Linear Projectors in the Max-Plus Algebra Abstract 1
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Linear Projectors in the max-plus Algebra G. Cohen S. Gaubert J.-P. Quadrat Centre Automatique et Systemes` INRIA Ecole´ des Mines de Paris Rocquencourt Fontainebleau, France Le Chesnay, France [email protected] Stephane.Gaubert Jean-Pierre.Quadrat @inria.fr Abstract In this paper, we consider the linear projection problem. Consider three semimodules , , and two linear maps In general semimodules, we say that the image of a linear B, C: U X Y operator B and the kernel of a linear operator C are direct B C factors if every equivalence class modulo C crosses the . (2) image of B at a unique point. For linear maps represented U → X → Y by matrices over certain idempotent semifields such as the We say that im B and ker C are direct factors if for all x (max, )-semiring, we give necessary and sufficient con- , there is a unique im B such that Cx C. When∈ ditions+ for an image and a kernel to be direct factors. We X ∈ C = it is the case: 1. the map 5B : , x z, which is characterize the semimodules that admit a direct factor (or C 2 C XC → X 7→C C linear, satisfies (5B ) 5B , 5B B B, C5B C (5B equivalently, the semimodules that are the image of a lin- is the projector onto im=B, parallel to= ker C); 2.= we have ear projector): their matrices have a g-inverse. We give the isomorphism / ker C im B; in particular, if and simple effective tests for all these properties, in terms of are free finitelyX generated' semimodules, then theU linear matrix residuation. mapX B, which can be identified with a matrix, yields a parametrization of the ‘abstract’ object / ker C. In [5], a first answer to the projectionX problem was 1 Introduction given, in a nonlinear setting. If , , are complete lattices, and if B, C are (possiblyUnonlinearX Y ) residuated Classical linear control theory is built on a firm algebraic maps (see 2.2 below), it was shown that im B and ker C ground: vector spaces, and modules. There is some ev- are direct factors§ iff, setting 5 B (C B)] C, (where idence that the construction of a ‘geometric approach’ F ] denotes the residuated map= of a map F), we have of (max, )-linear discrete event systems, in the spirit + C 5 C and 5 B B. Even in the simplest case of Wonham [14], requires the analogue of module the- of =-linear operators= over free finitely generated - ory, for semimodules over idempotent semirings, such as Rmax Rmax semimodules (that is, when B and C are matrices with the ‘(max, ) semiring’ Rmax (R , max, ). ] + = ∪ {∞} + entries in max), the operator 5 B (C B) C is a By comparison with modules, the theory of semimodules R complicated object (a min-max function= in the sense of over idempotent semirings is an essentially fresh subject, Olsder and Gunawardena — see e.g. [10]), and the test in which even the most basic questions are yet unsolved. C 5 C, 5 B B, is computationally difficult (an Clearly, the image of a linear map F : example= of non trivial= direct factors was given in [5], for should be defined as usual: im F F(x) Xx → Y. ( )2, ( )3, the proof that im B But what is the kernel of F? Some= authors{ [7,| 12]∈ X de-} Rmax Rmax andU = kerYC=are direct factorsX = involved a tedious computa- fine ker F x F(x) ε , where ε is the zero tion of equivalence classes, together with a geometrical element of =.{ This∈ notionX | is essentially= } non pertinent for argument). -linearY maps, since ker F is in general trivial, even Rmax In this paper, we give a much simpler test for matrices: for ‘strongly’ non injective maps. im B and ker C are direct factors iff there exist two matri- Consider now the following alternative definition ces L, K such that B LCB and C CBK (we denote = = 2 with the same symbol B the matrix B and the linear map ker F (x, x 0) F(x) F(x 0) . (1) C = ∈ X = x Bx). Then, 5B LC BK. The existence of 7→ = = Clearly, ker F is a semimodule congruence, and we can the matrices K, L can be checked very simply (in polyno- define the quotient semimodule / ker F. Now, triv- mial time) using residuation of matrices (and not of linear ially, the canonical isomorphism theoremX holds: im F maps). / ker F. Thus, in general semimodules, (1) seems' to As a by-product, we solve the following problem, Xbe the appropriate definition. For control applications, it which was left open in [5]: given a matrix B, does there is indeed appropriate, since F typically represents an ob- exist a projector onto im B?; or, equivalently does there servation map, by which we wish to quotient some state exist a matrix C such that im B and ker C are direct fac- space. tors? The answer is positive iff B admits a g-inverse, that is, iff B BXB, for some matrix X. The existence of a For the sake of symmetry, we will complete an idem- g-inverse= can also be checked (simply) in polynomial time potent semifield with a maximal element (for “top”), using matrix residuation. which satisfies aD , a ,> and a The proofs are critically based on a linear extension a , a >=>( ) ∀ε ∈. WeD∪ denote {>} >= theorem, which states that a linear form F on a finitely this> dioid,=> and∀ we∈ willD∪{>} call it\{ the} top completionD =ofD∪{>}. n D generated subsemimodule of (Rmax) can be represented In , the product also distributes with respect to : by a row vector G: F(x) Gx. This result was D ∧ = a(c d) ac ad , proved by Kim [8, Lemma 1.3.2] for matrices with en- a, b, c , ∧ = ∧ (4) ∀ ∈ D (c d)a ca da tries in the Boolean semiring. Cao, Kim and Roush [4, ∧ = ∧ Th. 4.7.4] proved a variant of this result for the semiring (this property does not hold in general dioids). ([0, 1], max, ). As in the case of [8, 4], the proof con- sists in proving that the maximal linear subextension is an extension. This seems to require very strong properties on 2.2 Residuation the dioid (lattice distributivity, invertibility of product). Definition 1. We say that a dioid is residuated if Note that certain results pertaining to kernels rely upon D a linear extension theorem whereas this theorem is not re- 1. for all a and b in , x ax b admits a maxi- D { ∈ D | } quired to prove dual results on images mal element denoted a b; \ In 2, we introduce the algebraic notions used in the 2. x xa b admits a maximal element de- paper.§ In 3, we prove the linear extension theorem, and noted{ ∈bD/a;| } derive factorization§ theorems for linear maps. In 4, we § characterize direct factors. In 5, we relate the existence 3. ( , ) is a lattice. of projectors to the existence of§ g-inverses. D Then, the maximal element of x axc b ex- ists and can be denoted a b/c which{ ∈ canD be| read indiffer-} 2 Algebraic Preliminaries ently as (a b)/c or a (b/\c). The top\ completion\ of an idempotent semifield is S 1 S residuated, with a x a x if a is invertible, ε x , We briefly and informally recall the few algebraic results and x ε if \x = , (similar\ formulæ=> needed here. More details can be found in [1] for dioids for />\). = 6= > >\>=> and ordered sets, and in [7] for semirings and semimod- Consider the following linear equations in X: ules. A seminal reference in residuation theory is [3]. See also [6]. AX B , (5a) = A semiring is a set equipped with two laws , , XC D , (5b) such that: ( , ) is a commutativeS monoid (the zero is de- = noted ε); ( S, ) is a (possibly noncommutative) monoid AXC F , (5c) S = (the unit is denoted e); is right and left distributive over where X, A, ... are (possibly rectangular) matrices with ; and the zero is absorbing. A semiring in which non entries in a residuated dioid . zero elements have an inverse is a semifield. A semiring We extend the and / notationD to matrices: is idempotent if a , a a a. Idempotent semir- \ S ∀ ∈ S = ings are also called dioids. In this paper, we will mostly A B def X AX B , (6a) consider dioids such as Rmax, which is also a semifield. \ = { | } D/C def _ X XC D , (6b) = { | } 2.1 Order properties of dioids def A F/C _ X AXC F . (6c) \ = { | } A dioid (or more generally, an idempotent additive _ monoid) is equipped with the natural order relation: Explicitly, we have the following formulæ, which relate the residuation of matrices to the residuation of scalars: a b a b b . (3) ⇐⇒ = (A B)ij Aki Bkj , (7a) \ = \ Then, a b coincides with the upper bound a b for the k ∨ ^ natural order . Note that ε is the bottom element of : (D/C)ij Dil/Cjl , (7b) = x ,ε x. D l ∀ ∈ D ^ Moreover, if is a semifield, ( , ) is a lattice. In- (A F/C)ij Aki Fkl/C jl . (7c) D D 1 1 1 \ = \ deed, for all non zero a, b, a b (a b ) kl ^ (a 1 b 1) 1;ifaor b is zero, ∧a b= ε. This∨ shows that= To decide whether the matrix equations (5) have a so- ( , ) is a lattice. We say that the∧ idempotent= semifield lution, it suffices to check that the maximal subsolution isD distributive if the lattice ( , ) is distributive [1].