Astrtion of Nondeterministi Automt Rong Su My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 1
Outline Motivtion Automton Astrtion Relevnt Properties Conlusions My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 2
Key Conepts of RW Supervisory Control Theory (SCT) Controllility Oservility Nonlokingness Cheking nonlokingness is omputtionlly intensive Let L m (S/G) = L m (G 1 ) L m (G n ) L m (S 1 ) L m (S r ) Let L(S/G) = L(G 1 ) L(G n ) L(S 1 ) L(S r ) Chek whether or not L m (S/G) = L(S/G) We hve the stte-spe explosion issue here! My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 3
A Few Attempts to Del with Nonlokingness Stte-feedk Control nd Symoli Computtion, e.g. supervisory ontrol of stte tree strutures (STS) Astrtion-Bsed Synthesis, e.g. oordinted modulr supervisory ontrol (MSC) hierrhil supervisory ontrol (HSC) Synthesis sed on Struturl Deoupling, e.g. interfe-sed supervisory ontrol (IBSC) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 4
Prolems Assoited with These Attempts STS is entrlized, not suitle for very lrge systems Current hierrhil nd modulr pprohes need oservers The oserver property is too strong! G 14 24 11 12 21 22 27 = {11,12,14,21,22,24,27} = {11,21} nd To mke P: * * n L m (G)-oserver we need = Interfes re very diffiult to design My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 5
Our Gol To define n strtion over (nondeterministi) FSAs, It hs the following property similr to wht n oserver hs, nmely for ny G nd n S whose lphet is the sme s (G), GS is nonloking if (nd only if) (G)S is nonloking It hs no speil requirement on trget lphet s n oserver does My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 6
Outline Motivtion Automton Astrtion Relevnt Properties Conlusions My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 7
Nondeterministi Finite-Stte Automton A finite-stte utomton G=(X,,, x 0, X m ) is nondeterministi if : X 2 X i.e stte my hve more thn one trnsition with the sme event lel x x 1 x 2 From now on we ssume ll utomt re nondeterministi My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 8
Automton Produt Let G i =(X i, i, i,x 0,i,X m,i )( i ) with i=1,2. The produt of G 1 nd G 2, written s G 1 G 2, is n utomton G 1 G 2 =(X 1 X 2, 1 2, 1 2, (x 0,1,x 0,2 ),X m,1 X m,2 ) where 1 2 :X 1 X 2 ( 1 2 )2 X1X2 is defined s follows, ( )(( x 1 2 1, x 2 ), ) : 1( x1, ) { x2} { x1} 2( x2, ) 1( x1, ) 2( x 2, ) if if if 1 2 1 2 1 2 My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 9
The Conept of Equivlene Reltion Given set X, let R e inry reltion on X, nmely R X X For ny (x,x)r, we write xrx. We sy R is n equivlene reltion on X, if R is reflexive, i.e. (xx) xrx R is symmetri, i.e. (x,yx) xry yrx R is trnsitive, i.e. (x,y,zx) xry yrz xrz Let E(X) e the olletion of ll equivlene reltions on X E(X) is omplete lttie My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 10
The Conept of Mrking Wek Bisimilrity Given G=(X,,,x 0,X m ), let, R X X e n equivlene reltion. R is mrking wek isimultion reltion over X with respet to if R X m X m (X X m ) (X X m ) For ll (x,x)r nd s *, if (x,s) then there exists s * suh tht (x,s) P(s)=P(s) (y(x,s))(y(x,s)) (y,y)r where P : * * is the nturl projetion The lrgest mrking wek isimultion is mrking wek isimilrity, written s My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 11
Automton Astrtion Let G=(X,,,x 0,X m ) nd For eh xx let [x] := {xx (x,x)}, nd X/ := {[x] xx}. G/ = (X,,,x 0,X m ) is n utomton strtion of G w.r.t. if X = X/, X m = {[x]x [x] X m }, x 0 = [x 0 ]X :X 2 X, where for ny [x]x nd, ([x],):={[x]x (y[x],y[x])(u,u(-) * ) y(y,uu)} My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 12
Exmple 0 1 2 3 4 5 2 3,4 0 1 5 u 6 = {,,,, u} = {, } 6 G G/ My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 13
Outline Motivtion Automton Astrtion Relevnt Properties Conlusions My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 14
Effet of Silene Pths 2 G 0 1 G 0 1 2 = {,} = {} = {,} = {} G/ 0 1 G/ 0 1 2 My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 15
Astrtion my rete unwnted ehviours. To void this, we introdue the onept of stndrdized utomt. My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 16
The Stndrdized Automt Suppose G = (X,,,x 0,X m ). Bring in new event symol. will e treted s unontrollle nd unoservle. An utomton G = (X,{},,x 0,X m ) is stndrdized if x 0 X m (xx) (x,) x = x 0 () (x 0,)= (xx)({}) x 0 (x,) Let () e the olletion of ll stndrdized utomt over. My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 17
Exmple of Stndrdized Automton d G : efore stndrdiztion G : fter stndrdiztion d My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 18
Mrking Awreness G() is mrking wre with respet to, if (xx-x m )(s * ) (x,s)x m P(s) where P: * * is the nturl projetion. My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 19
Automton Astrtion vs Nturl Projetion Let B(G) ={s * (x(x 0,s))(s * ) (x,s) X m = }. Let N G (x) ={s * (x,s) X m }. In prtiulr, N(G):= N G (x 0 ). Proposition 1 Let G(),, nd P: * * e the nturl projetion. Then P(B(G)) B(G/ ) nd P(N(G))=N(G/ ) i.e. utomton strtion my potentilly rete more loking ehviours If G is mrking wre with respet to ', then P(B(G))= B(G/ ) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 20
/ () () / () () N N B B P * 2 * 2 P * 2 * 2 When G is mrking wre with respet to ' My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 21
Nonloking Preservtion nd Equivlene Let G 1, G 2 (). G 1 is nonloking preserving w.r.t. G 2, denoted s G 1 G 2, if B(G 1 )B(G 2 ) nd N(G 1 )=N(G 2 ) For ny sn(g 1 ), nd x 1 1 (x 1,0,s), there exists x 2 2 (x 2,0,s) suh tht N G2 (x 2 ) N G1 (x 1 ) x 1 X 1,m x 2 X 2,m G 1 is nonloking equivlent to G 2, denoted s G 1 G 2, if G 1 G 2 nd G 2 G 1 My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 22
Proposition 2 (Nonloking Invrine under produt) For ny, G 1,G 2 () nd G 3 (), if G 1 G 2 then G 1 G 3 G 2 G 3 if G 1 G 2 then G 1 G 3 G 2 G 3 My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 23
Proposition 3 (Nonloking Invrine under strtion) For ny nd G 1,G 2 (), if G 1 G 2 then G 1 / G 2 / if G 1 G 2 then G 1 / G 2 / My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 24
Proposition 4 (Chin Rule of Automton Astrtion) Suppose nd G(). Then (G/ )/ G/. My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 25
Proposition 5 (Distriution of Astrtion over Produt) Let G i ( i ), where i=1,2, nd 1 2. If 1 2, then (G 1 G 2 )/ (G 1 / 1 )(G 2 / 2 ). If 1 2 nd G i (i=1,2) is mrking wre w.r.t. i, then (G 1 G 2 )/ (G 1 / 1 )(G 2 / 2 ) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 26
Exmple 1 0 1 2 3 0 1 G 1 G 2 1 ={,,,} 2 ={,} ={,} 0 1 2 3 0 1,2 3 G 1 G 2 (G 1 G 2 )/ My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 27
Exmple 1 (ont.) 0 1 2 3 0 1 G 1 G 2 0 1 2 3 0 1 0 1 2 3 G 1 / 1 G 2 / 2 (G 1 / 1 ) (G 2 / 2 ) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 28
Exmple 1 (ont.) Clerly, (G 1 G 2 )/ (G 1 / 1 )(G 2 / 2 ) Thus, the ondition of mrking wreness is only suffiient. My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 29
Exmple 2 G 1 G 2 1 ={,} 2 ={,,} ={,,} My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 30
Exmple 2 (ont.) G 1 G 2 (G 1 G 2 )/ My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 31
Exmple 2 (ont.) G 1 / 1 G 2 / 2 (G 1 / 1 )(G 2 / 2 ) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 32
Clerly, (G 1 G 2 )/ (G 1 / 1 )(G 2 / 2 ) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 33
Exmple 3 G 1 G 2 1 ={,,} 2 ={,} ={,} My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 34
Exmple 3 (ont.) G 1 G 2 (G 1 G 2 )/ My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 35
Exmple 3 (ont.) G 1 / 1 G 2 / 2 (G 1 / 1 )(G 2 / 2 ) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 36
Clerly, (G 1 G 2 )/ (G 1 / 1 )(G 2 / 2 ) But, it is not true tht (G 1 G 2 )/ (G 1 / 1 )(G 2 / 2 ) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 37
Exmple 3 (revisit) G 1 G 2 1 ={,,} 2 ={,} ={,,} My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 38
Exmple 3 (ont.) G 1 G 2 (G 1 G 2 )/ My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 39
Exmple 3 (ont.) G 1 / 1 G 2 / 2 (G 1 / 1 )(G 2 / 2 ) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 40
We n hek tht, (G 1 G 2 )/ (G 1 / 1 )(G 2 / 2 ) My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 41
Min Result Theorem: Given nd, let G() nd S(). Then B((G/ )S)= B(GS)= G is mrking wre w.r.t. [B((G/ )S)= B(GS)=] My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 42
A Computtionl Chllenge Let { i ii={1,2,,n}} e olletion of lol lphets. For ny JI, let J := jj j. Let G i ( i ) for eh ii, nd I. We wnt to ompute ( ii G i )/ effiiently. My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 43
Sequentil Astrtion over Produt (SAP) For k=1,2,,n, J(k) := {1,2,,k} nd T(k) := Jk ( I-Jk ) If k=1 then W 1 :=G 1 / T(1) If k>1 then W k :=(W k-1 G k )/ T(k) Proposition 6 Suppose W n is omputed y SAP. Then ( ii G i )/ W n. My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 44
Conlusions Advntges of this pproh It possesses the good spets of n oserver It does not hve the d spets of n oserver Potentil disdvntges of this pproh Astrtion retes more trnsitions, whih might omplite synthesis The mrking wreness ondition is suffiient ut not neessry My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 45