Reachability problems for products of matrices in semirings
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Reachability problems for products of matrices in semirings Stéphane Gaubert — Ricardo Katz. N° 4944 September 2003. ISSN 0249-6399. ISRN INRIA/RR--4944--FR+ENG. THÈME 4. apport de recherche.
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(144). -. ]. a i1 , . . . , a ik ∈ Σ r µ(ai1 ai2 . . . aik ) µ(ai1 +1 ai2 . . . aik ) . . . µ ¯ ◦ δ(ai1 . . . aik ) = µ(ar ai2 . . . aik ). ¡µ*/£. . . ) n,(r−1)n ) n,(r−1)n . . . ) n,(r−1)n ) (i1 −1)n,(r−1)n. . . ZKtU1hgdg<etZbdoqrh{jbix8egdhs{4jlUZKtr1Zx8bihgdbn«hkjbdmso(mK¡ p£4jlm i = r ¡µ/£ µ(a z) . ∀z ∈ Σ , µ ¯ ◦ δ(a z) = ¢ ¢ º ¾ ¿ º º ¿,¿ 7O3TVUbdtZYX1Z uu bdoQbitf&ms{l{lmk4Z u M{lmsX jUZHr{lmcm$m'} VT$D~/TVU¢ - hso u } ± c²TVU¢ - À¢¬T m!hocv!X[ms{lrUbitX µ : Σ → \ ²Zqhstltm*x8bihkjlZRjlUZ(X[mp{rUbdtX µˆ : \ {b, c} → u bdhs (µ(a ), . . . , µ(a )) . ¡ s/£ I µ ˆ(b) = , µ ˆ(c) = I T,mKtbnX[rgdbMv_omsjlhkjlbnmpotw4Z$$bdgdgetZ3hx8mpo/pZYo/jbdmso_mxYvcxYgnbixVbdo u Z8y*bdom&jUZ3gnZYjjlZY{"tm Σ ctm]jlUhkj Z8j"x¢ ZAtUhgdgetZfjlUZ_j{lbncbihgD¦hsx8jjlUhkjKhso/vª²ms{ u v ∈ {b, c} xwho1Z a ${lbnjjZw=o'a¡Meobia/eZYgdv=£Vhsatw ¡ - £ v = c b . . . c bc $UZY{lZ k ≥ 0 hso u 0 ≤ i , . . . , i $bnjUjUZKxYmsocsZwo/jbdmsoRjlUhkj v = c = c $UZwo k = 0 ¢ ¬g t O & v ∈ {b, c} ` W , c U / U¨¡ - £ ]_ %U%] 4º u bih [µ(a a . . . a ), . . . , µ(a a . . . a )]ˆµ(b ) . ¡ pp£ µ ˆ(v) = ms{$bdot0j"hoxYZs$UZYo r = 3 ¡¦/s£²tjlhkjlZwt²jlUhkj ) (i1 −1)n,n. ∗ r. r ) (r−1)n,n. r. 1. ) n,(r−1)n ) (r−1)n,(r−1)n. . ∗. ∗ r. rn×rn. ) (r−1)n,n. (r−1)n ) n,(r−1)n. nn. 1. r+1. 1. r ∗. 1. . r. 2. r+2. i1. . n×n. &". ik. ik+1. ik+1. k+1. i1. ∗. i1 i2 1 2. ik+1 k+1. i1 i2 r r+1. ik+1 r+k. k. /. . µ ˆ(c2 bc7 bc9 bcb2 c11 ) = . µ(a21 a72 a93 a1 a11 3 ). ). µ(a22 a73 a91 a2 a11 1 ). ) ). . ) ). µ ) ˆ(b 2 7 9 11 µ(a3 a1 a2 a3 a2 ). 5. ) .. mso1tb u ZY{²jlUZ tZYX[bds{lmser \ R!J¤ w( 7 V m gnm*x"¸ u bihsmpohg&XAhkjl{bix8Zt|$bnjU u bdhssmso1hgkgdm*x"¸ctm u bdX_Zwotbnmpo n YjlmssZYjUZw{$bnjU]jU⊂Z|s{lmser ³pZYoZw{lhjZ u cv3jlUZXAhkjl{bny B := µˆ(b)r ¡ bdt3bnocsZw{jlbngnZtbnoxYZ B = B £8¢²`cbdox8Z B B ⊂ (*Mmp{3hgdg (D, R), (D , R ) ∈ × ¤4Z U1Bh:sZ ¡ s/£ $UZY{lZ DRD R ∈ , RR ∈ . DRD R = DRD R RR , ¡ o¤msjUZw{A4mp{ u tYDjlUZQtZwX[bnp{mper¬m XAhkj{lbix8Zwt_ms jUZqMms{lX 4$UZY{lZ (D, R) ∈ × ¤|bit hqtZYX[b u bn{lZwx8jKr{lm u exFjKms c v ¨¢ £TVUZwo ms{lXfegdh!¡¦/s£ bitDR r{lmksZ u cvhsobdX[X[Z u bihkjlZAbno u exF´ jlbnmpo,jUhso¸*t jm¥¡ sp£F hso u jmqjUZRmstZY{lkhkjlbnmpoQjU1hkj D 7→ B DB hsxFj"tmso D ∈ cvxYv*x8gdbdx r&ZY{lXfejlhkjlbnmpo(ms u bihpmsohsg+gnm*x"¸*tw¢ .. −1. 0. 0. rn×rn. −1. T. 0. −1. 0. 0. 0. −1. . /. −1. ¨7¨ª© «¬¦¬,¬. 0. 0.
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