (.
(211). In algebraic terms, the height generating series is rational over the (max,+) semiring ( is the free monoid on , basic properties of rational series can be found e.g. in [38]).. . a. . c a b. c a. b. a. U . . .
(212).
(213). . ( . . . . B . - .
(214). . - . . . . ,. . . , . U. . . .
(215). (. ,.
(216). . . . 3 %. .
(217). % . . . % %. . . . U . 3 % ,S . . 3 3 ,S %. . %. . . .
(218). (. 3 ,S. .
(219). . 3 . . . .
(220). . . . 3 . 3 3 . . . . . 3. . Fig. 2. Heap of Pieces. Let us mention an open problem. If an infinite sequence of pieces H is taken at ran o r dom, say in an independent identically distributed way with the uniform distributionon , it is known << <. RR n ˚ 3088. << <. .
(221) 10. St´ephane Gaubert , Max Plus. . [14,2] that there exists an asymptotic growth rate
(222) % o $ &( e.
(223) 1. . . (. . < <<. . r. . q o. :
(224). a.s.. (16). The effective computation of the constant
(225) (Lyapunov exponent) is one of the main open problems in (max,+) algebra. The Lyapunov exponent problem is interesting for general random matrices (not
(226) S only for special matrices associated with pieces), but the heap case (even with unit height, .
(227) ) is typical and difficult enough to begin with. Existing results on Lyapunov exponents can be found in [2]. See also the paper of Gaujal and Jean-Marie in [24], and [6].. 2.4. An Algebra for Decision . +O P. . . " #$. % &(
(228). , has been invented by Simon [39] to solve The “tropical” semiring the following classical problem posed by Brzozowski: is whether a rational language N [ it decidable has the Finite Power Property (FPP): , N ' N N 33 3 N . The problem was solved independently by Simon and Hashiguchi.. !". . FACT 7 (S IMON ). The FPP problem for rational languages reduces to the finiteness problem for OP finitely generated semigroups of matrices with entries in , which is decidable.. . Other (more difficult) decidable properties (with applications to the polynomial closure and star height problems) are the finite section problem, which asks, given a finitely generated semigroup of matrices over the tropical semiring, whether the set of entries in position , is finite; and the more general limitation problem, which asks whether the set of coefficients of a ra +O P tional series in , with noncommuting indeterminates, is finite. These decidability results due to +O P Hashiguchi [25], Leung [29] and Simon [40] use structural properties of long optimal words in automata (involving multiplicative rational expressions), and combinatorial arguments. By comparison with basic Discrete Event System and Markov Decision applications, which essentially involve o e & semigroups with a single generator ( ), these typically noncommutative problems represent a major jump in difficulty. We refer the reader to the survey of Pin in [24], to [40,25,29], and to the references therein. However, essential in the understanding of the noncommutative case is (
(229) the one generator case, covered by the Perron-Frobenius theory detailed below. Let us point out an open problem. The semigroup of linear projective maps / / is the quotient of the semigroup of matrices / / by the proportionality relation:
(230)
(231) .
(232)
(233) (i.e..
(234)
(235) ). We ask: can we decide whether a finitely generated semigroup of linear projective maps is finite ? The motivation is the following. If the image of a finitely generated semigroup with
(236) I I generators morphism / / by e the canonical
(237) / / / / is finite, then the
(238) < << Lyapunov exponent
(239) a.s. limo o (same probabilistic assumptions r J r as for (16), L `^acb , by definition) can be computed from a finite Markov Chain on the associated projective linear semigroup [19,20].. " . . " . . . . " . . . . . " " . . . &(. . . . . . . . . . INRIA.
(240) Methods and Applications of (max,+) Linear Algebra. 3 3.1. . Solving Linear Equations in the. 11. 2
(241) . Semiring. A hopeless algebra?. The general system of. 6 )(
(242). -linear equations with unknowns G. ]*. G "() G. . . . I
(243). . /. . M] . (. 6. . r. . . << <* I
(244). /. G writes: <. (17). Unlike in conventional algebra, a square linear system ( generically solvable (consider I) isnot (.
(245) (
(246) 0G "
(247) G ,% , which has no solution, since for all G G G % ). , There are several ways to make this hard reality more bearable. One is to give general structural results. Another is to deal with natural subclasses of equations, whose solutions can be obtained by efficient methods. The inverse problem G" ( can be dealt with using residuation. The spectral (
(248) problem G#
(249) G (
(250) scalar) is solved using the analogue of Perron-Frobenius theory. solved via rational methods familiar in language theory The fixed point problem G G5( can be [ H (introducing the “star” operation X . &333 ). A last way, which has*the seduction of forbidden things, is to say: “certainly, the solution of 0GX 1G .% is G . For if this equation has no ordinary solution, the symmetrized equation (obtained by putting each occurrence of the unknown in
(251) the . other side of the equality) G .1 =G % has the unique solution G N is the requested solution.” Whether or not this argument is valid is the object of Thus, G symmetrization theory. All these approaches rely, in one way or another, on the order structure of idempotent semirings that we next introduce.. . . . . ". . 3.2. Natural Order Structure of Idempotent Semirings. An idempotent semiring can be equipped with the following natural order relation
(252). W. (. <. 2"(
(253) (. (18). We will write ( when
(254) ( and & T ( . The natural order endows with a sup-semilattice structure, for which "( & (2h` acb ( (this is the least upper bound of the set ( ),VU and
(255) , ( W ( isU the bottom laws preserve this order, i.e. ( U element). U U The semiring )(
(256).
(257) (
(258) () )
(259) % & ( .
(260) For the +O P semiring , the natural order
(261) coincides with the usual one. For the semiring , the natural order is the opposite of the usual one. Since addition coincides with the sup for the natural order, there is a simple way to define infinite sums, in an idempotent semiring, setting G u ` c a b G , for any possibly infinite (even of elements of , when the sup exists. We say that the idempotent non denumerable) family G semiring is complete if any family has a supremum, and if the product distributes over infinite sums.
(262) When is complete,
(263) becomes automatically a complete lattice, the greatest lower bound being *
(264) equal to G `^acb semiring is not complete (a FG . The complete idempotent semiring must have a maximal element), but it can be embedded in the complete NI semiring .. . . " ". " ". RR n ˚ 3088. . " ". . ". . ". . .
(265) 12. 3.3. St´ephane Gaubert , Max Plus. Solving. . using Residuation. . .
(266) ( always does (take G # ). Thus, a natural way of In general, GL#( has no solution4, but G & attacking G *( is to relax the equality and study the set of its subsolutions. This can be formalized in terms of residuation [5], a notion borrowed from ordered sets theory. We say that a monotone map
(267) is residuated if for all , the set g from an ordered set to an ordered set G g G
(268) has a maximal element, denoted g . The monotone map g , called residual or residuated map of g , is characterized alternatively by g g Id g g Id. An idempotent semiring is residuated if the right and left multiplication maps
(269) G -G , , are residuated, for all G G , . A complete idempotent semiring is automatically residuated. We set. . . .
(270) . . def. ( . .
(271). .
(272). (. . . )(
(273). G. . . . DG
(274) .( . .
(275) . . (. I. . . . . def. . . . . . .
(276). (. . . . . G. F<. G
(277) .(. . . . .
(278). In the completed semiring , (
(279) ( is equal to ( when L
(280) T , and is. equal to if X
(281) . The residuated character is transfered from scalars to matrices as follows.. . . Proposition 2 (Matrix residuation). Let be a complete idempotent semiring. Let / . The
(282)
(283) def map
(284) is given by -G G / , is residuated. For any / ,
(285) . . " r /. In the case of. . ' I. . , this reads: . "
(286). I. . . r . . . with the convention dual to that of , +O P in (19) a matrix product in the semiring the opposite of .. . % &. " . . /.
(287) (. . . . . G . .
(288). . . . (19). G , for any % &(
(289). . ". . . . . . . We recognize , involving the transpose of. ). Let denote a complete idempotent semiring, and let % / , . has a solution iff in time 6 (scalar operaCorollary 3 allows us to check the existence of a solution G of GR . . Corollary 3 (Solving G / . The equation G . . .
(290). .
(291). ) *
(292). tions are counted for one time unit). In the case, a refinement (due to the total order) allows us to decide the existence of a solution by inspection of the minimizing sets in (19), see [15,44]. 4. +. $&%' + $. '. . It is an elementary exercise to check that the map *1 , 3 , 3 , is surjective (resp. injective) iff the matrix contains a monomial submatrix of size (resp. ), a very unlikely event — recall that a square matrix is monomial if there is exactly one non zero element in each row, and in each column, or (equivalently) if it is a product of a permutation matrix and a diagonal matrix with non zero diagonal elements. This implies that a matrix has a left or a right inverse iff it has a monomial submatrix of maximal size, which is the analogue of a well known result for nonnegative matrices [4, Lemma 4.3].. . . INRIA.
(293) Methods and Applications of (max,+) Linear Algebra. 3.4. . 13. Basis Theorem for Finitely Generated Semimodules over. . I
(294). A finitely generated semimodule #j <<<(>j NI
(295) of vectors of / :. r. . . . . . . ". . p.
(296). r. "j ". . / is the set of linear combinations of a finite family. r. <.
(297). <<(
(298). . . I . < . . 6. In matrix terms, can be identified to the column space or image of the J matrix A NI
(299) j <<< Mj def Im G G , . The row space of is the column space of "j Y r (the transpose of ). The family is a weak basis of if it is a generating family, minimal for inclusion. The following result, due to Moller [33] and Wagneur [42] (with variants) states that NI
(300) finitely generated subsemimodules of / have (essentially) a unique weak basis.. . . . . . ". . . . . I
(301). . Theorem 4 (Basis Theorem). A finitely generated semimodule basis. / has "aj weak << < >j , Any two weak bases have the same number of generators. For any two weak bases < <<* << <( D << <* r . , there exist invertible scalars
(302)
(303) and a permutation of such that j r r
(304) U W .. ". . " ". . The cardinality of a weak basis is called the weak rank of the semimodule, denoted rk . The weak column rank (resp. weak row rank) of the matrix is the weak rank of its column (resp. row) space. Unlike in usual algebra, the weak row rank in general differs from the weak column rank (this is already the case for Boolean matrices). Theorem 4 holds more in any idempotent semiring
(305) Wgenerally !
(306) satisfying +
(307) T + X
(308) ( and the following axioms: and , W ! , ( . The axioms needed to set up a general rank theory in idempotent semirings are not currently understood. Unlike in vector spaces, there exist finitely generated
(309) 6 semimodules weak rank, if the dimension of the ambient space is at least ; and not / of arbitrarily large I
(310) 6 all subsemimodules of / are finitely generated, even with * .. .
(311) .
(312) . . * .
(313). Example 5 (Cuninghame-Green [15],Th. 16.4). The weak column rank of the AJ. ". #. . . %. <<< % <<< % % <<< . . . %. . . . . . matrix. is equal to for all . This can be understood geometrically using a representation due to I
(314) Mairesse. We visualize the set of vectors with finite entries of a semimodule by the 8 0
(315) H subset of , obtained by projecting orthogonally, on any plane orthogonal to . Since
(316) is invariant by multiplication by any scalar
(317) , i.e. by the usual addition of the vector
(318)
(319)
(320) , the semimodule is well determined by its projection. We only loose the points with entries which are H sent to some infinite end of the plane. The semimodules Im Im H Im are shown on Fig 3. r by gray regions. 8 The generators are represented by bold points, and the semimodules The broken line j j between any two generators represents Im. . This picture should make it clear that a weak I
(321) H basis of a subsemimodule of may have 8 may have as many generators as a convex set of extremal points. The notion of weak rank is therefore a very coarse one.. . . . RR n ˚ 3088. . . . .
(322) 14. St´ephane Gaubert , Max Plus. . Im. ,. + $. ,. Im. + $. ,. Fig. 3. An infinite ascending chain of semimodules of. . . . Im. 3 ,. + $. (see Ex. 5).. A NI
(323) Let . A weak basis of the semimodule Im can by a greedy algorithm. /
(324) 6 be computed .
(325) Let denote the -th column of , and let denote the J matrix obtained by delet
(326) ing column . We say that column of is redundant if Im , which can be checked by
(327) when 6 isH redundant, we do not change the semimodule Im . Corollary 3. Replacing by
(328) Continuing this process, we terminate in time with a weak basis.. . !. . . . . . . . . . . Application 6 (Controllability). The fact that ascending chains of semimodules need not stationnarize yields pathological features in terms of Control. Consider the controlled dynamical system:. . X I
(329). . G. . %.
(330). . . . . G. ie. (
(331). G. . T. X I
(332). ie. .
(333). . j i e(
(334). j i e(
(335) Ie.
(336) . . X I
(337). . . T M e. . . <<. . . <. (20) <. <<. k , and is a sequence of where / / ,
(338) d /
(339) control vectors. Givenj a state , the accessibility problem (in time ) asks whether there / fk
(340) d . Clearly, d is accessible in time k iff it belongs is a control sequence such that G to the < <<* m r
(341) . Corollary 3 allows us to decide m
(342)
(343) image of the controllability matrix d , for any the accessibility of . However, unlike in conventional algebra (in which Im m Im k . 6 / , thanksk to Cayley-Hamilton theorem), the semimodule of accessible states Im m may grow indefinitely as .. . 3.5. . . . . . . Solving. by Elimination. The following theorem is due to Butkoviˇc and Heged¨us [9]. It was rediscovered in [18, Chap. III].. . . . I
(344) Theorem 7 (Finiteness Theorem). Let
(345) . The set / neous system G
(346) G is a finitely generated semimodule.. . . of solutions of the homoge-. This is a consequence of the following universal elimination result.. . . . . Theorem 8 (Elimination & of Equalities in Semirings). Let denote an arbitrary Let T T semiring. and any row vectors ( , the hyperplane G
(347) / . If for any O -G
(348) (4G is a finitely generated semimodule, then G G1
(349) G is a finitely generated semimodule.. . . . . . INRIA.
(350) Methods and Applications of (max,+) Linear Algebra I. . 15. T.
(351). The fact that hyperplanes of are finitely generated can be checked by elementary means H (but the number of generators can be of order O I). Theorem 8 can be easily proved by induction on the number of equations (see [9,18]). In the case, the resulting naive algorithm has a doubly exponential complexity. But it is possible to incorporate the construction of weak bases in the algorithm, which much reduces the execution time. The making (and complexity analysis) of efficient algorithms for G1
(352) G is a major open problem. When only a single solution is needed, the algorithm of Walkup and Borriello (in [24]) seems faster, in practice. There is a more geometrical way to understand the
(353) H finiteness theorem. Consider the following I
(354) I
(355) correspondence between semimodules of r / (couples of row vectors) and / r (column vectors), respectively:. . . .
(356) #. . . . (.
(357). . r. /. H
(358). . .
(359). . r /
(360). . . H. . . . G DGR.(4G. A NI
(361) . . G. . . /. . r. . . . . -G *(4G. . . . . . ( I. .
(362)
(363). /. . . r. <. . . (21). . is a finitely generated semimodule (i.e. if all the row vectors ( belong Theorem 7 states that if to the row space of a matrix is finitely generated. Conversely, if is
(364) ) then, its orthogonal
(365) finitely generated, so does (since the elements ( of are the solutions of a finite system of
(366) linear equations). The orthogonal semimodule is exactly the set of linear equations ( = -G1 NI
(367) (4G satisfied by all the G . Is a finitely generated subsemimodule / r defined by its equations ? The answer is positive [18, Chap. IV,1.2.2]:. . . . . . . . . . Theorem 9 (Duality Theorem). For all finitely generated semimodules . .
(368). . . . . . . . NI.
(369). /. r, .
(370). . . In general, . The duality theorem is based on the following analogue of the Hahn
(371) semimodule, and T , Banach theorem, stated in [18]: if / r is a finitely generated
(372) $ I
(373)
(374) H there exist ( such that and . * T ( 1 G . ( G G r /
(375) The kernel of a linear operator should be defined as . When is M G KG the projector on the image of a linear operator
(376) , parallel to , defined? The answer is given in [12].. . . 3.6. Solving.
(377) . . . using Rational Calculus. . . Let denote a complete idempotent semiring, and let G GA5( is 9( , where the star operation is given by:. . ". ". ". def. . . . / /. /. . / . (. . . . / . The least solution of. <. (22). #"" . Moreover, G this is most well known (see e.g. [38]), and ( satisfies the equation G )G+ ( *. All.
(378) we will only insist on the features special to the case. We can interpret as the maximal 2 weight of a path from to of any length, in the graph associated with . We next characterize the NI
(379) I
(380) convergence of in ( is a priori defined in value which / / / / , but the breaks the semifield character of is undesired in most applications). The following fact is standard (see e.g. [2, Theorem 3.20]).. ". RR n ˚ 3088. . ". .
(381) 16. St´ephane Gaubert , Max Plus. . N . . .
(382). " . I. r. . Proposition 10. Let belong to / / . The entries[ of positive weight in the graph2 of . Then, 2 ".33 34 /. ". . " . 76. .
(383). iff there are no circuits with. .. can be computed in time ) ( using classical universal Gauss algorithms (see The matrix 8 .
(384) ie
(385) e.g.fe[21]). Special algorithms exist for the semiring. For instance, the sequence G 0
(386)
(387) 6 76
(388) 6 50
(389) G ( , G % * stationarizes before step (with G
(390) G ( ) iff ( is finite. This allows us to compute ( very simply. A complete account of existing algorithms can be found in [21].. . 3.7. ". ".
(391) . The. ". Perron-Frobenius Theory ) (
(392). The most ancient, most typical, and probably most useful results are relative to the spec
(393) tral problem G
(394) G . One might argue that 90% of current applications of algebra are based on a complete understanding of the spectral problem. The theory is extremely similar to the well (
(395) known Perron-Frobenius theory (see e.g. [4]). The case turns out to be very appealing, and (
(396) specslightly more complex than the conventional one (which is not surprising, since the tral problem is a somehow degenerate limit of the conventional one, see 2.2). The main discrepancy is the existence of two graphs which rule the spectral elements of , the weighted graph canonically 2 associated with a matrix , and one of its subgraphs, called critical graph. First, let us import the notion of irreducibility from the conventional Perron-Frobenius theory. We say that has access to if there is a path from to in the graph of , and we write . The
(397) classes of are the equivalence classes for the relation and . A matrix with a single class is irreducible. A class is upstream (equivalently is downstream ) if a node of has access to a node of . Classes with no other downstream classes are final, classes with no other upstream classes are initial.(
(398) The following famous result has been proved again and again, with various degrees of generality and precision, see [37,41,15,44,22,2,31].. . . . . . . . " . . " . ". Theorem 11 (“
(399) Perron-Frobenius Theorem”). An irreducible matrix has a unique eigenvalue, equal to the maximal circuit mean of :. . . I.
(400). . . . /. o#p. tr. o. . r.
(401) . " " " " . . . . r. o. . /. 33 3. e. . " ". <. . I
(402). /. /. (23). We have the following refinements in terms of inequalities [18, Chap IV], [3].. . . Lemma 12 (“Collatz-Wielandt Properties”). For any. Moreover, if. . . . .
(403). . .
(404). . . is irreducible, I .
(405). . . % &(.
(406). . I. j. j. . . . . N . . . .
(407).
(408). I . / . / . . . .
(409). /. /,. j . . #. . j.
(410)
(411). j
(412). j. .<. .<. (24). (25). INRIA.
(413) Methods and Applications of (max,+) Linear Algebra. 17. . . The characterization (25) implies in particular that, for an irreducible matrix , mal value of the linear program . . % &.
(414). $. s.t.. " . j. . ". j. . NI .
(415). is the opti-. <
(416). This was already by Cuninghame-Green [15]. The standard way76 to
(417) compute the maximal I noticed
(418) circuit mean is to use Karp algorithm [27], which runs in time 8 . The specialization of Howard algorithm (see e.g. [35]) to deterministic Markov Decision Processes with average reward, yields an algorithm whose average execution time is in practice far below that of Karp algorithm, but no polynomial bound is known for the execution time of Howard algorithm. Howard algorithm is also well adapted to the semi-Markov variants (12). Unlike in conventional Perron-Frobenius theory, an irreducible matrix may have several (non proportional) eigenvectors. The characterization of< the uses the notion of critical graph. An
(419) <<( eigenspace o
(420) whose mean weight attains the in (23). arc is critical if it belongs to a circuit $ r Then, the nodes are critical. Critical nodes and arcs form <the graph. A critical class is <<( critical denote a strongly connected component of the critical graph. Let the critical classes. Let def
(421)
(422) r I
(423) . I
(424) * (i.e. ). Using Proposition 10, r the existence of is guaranteed. If is in a critical class, we call the column V of critical. The following result can be found e.g. in [2,16].. . ! . . . . . . . " . . " . . . . . .
(425). ". " " ". ". Theorem 13 (Eigenspace). Let / / denote an irreducible matrix. The critical columns of span the eigenspace of . If we select only one column, arbitrarily, per critical class, we obtain a weak basis of the eigenspace.. ". . " " . $. " . . Thus, the cardinality of a weak basis is equal to the number of critical classes. For any two within V V the same critical class, the critical columns and are proportional. NI(
(426) We nexto show and the eigenvectors determine the asymptotic bee how the eigenvalue havior of as . U The cyclicity of a critical class is by definition the of the lengths of its circuits. The cyclicity of is the lcmof the cyclicities its critical classes. Let us pick arbitrar < <<*of ily an index within each critical class , for , and let. denote the column and row of index of (. are right and left eigenvectors of , respectively). The following result follows from [2].. . . . " . Theorem 14 (Cyclicity). Let that U. . . . . e. . I. . . [ .
(427). /. . /. U. . . be an irreducible matrix. There is an integer . . W. . . o . q. . . I .
(428). o . [. such (26). < where (27) [ p r NI H The matrix which satisfies ,o called the spectral projector of . " N.I J U o" inis conventional The cyclicity theorem, which writes q algebra, implies that where is the cyclicity of . Moreover, if . e. . W. o. I. . ,.
(429). o. . . . RR n ˚ 3088.
(430).
(431).
(432) 18. . o. St´ephane Gaubert , Max Plus. e. . I. . .
(433). . . . .
(434). . , independently of G / , and that a periodic regime o I
(435) ois attained in finite time.U The limit behavior is known a priori. Ultimately, the sequence visits << <( periodically accumulation points, which are r , where is the spectral projector of . The length of the transient behavior [ can be arbitrarily large. In terms of Markov Decision, Theorem 14 says that optimal long trajectories stay almost all the time on the critical graph (Turnpike H theorem). Theorem 14 is illustrated in Fig. 4, which shows the images of a cat (a region of the H plane) by the iterates of ( 8 , etc.),
(436) and , where G grows as. J. . . . . . . @ @ @ % % % * % EC
(437) & %C
(438) %%EC < (28) We have the spectral projector is rank one (its . Since has a unique critical H circuit, column and row spaces are lines). We find that : every point of the plane is sent in at most . . .
(439). two steps to the eigenline .G" interpretations exist for
(440) and .. Im. +. +. . . G , then it is translated by . +. .
(441). . . Im. Im. . + Fig. 4. A cat in a. at each step. Similar. . . dynamics (see (28)). ,- .0/(1243. . . . I(. .
(442). (
(443). . . Let us now consider a reducible matrix . Given a class , we denote by the j j eigenvalue of the restriction of the matrix to . The support of a vector is the set supp j implies . T . A set of nodes is closed if . We say that a class is final in if there is no other downstream class in .. " . ". . . . NI. .
(444). . . . Theorem 15 (Spectrum of reducible matrices). A matrix an eigenvector with / I/ #has D << < V6
(445)
(446) iff is closed,
(447) is equal to support and eigenvalue for any class that I
(448) is final in , and
(449) for any other class in .. . . The proof can be found in [43,18]. See also [3]. In particular, of initial classes are auto I eigenvalues
(450) (given by (23)) is also automatically matically eigenvalues of . The maximal circuit mean an eigenvalue of (but the associated eigenvector need not be finite). A weak basis of the eigenspace is given in [18, Chap. IV,1.3.4].. . . . . . Example 16 (Taxicab The matrix MDP, in > Fig 1, has classes, Iof the I shown U 9U
(451) EU ]eigenproblem). H
(452) taxicab H , H namely . Since , there are no finite . r. r. 8. r. INRIA.
(453) Methods and Applications of (max,+) Linear Algebra. . . . 19. . . . H ). The only other closed set is eigenvectors (which have I support
(454)
(455) > . r r , whichofis initial. Thus the only eigenvalue of . Let denote the restriction to U is
(456) VU
(457) EU ]U r r H . There are two critical circuits and H , and thus two critical classes U , H U r r r H A weak basis of the eigenspace of is given by the columns and (e.g.) of. . . . U. . U. "
(458). . . . U. r . H. U . r. . . %. %. . . r. H. . % . . . %. . B. Completing these two columns by a in row , we obtain a basis of the eigenspace of . The non existence of a finite eigenvector is obvious in terms of control. If such an eigenvector existed, by Fact 2, the optimal reward of the taxicab would be independent of the starting point. But, if the taxi driver starts from City 3, he remains blocked there with an income of $ per journey, whereas if he starts from any other node, he should clearly > either run indefinitely in City 1, either shuttle from the airport to City 2, with an average income of $ per journey (these two policies can be obtained by applying Fact 2 to the MDP restricted to , taking the two above eigenvectors).. r. The following extension to the reducible case of the cyclicity theorem is worth being mentioned.. " . Theorem 17 (Cyclicity, reducible case). Let$ N , and a family of scalars
(459) , . e. . [. . e. . . . . . I. . 6 U. . ,% .
(460). W. . /. / . There exist two integers . o" . U " . . and. U. . , such that. " o"
(461). q. [. . (29). " . The scalars
(462) " are taken from the set of eigenvalues of the Characterizations exist for and
(463) " I for all . If do not belong to the classes of . If belong to the same class ,
(464) o r J o" may have distinct accumulation points, same class, the theorem impliese that the sequence U U. $. .
Figure
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