The Cayley-Hamilton Theorem For Finite Automata. Radu Grosu SUNY at Stony Brook
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1 The Cyley-Hmilton Theorem For Finite Automt Rdu Grosu SUNY t Stony Brook
2 How did I get interested in this topic?
3 Convergence of Theories Hyrid Systems Computtion nd Control: - convergence etween control nd utomt theory. Hyrid Automt: n outcome of this convergence - modeling formlism for systems exhiiting oth discrete nd continuous ehvior, - successfully used to model nd nlyze emedded nd iologicl systems.
4 voltge(mv) Lck of Common Foundtion for HA v V E x Ax Bu v Cx v V R Mode dynmics: - Liner system (LS) v V / R t 0 Mode switching: - Finite utomton (FA) v V U Stimulted s / di t s v V U Different techniques: - LS reduction - FA minimiztion time(ms) LS & FA tught seprtely: No common foundtion!
5 Min Conjecture Finite utomt cn e conveniently regrded s time invrint liner systems over semimodules: - liner systems techniques generlize to utomt Exmples of such techniques include: - liner trnsformtions of utomt, - minimiztion nd determiniztion of utomt s oservility nd rechility reductions - Z-trnsform of utomt to compute ssocited regulr expression through Gussin elimintion.
6 Miniml DFA re Not Miniml NFA (Arnold, Dicky nd Nivt s Exmple) x 1 x 2 c x 3 x 4 c x 1 x 2 x 3 c L = (* + c*)
7 Miniml NFA: How re they Relted? (Arnold, Dicky nd Nivt s Exmple) x 1 c x 2 x 3 x 4 c c x 5 x 1 c c x 2 x 3 x 4 c x 5 L = +c + +c + c+c No homomorphism of either utomton onto the other.
8 Miniml NFA: How re they Relted? (Arnold, Dicky nd Nivt s Exmple) x 1 c x 2 x 3 x 4 c c x 8 x 1 c c x 5 x 6 x 7 c x 8 Crrez s solution: Tke oth in terminl NFA. Is this the est one cn do? No! One cn use use liner (similrity) trnsformtions.
9 Oservility Reduction HSCC 09 (Arnold, Dicky nd Nivt s Exmple) A x 1 c x 2 x 3 x 4 c c x 5 A x 1 c c x 23 x 24 x 34 c x 5 Define liner trnsformtion x t = x t T: T x 1 x 2 x 3 x 4 x 5 x x x x x A = [AT] T (T 1 AT) x t 0 = x t 0 T C = [C] T (T 1 C)
10 Rechility Reduction HSCC 09 (Arnold, Dicky nd Nivt s Exmple) A x 1 c c x 2 x 3 x 4 c x 5 A x 1 c x 23 x 24 x 34 c c x 5 Define liner trnsformtion x t = x t T: T x 1 x 2 x 3 x 4 x 5 x x x x x A t = [A t T] T (T 1 A t T) x 0 t = x 0 t T C = [C] T (T 1 C)
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14 Oservility nd minimiztion
15 Finite Automt s Liner Systems Consider finite utomton M = (X,,,S,F) with: - finite set of sttes X, finite input lphet, - trnsition reltion X X, - strting nd finl sets of sttes S,F X
16 Finite Automt s Liner Systems Consider finite utomton M = (X,,,S,F) with: - finite set of sttes X, finite input lphet, - trnsition reltion X X, - strting nd finl sets of sttes S,F X Let X denote row nd column indices. Then: - defines mtrix A, - S nd F define corresponding vectors
17 Finite Automt s Liner Systems Now define the liner system L M = [S,A,C]: x t (n+1) = x t (n)a, x 0 = S y t (n) = x t (n)c, C = F
18 Finite Automt s Liner Systems Now define the liner system L M = [S,A,C]: x t (n+1) = x t (n)a, x 0 = S y t (n) = x t (n)c, C = F Exmple: consider following utomton: x 3 x 1 x 2 x 0 = A = C = 0
19 Semimodule of Lnguges ( * ) is n idempotent semiring (quntle): - (( * ),+,0) is commuttive idempotent monoid (union), - (( * ),,1) is monoid (conctention), - multipliction distriutes over ddition, - 0 is n nnihiltor: 0 = 0 (( * )) n is semimodule over sclrs in ( * ): - r(x+y) = rx + ry, (r+s)x = rx + sx, (rs)x = r(sx), - 1x = x, 0x = 0 Note: No dditive nd multiplictive inverses!
20 Semimodule of Lnguges ( * ) is n idempotent semiring (quntle): - (( * ),+,0) is commuttive idempotent monoid (union), - (( * ),,1) is monoid (conctention), - multipliction distriutes over ddition, - 0 is n nnihiltor: 0 = 0 (( * )) n is semimodule over sclrs in ( * ): - r(x+y) = rx + ry, (r+s)x = rx + sx, (rs)x = r(sx), - 1x = x, 0x = 0 Note: No dditive nd multiplictive inverses!
21 Oservility Let L = [S,A,C]. Oserve its output upto n-1: [y(0) y(1)... y(n-1)] = x 0 t [C AC... A n-1 C] = x 0 t O (1) If L opertes on vector spce: - L is oservle if: x 0 is uniquely determined y (1), - Oservility mtrix O: hs rnk n, - n-outputs suffice: A n C = s 1 A n-1 C + s 2 A n-2 C s n C If L opertes on semimodule: - L is oservle if: x 0 is uniquely determined y (1)
22 Oservility Let L = [S,A,C]. Oserve its output upto n-1: [y(0) y(1)... y(n-1)] = x 0 t [C AC... A n-1 C] = x 0 t O (1) If L opertes on vector spce: - L is oservle if: x 0 is uniquely determined y (1), - Oservility mtrix O: hs rnk n, - n-outputs suffice: A n C = s 1 A n-1 C + s 2 A n-2 C s n C (Cyley-Hmilton Theorem) If L opertes on semimodule: - L is oservle if: x 0 is uniquely determined y (1)
23 Oservility Let L = [S,A,C]. Oserve the output upto n-1: [y(0) y(1)... y(n-1)] = x 0 t [C AC... A n-1 C] = x 0 t O (1) If L opertes on vector spce: - L is oservle if: x 0 is uniquely determined y (1), - Oservility mtrix O: hs rnk n, - n-outputs suffice: A n C = s 1 A n-1 C + s 2 A n-2 C s n C If L opertes on semimodule: - L is oservle if: x 0 is uniquely determined y (1)
24 The Cyley-Hmilton Theorem ( A n = s 1 A n-1 + s 2 A n s n I )
25 Permuttions Permuttions re ijections of {1,...,n}: - Exmple: = {(1,2),(2,3),(3,4),(4,1),(5,7),(6,6),(7,5)} The grph G() of permuttion : - G() decomposes into: elementry cycles, The sign of permuttion: - Pos/Neg: even/odd numer of even length cycles, - P n / P n : ll positive/negtive permuttions.
26 Permuttions re ijections of {1,...,n}: - Exmple: = {(1,2),(2,3),(3,4),(4,1),(5,7),(6,6),(7,5)} The grph G() of permuttion : - G() decomposes into: elementry cycles 1 4 Permuttions The sign of permuttion: Pos/Neg: even/odd numer of even length cycles - P n / P n : ll positive/negtive permuttions.
27 Permuttions re ijections of {1,...,n}: - Exmple: = {(1,2),(2,3),(3,4),(4,1),(5,7),(6,6),(7,5)} The grph G() of permuttion : - G() decomposes into: elementry cycles 1 4 Permuttions The sign of permuttion : Pos/Neg: even/odd numer of even length cycles - P n / P n : ll positive/negtive permuttions
28 Eigenvlues in Vector Spces The eigenvlues of squre mtrix A: - Eigenvector eqution: x t A = x t s The chrcteristic eqution of A: eigenvector - The chrcteristic polynomil: cp A (s) = si-a - The chrcteristic eqution: cp A (s) = 0 The determinnt of A: eigenvlue - The determinnt: A = (A) - (A) P n P n, n i1 - Permuttion ppliction: (A) = A(i,(i))
29 Mtrix-Eigenspces in Vector Spces The eigenvlues of squre mtrix A: - Eigenvector eqution: x t (si-a) = 0 The chrcteristic eqution of A: - The chrcteristic polynomil: cp A (s) = si-a - The chrcteristic eqution: cp A (s) = 0 The determinnt of A: - The determinnt: A = (A) - (A) P n P n, n i1 - Permuttion ppliction: (A) = A(i,(i))
30 Mtrix-Eigenspces in Vector Spces The eigenvlues of squre mtrix A: - Eigenvector eqution: x t (si-a) = 0 The chrcteristic eqution of A: - The chrcteristic polynomil: cp A (s) = si-a - The chrcteristic eqution: cp A (s) = 0 The determinnt of A: - The determinnt: A = (A) - (A) P n P n, n i1 - Permuttion ppliction: (A) = A(i,(i))
31 Mtrix-Eigenspces in Vector Spces The eigenvlues of squre mtrix A: - Eigenvector eqution: x t (si-a) = 0 The chrcteristic eqution of A: - The chrcteristic polynomil: cp A (s) = si-a - The chrcteristic eqution: cp A (s) = 0 The determinnt of A: - The determinnt: A = (A) - (A) P n P n, n i1 - Weight of permuttion: (A) = A(i,(i))
32 The Cyley-Hmilton Theorem (CHT) A stisfies its chrcteristic eqution: cp A (A) = A = si-a = s s s- 33 si-a = s 3-33 s 2-12 s = 0 s = 33 s s A I = 33 A A I
33 The Cyley-Hmilton Theorem (CHT) A stisfies its chrcteristic eqution: cp A (A) = A = si-a = s s s- 33 si-a = s 3-33 s 2-12 s = 0 s = 33 s s A I = 33 A A I
34 The Cyley-Hmilton Theorem (CHT) A stisfies its chrcteristic eqution: cp A (A) = A = si-a = s s s- 33 si-a = s 3-33 s 2-12 s = 0 s = 33 s s A I = 33 A A I
35 The Cyley-Hmilton Theorem (CHT) A stisfies its chrcteristic eqution: cp A (A) = A = si-a = s s s- 33 si-a = s 3-33 s 2-12 s = 0 s = 33 s s A I = 33 A A I
36 The Cyley-Hmilton Theorem (CHT) A stisfies its chrcteristic eqution: cp A (A) = A = si-a = s s s- 33 si-a = s 3-33 s 2-12 s = 0 s = 33 s s A I = 33 A A I cycle cycle cycle cycle cycle
37 The Cyley-Hmilton Theorem (CHT) A stisfies its chrcteristic eqution: cp A (A) = 0 Implicit ssumptions in CHT: - Sutrction is ville - Multipliction is commuttive Does CHT hold in semirings? - Sutrction not indispensile (Rutherford, Struing) - Commuttivity still prolemtic
38 The Cyley-Hmilton Theorem (CHT) A stisfies its chrcteristic eqution: cp A (A) = 0 Implicit ssumptions in CHT: - Sutrction is ville - Multipliction is commuttive Does CHT hold in semirings? - Sutrction not indispensile (Rutherford, Struing) - Commuttivity prolemtic
39 CHT in Commuttive Semirings (Struing s Proof) Lift originl semiring to the semiring of pths: - Mtrix A is lifted to mtrix G A of pths A = G A = 0 (1,2) 0 (2,1) 0 (2,3) (3,1) 0 (3,3)
40 CHT in Commuttive Semirings (Struing s Proof) Lift originl semiring to the semiring of pths: - Mtrix A is lifted to mtrix G A of pths - Permuttion cycles lifted cyclic pths = {(1,2),(2,1)} = (1,2)(2,1)
41 CHT in Commuttive Semirings (Struing s Proof) Lift originl semiring to the semiring of pths: - Mtrix A is lifted to mtrix G A of pths - Permuttion cycles lifted cyclic pths Prove CHT in the semiring of pths: n n-q G A = n-q G A (CHT holds?) q 0 P q n q0 P q
42 Lift originl semiring to the semiring of pths: - Mtrix A is lifted to mtrix G A of pths - Permuttion cycles lifted cyclic pths Prove CHT in the semiring of pths: - Show ijection etween pos/neg products 0 2 G A = G A P 3 CHT in Commuttive Semirings (Struing s Proof) P 1 (3,3)(1,2)(2,1) (3,3)(1,2)(2,1) 1 (3,1) (1,2) (2,1) 3 2 (2,3) (3,3) 1 (3,1) (1,2) (2,1) 3 2 (2,3) (3,3)
43 CHT in Commuttive Semirings (Struing s Proof) Lift originl semiring to the semiring of pths: - Mtrix A is lifted to mtrix G A of pths - Permuttion cycles lifted cyclic pths Prove CHT in the semiring of pths: - Show ijection etween pos/neg products Port results ck to the originl semiring: - Apply products: (A) - Pth ppliction: ( 1... n )(A) = A( 1 )...A( n )
44 CHT in Idempotent Semirings Lift originl semiring to the semiring of pths: - Mtrix A: order in pths importnt - Permuttion cycles: rottions re distinct
45 CHT in Idempotent Semirings Lift originl semiring to the semiring of pths: - Mtrix A: order in pths importnt - Permuttion cycles: rottions re distinct = {(1,2),(2,1)} = (1,2)(2,1) (2,1)(1,2)
46 CHT in Idempotent Semirings Lift originl semiring to the semiring of pths: - Mtrix A: order in pths importnt - Permuttion cycles: rottions re distinct Prove CHT in the semiring of pths: - Products G n- : cycles to e properly inserted
47 CHT in Idempotent Semirings Lift originl semiring to the semiring of pths: - Mtrix A: order in pths importnt - Permuttion cycles: rottions re distinct Prove CHT in the semiring of pths: - Products G n- : cycles to e properly inserted G n- = G n- + G G n G n-
48 CHT in Idempotent Semirings Lift originl semiring to the semiring of pths: - Mtrix A: order in pths importnt - Permuttion cycles: rottions re distinct Prove CHT in the semiring of pths: - Products G n- : cycles to e properly inserted Port results ck to the originl semiring: - Apply products: G n- (A)
49 Theorem: G n n- = G A Proof: CHT in Idempotent Semirings n q1 n P q LHS RHS: Let LHS - Pidgeon-hole: hs t lest one cycle in s - Structurl: is simple cycle of length k - Remove in : [s/ ] is in G n- - Shuffle-product: G n- reinserts RHS LHS: Let RHS - No wrong pth: The shuffle is sound - Idempotence: Tkes cre of multiple copies
50 Theorem: G n n- = G A Proof: CHT in Idempotent Semirings n q1 n P q LHS RHS: Let LHS - Pidgeon-hole: hs t lest one cycle in s - Structurl: is lso simple cycle - Remove in : [s/ ] is in G n- - Shuffle-product: G n- reinserts RHS LHS: Let RHS - No wrong pth: The shuffle is sound - Idempotence: Tkes cre of multiple copies
51 Theorem: G n n- = G A Proof: CHT in Idempotent Semirings n q1 n P q LHS RHS: Let LHS - Pidgeon-hole: hs t lest one cycle in s - Structurl: is lso simple cycle - Remove in : [s/ ] is in G n- - Shuffle-product: G n- reinserts RHS LHS: Let RHS - No wrong pth: The shuffle is sound - Idempotence: Tkes cre of multiple copies
52 CHT in Idempotent Semirings Define: (i,i) = if (i,i) = 0 0 if (i,i) = Theorem: clssic CHT cn e derived y using: - G n- n- n- = G + G n- n- - ppliction of CHT to G nd G Mtrix CHT: cn e regrded s constructive version of the pumping lemm.
53 CHT in Idempotent Semirings Define: (i,i) = if (i,i) = 0 0 if (i,i) = Theorem: clssic CHT cn e derived y using: - G n- n- n- = G + G n- n- - ppliction of CHT to G nd G Mtrix CHT: cn e regrded s constructive version of the pumping lemm.
54 CHT in Idempotent Semirings Define: (i,i) = if (i,i) = 0 0 if (i,i) = Theorem: clssic CHT cn e derived y using: - G n- n- n- = G + G n- n- - ppliction of CHT to G nd G Mtrix CHT: cn e regrded s constructive version of the pumping lemm.
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56 Finite Automt s Liner Systems Now define the liner system L M = [S,A,C]: x t (n+1) = x t (n)a, x 0 = S() y t (n) = x t (n)c, C = F() Exmple: consider following utomton: L 1 x 3 x 1 x 2 A = A() + A() A() = 0 1 0, x 0( ) = A() = 0 0 0, C( ) =
57 Oservility Let L = [S,A,C] e n n-stte utomton. It's output: 0 [ O t n-1 t 0 0 [y(0) y(1)... y(n-1)] = x C AC... A C] = x (1) L is oservle if x is uniquely determined y ( 1). Exmple: the oservility mtrix O of L is: 1 O = n AC 3 ε x x x L 1 x 3 x 1 x 2
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