Finite Automata as Time-Inv Linear Systems Observability, Reachability and More

Transcription

1 Finite Automt s Time-Inv Liner Systems Oservility, Rechility nd More Rdu Grosu Deprtment of Computer Science, Stony Brook University Stony Brook, NY , USA Astrct. We show tht regrding finite utomt (FA) s discrete, time-invrint liner systems over semimodules, llows to: (1) express FA minimiztion nd FA determiniztion s prticulr oservility nd rechility trnsformtions of FA, respectively; (2) express FA pumping s property of the FA s rechility mtrix; (3) derive cnonicl forms for FAs. These results re to our knowledge new, nd they my support fresh look into hyrid utomt properties, such s minimlity. Moreover, they my llow to derive generlized notions of chrcteristic polynomils nd ssocited eigenvlues, in the context of FA. 1 Introduction The technologicl developments of the pst two decdes hve nurtured fscinting nd very productive convergence of utomt- nd control-theory. An importnt outcome of this convergence re hyrid utomt (HA), populr modeling formlism for systems tht exhiit oth continuous nd discrete ehvior [3,11]. Intuitively, HA re extended finite utomt whose discrete sttes correspond to the vrious modes of continuous dynmics system my exhiit, nd whose trnsitions express the switching logic etween these modes. HA hve een used to model nd nlyze emedded systems, including utomted highwy systems, ir trffic mngement, utomotive controllers, rootics nd rel-time circuits. They hve lso een used to model nd nlyze iologicl systems, such s immune response, io-moleculr networks, gene-regultory networks, protein-signling pthwys nd metolic processes. The nlysis of HA typiclly employs comintion of techniques orrowed from two seemingly disjoint domins: finite utomt (FA) theory nd liner systems (LS) theory. As consequence, typicl HA course first introduces one of these domins, next the other, nd finlly their comintion. For exmple, it is not unusul to first discuss FA minimiztion nd lter on LS oservility reduction, without ny forml link etween the two techniques. In this pper we show tht FA nd LS cn e treted in unified wy, s FA cn e conveniently represented s discrete, time-invrint LS (DTLS). Consequently, mny techniques crry over from DTLS to FA. One hs to e creful however, ecuse the DTLS ssocited to FA re not defined over vector spces, ut over more generl semimodules. In semimodules for exmple, the row rnk of mtrix my differ from its column rnk. R. Mjumdr nd P. Tud (Eds.): HSCC 2009, LNCS 5469, pp , Springer-Verlg Berlin Heidelerg 2009

2 FA s Time-Inv LS Oservility, Rechility nd More 195 In prticulr, we show tht: (1) deterministic-fa minimiztion nd nondeterministic-fa determiniztion [2] re prticulr cses of oservility nd rechility trnsformtions [5] of FA, respectively; (2) FA pumping [2] is property of the rechility mtrix [5] ssocited to n FA; (3) FA dmit cnonicl FA in oservle or rechle form, relted through stndrd trnsformtion. While the connection etween LS nd FA is not new, especilly from lnguge-theoretic point of view [2,6,10], our oservility nd rechility results for FA re to our knowledge new. Moreover, our tretment of FA s DTLS hs the potentil to led to knew understnding of HA minimiztion, nd of other properties common to oth FA nd LS. The rest of the pper is orgnized s follows. Section 2 reviews oservility nd rechility of DTLS. Section 3 reviews regulr lnguges, FA nd grmmrs, nd introduces the representtion of FA s DTLS. Section 4 presents our new results on the oservility of FA. Section 5 shows tht these results cn e used to otin y dulity similr results for the rechility of FA. In Section 6 we ddress pumping nd minimlity of FA. Finlly, Section 7 contins our concluding remrks nd directions for future work. 2 Oservility nd Rechility Reduction of DTLS Consider discrete, time-invrint liner system (DTLS) with no input, only one output, nd with no stte nd mesurement noise. Its [I,A,C], stte-spce description in left-liner form is then given s elow [5]: 1 x(0) = I, x T (t +1)=x T (t)a, y(t) =x T (t)c where x is the stte vector of dimension n, y is the (sclr) output, I is the initil stte vector, A is the stte trnsition mtrix of dimension n n, C is the output mtrix of dimension n 1, nd x T is the trnsposition of x. Oservility. A DTLS is clled oservle, ifitsinitil stte I cn e determined from sequence of oservtions y(0),...,y(t 1) [5]. Rewriting the stte-spce equtions in terms of x(0) = I nd the given output up to time t 1 one otins the following output eqution: [y(0) y(1)... y(t 1)] = I T [C AC... A t 1 C]=I T O t Let X e the stte spce nd W =spn[cac...a k C...]etheA-cyclic suspce (A-CS) of X generted y C. SinceC 0, the dimension of W is 1 k n, nd [CAC...A k 1 C]issisforW [7]. 2 As consequence, for ech t k, there exist sclrs 0... k 1 such tht A t C =(C) (A k 1 C) k 1. If k<nthen setting x T O t = k 1 i=0 (Ai C)f i (,...,x n )to0resultsink liner equtions f i (,...,x n )=0 in n unknowns, s A i C re linerly independent for i [0,k 1]. Hence, there exist n k linerly independent vectors x, such tht x T O t = 0, i.e. the dimension of the null spce N (O t )=N(O n )isn k nd the rnk ρ(o t )=ρ(o n )=k. Ifk = n then N (O n )={0}. ThesetN (O n ) is clled the unoservle spce of the system ecuse y(s) =0 for ll s if x(0) N(O), nd the mtrix O = O n is clled the oservility mtrix. 1 The left-liner representtion is more convenient in the following sections. 2 This fct is used y the Cyley-Hmilton theorem.

3 196 R. Grosu Bsis: [ n... n ] n 1 2 k Bsis: [q 1 q2... qk] Q 1 x T x T A A x T x T A A Q A = Q = ith column: Representtion of Aq i with respect to [ q... q ] [ q... q ] A = Q 1 A Q q 1 2 k q 1 2 k Fig. 1. Similrity trnsformtions If ρ(o)=k<n then the system cn e reduced to n oservle system of dimension k. The reduction is done s follows. Pick columns C,AC,...,A k 1 C in O nd dd n k linerly independent columns, to otin mtrix Q. Then pply the similrity trnsformtion x T = x T Q, to otin the following system: x T (t +1) = x T (t)aq = x T (t)q 1 AQ = x T (t)a y(t) = x T (t)c = x T (t)q 1 C = x T (t)c The trnsformtion is shown in Figure 1, where n i re the stndrd sis vectors for n-tuples (n i is 1 in position i nd 0 otherwise), nd q i re the column vectors in Q. Ech column i of A is the representtion of Aq i in the sis Q, ndc is the representtion of C in Q. Sinceq 1,...,q k is sis for W,llA i,j nd C i for j k i re 0. Hence, the new system hs the following [ form: ] [x T o (t +1) x T o (t +1)] = [x T o (t) x T Ao A o (t)] 12 [ 0 ] A o y(t) = [x T o (t) xt o (t)] Co 0 where x o hs dimension k, x o hs dimension n k, nda o hs dimension k k. Insted of working with the unoservle system [A, C] one cn therefore work with the reduced, oservle system [A o, C o ] tht produces the sme output. Rechility. Dully, the system S =[I,A,C] is clled rechle, ifitsfinl stte C cn e uniquely determined from y(0),...,y(t 1). Rewriting the sttespce eqution in terms of C, one otins the following eqution: [y(0) y(1)... y(t 1)] T =[I (I T A) T... (I T A t 1 ) T ] T C = R t C Since R t C =(C T Rt T )T nd (I T A t 1 ) T =(A T ) t 1 I, the rechility prolem of S =[I,A,T] is the oservility prolem of the dul system S T =[C T,A T,I T ]. Hence, in order to study the rechility of S, one cn study the oservility of S T insted. As for oservility, ρ(r t )=ρ(r n ), where n is the dimension of the stte spce X. MtrixR = R n is clled the rechility mtrix of S. Let k = ρ(r). If k = n then the system is rechle. Otherwise, there is n equivlence trnsformtion x T = x T Q which trnsforms S into rechle system S r =[I r, A r, C r ]ofdimensionk. The rechility trnsformtion of S is the sme s the oservility trnsformtion of S T. 3 FA s Left-Liner DTLS Regulr expressions. A regulr expression (RE) R over finite set Σ nd its ssocited semntics L(R) re defined inductively s follows [2]: (1) 0 RE nd

4 FA s Time-Inv LS Oservility, Rechility nd More 197 () () (c), 3 5 Fig. 2. () DFA M 1.()DFAM 2.(c)DFAM 3. L(0) = ; (2)ɛ RE nd L(ɛ)={ɛ}; (3)If Σ then RE nd L()={}; (4) If P, Q RE then: P + Q RE nd L(P + Q)=L(P ) L(Q); P Q RE nd L(P Q)=L(P ) L(Q); P RE nd L(P )= n N L(P ) n. The denottions of regulr expressions re clled regulr sets. 3 For exmple, the denottion L(R 1 ) of the regulr expression R 1 = +, is the set of ll strings (or words) consisting of more thn one repetition of or of, respectively. It is custom to write + for,sor 1 = Alnguge L is suset of Σ nd consequently ny regulr set is (regulr) lnguge. If two regulr expressions R 1,R 2 denote the sme set one writes R 1 = R 2.In generl, one cn write equtions whose indetermintes nd coefficients represent regulr sets. For exmple, X = Xα+ β. Itslest solution is X = βα [2]. The structure S =(Σ, +,, 0,ɛ)issemiring, s it hs the following properties: (1) A =(Σ, +, 0) is commuttive monoid; (2) C =(Σ,,ɛ)ismonoid; (3) Conctention left (nd right) distriutes over sum; (4) Left (nd right) conctention with 0 is 0. Mtrices M m n (S) over semiring with the usul mtrix sum nd multipliction lso form semiring, ut note tht in semiring there is no inverse opertion for ddition nd multipliction, so the inverse of squre mtrix is not defined in clssic sense. If V = M m 1 (A) ndsclr multipliction is conctention then R =(V, S, ) isns-right semimodule [10]. Finite utomt. A finite utomton (FA) M =(Q, Σ, δ, I, F ) is tuple where Q is finite set of sttes, Σ is finite set of input symols, δ : Q Σ P(Q) is the trnsition function mpping ech stte nd input symol to set of sttes, I Q is the set of initil sttes nd F Q is the set of finl sttes [2]. If I nd δ(q, ) re singletons, the FA is clled deterministic (DFA); otherwise it is clled nondeterministic (NFA). Three exmples of FAs re shown in Figure 2. Let δ extend δ to words. A word w Σ is ccepted y FA M if for ny q 0 I, thesetδ (q 0,w) F. ThesetL(M) of ll words ccepted y M is clled the lnguge of M. For exmple, L(M 1 )=L( ). Grmmrs. A left-liner grmmr (LLG) G =(N,Σ,P,S) is tuple where N is finite set of nonterminl symols, Σ is finite set of terminl symols disjoint from N, P N (N Σ) is finite set of productions 4 of the form A Bx or A x with A, B N nd x Σ {ɛ}, nds N is the strt symol [2]. Aword 1... n is derived from S if there is sequence of nonterminls N 1...N n in N such tht S N 1 1 nd N i 1 N i i for ech i [2,n]. The set L(G) of ll words derived from S is clled the lnguge of G. 3 The conctention opertor is usully omitted when writing regulr expression. 4 It is custom to write pirs (x, y) P s x y.

5 198 R. Grosu Equivlence. FAs, LLGs nd REs re equivlent, i.e. L = L(M) for some FA M if nd only if L = L(G) for some LLG G nd if nd only if L = L(E) forsome RE E [2]. In prticulr, given n FA M =(Q, Σ, δ, I, F ) one cn construct n equivlent LLG G =(Q {y},σ,p,y)wherep is defined s follows: (1) y q for ech q F,(2)q ɛ, forq I, nd(3)r q if r = δ(q, ). Replcing ech set of rules A α 1,...,A α n with one rule A α α n leds to more concise representtion. For exmple the LLG G 1 derived from M 1 is: ɛ, y +, +, + Ech nonterminl denotes the set of words derivle from tht nonterminl. One cn regrd G 1 s liner system S over REs. One cn lso regrd G 1 s discrete, time-invrint liner system (DTLS) S 1 defined s elow: x T (t +1)=x T (t)a, y(t) =x T (t)c I = ɛ 0 A = C = 0 ɛ ɛ The initil stte of S 1 is the sme s the initil stte of DFA M 1 nd it corresponds to the production ɛ of LLG G 1. The output mtrix C sums up the words in nd. It corresponds to the finl sttes of DFA M 1 nd to the production y + in LLG G 1.MtrixA is otined from DFA M 1 y tking v A ij if δ(x i,v)=x j nd A ij =0ifδ(x i,v) x j for ll v Σ. Theset of ll outputs of S 1 over time is t N {y(t)} = I T A C = L(M 1 ). Mtrix A cn e computed in R s descried in [6]. This provides one method for computing L(M). Alterntively, one cn use the lest solution of n RE eqution, nd pply Gussin elimintion. This method is equivlent to the rip-out-nd-repir method for converting n FA to n RE [2]. In the following, ll four equivlent representtions, RE, FA, LLG nd DTLS, of finite utomton, re simply referred to s n FA. The oservility/rechility prolem for n FA is to determine its initil/finl stte given y(t) for t [0,n 1]. In vector spces, these re unique if the rnk of O/R is n. Insemimodules however, the row rnk is generlly different from the column rnk. 4 Oservility Trnsformtions of FA Lck of finite sis. Let I e set of indices nd R e n S-semimodule. A set of vectors Q = {q i i I} in R is clled linerly independent if no vector q i Q cn e expressed s liner comintion j (I i) q j j of the other vectors in Q, for ritrry sclrs j S.Otherwise,Q is clled linerly dependent. The independent set Q is clled sis for R if it covers R, i.e. spn(q) =R [4]. E 0 E 1 E 2 C AC A 2 C E = O = ɛ 2 ɛ 2 x1

6 FA s Time-Inv LS Oservility, Rechility nd More 199 Now consider DFA M 1. Its oservility mtrix O is given ove. Ech row i of O consists of the words ccepted y M 1 strting in stte x i sorted y their length in incresing order. Ech column j of O is the vector A j 1 C, consisting of the ccepted words of length j 1 strtinginx i. The corresponding executions E 0, E 1 nd E 2 of DFA M 1 re lso given ove. The columns of O elong to the A-cyclic suspce of X generted y C (A- CS), which hs finite dimension in ny vector spce. In the S-semimodule R, where S is the semiring of REs, however, A-CS my not hve finite sis. For exmple, for DFA M 1 it is not possile to find REs r ij nd vectors A ij C such tht A i C = k j=1 C)r (Aij ij,fori j <i. Intuitively, strcting out the sttes of n FA from its executions, elimintes liner dependencies. The stte informtion included in E 1 nd E 2 llows to cpture their liner dependence: E 2 is otined from E 1 y sustituting the lst occurrence of sttes 2 nd 3 with the loops 22 nd 33, respectively. Regrding sustitution s multipliction with sclr, one cn therefore write E 2 = E 1 (22+33). Indexed oolen mtrices. In the ove multipliction we tcitly ssumed tht,e.g.(12)(33) = 0, ecuse -trnsition vlid in stte 3 cnnot e tken in sttes 1 nd 2. Treting independently the σ-successors/predecessors of n FA M =(Q, Σ, δ, I, F ), for ech input symol σ Σ, llows to cpture this intuition in stteless wy. Formlly, this is expressed with indexed oolen mtrices (IBM), defined s follows [12]: (1) C i =(i F ); (2) I i =(i I); (2) For ech σ Σ, (A σ ) ij =(δ(i, σ)=j); nd (3) A σ1...σ n = A σ1...a σn. For exmple, one otins the following mtrices for the DFA M 1 : I = 1 0 A = A = C = Indexing enforces word y word nlysis of cceptnce nd ensures, for exmple for M 1,thtA C = A (A C) = 0. Consequently, for every word w Σ the vector A w C hs row i equl to 1, if nd only if, w is ccepted strting in x i. Ordering ll vectors A wi C,forw i Σ i, in lexicogrphic order, results in oolen oservility mtrix O =[A w0 C...A wm C]. This mtrix hs n rows nd Σ n 1 columns. Its column rnk is the dimension of the A-CS W of the oolen semimodule B ecuse ll O ij B. Hence it is finite nd less thn 2 n 1. C A C A C A C A C A C A C O = x1 For exmple, mtrix O for M 1 is shown ove. It is esy to see tht vectors C, A C nd A C re independent. Moreover, A C = A C, A C = A C. Hence, ll vectors A w C,forw {, }, re generted y the sis Q =[C, A C, A C]. The structure of O is intimtely relted to the sttes nd trnsitions of the ssocited FA. Column C is the set of ccepting sttes, nd ech column A w C

7 200 R. Grosu (),c, 4 4 3,c () (c) Fig. 3. () FA M 4.()FAM 5.(c)FAM 6. is the set of sttes tht cn rech C y reding word w. Inotherwords,A w C is the set of ll w-predecessors of C. In the following we do not distinguish etween n FA nd its IBM representtion. The ltter is used to find pproprite ses for similrity trnsformtions nd prove importnt properties out FA. To this end, let us first review nd prove three importnt properties out the rnks of oolen mtrices. Theorem 1. (Rnk independence) If n 3 then the row rnk ρ r (O) nd the column rnk ρ c (O) of oolen mtrix O my e different. Proof. Consider the oservility mtrices 5 of FA M 4 nd M 5 shown in Figure 3: ρ r (O(M 4 )) = 3, ρ c (O(M 4 )) = 4, nd ρ r (O(M 5 )) = 4, ρ c (O(M 5 )) = 3. C A C A CA cc C A C A C O(M 4 )= x1 O(M 5 )= To ensure tht n FA is trnsformed to n equivlent DFA, it is convenient to introduce two more rnks: ρ d r (O) ndρd c (O). They represent the numer of distinct rows nd columns in O, respectively. Hence, these rnks consider only liner dependencies in which the sum is identicl to its summnds. Theorem 2. (Rnk ounds) The vrious row nd column rnks re ounded nd relted to ech other y the following inequlities: 1 ρ r (O) ρ d r (O) n, 1 ρ c(o) ρ d c (O) 2n 1, 1 ρ c (O) C n n/2 + n/2 Proof. First nd second inequlities re ovious. For the third oserve tht: (1) The set of comintions Ci n is independent; (2) It covers ll Cj n with j>i; (3) Only i 1 independent vectors my e dded to Ci n from ll Cj n,withj<i. The A-CS of B is very similr to the A-CS of vector spce. For exmple, let A k C e the set of ll vectors A w C with w = k. Then the following holds. Theorem 3. (Rnk computtion) If ll vectors in A k C re linerly dependent on sis Q for [CAC...A k 1 C], then so re ll the ones in A j C,withj k. Proof. The proof is identicl to the one for vector spces, except tht induction is on the length of words in A w C,ndA k C re sets of vectors. 5 We show only the sis columns of the oservility mtrix.

8 FA s Time-Inv LS Oservility, Rechility nd More 201,,, 4 4 () () Fig. 4. () FA M 7.()FAM 8. Oservility trnsformtions. The four rnks discussed ove suggest the definition of four equivlence trnsformtions x T = x T Q,whereQ consists of the independent (or the distinct), rows (or columns) in O, respectively. Ech stte q Q of the resulting FA M, is therefore suset of the sttes of M, ndech σ-trnsition to q in M is computed y representing its σ-predecessor A σ q in Q. Row-sis trnsformtions. These trnsformtions utilize sets of oservlyequivlent sttes in M to uild the independent sttes q Q of M. The length of the oservtions, necessry to chrcterize the equivlence, is determined y Theorem 3. The equivlence mong stte-oservtions itself, depends on whether ρ r (O) (liner equivlence) or ρ d r(o) (identity equivlence) is used. Using ρ r (O), one fully exploits liner dependencies to reduce the numer of sttes in M. For exmple, suppose tht = +,ndtht nd re independent. Then one cn replce the sttes, nd in M, with sttes q 1 = {, } nd q 2 = {, } in M. This generlizes to multiple dependencies, nd ech new stte q Q contins only one independent stte x. Consequently, the lnguge L(q)=L(x). Among the sttes q Q, thesttec is ccepting, nd ech q tht contins n initil stte in M is initil in M. The trnsitions mong sttes q Q re inferred from the trnsitions in M. The generl rule is tht q i σ qj, if ll sttes in q i re σ-predecessors of the sttes in q j. However, s Q is not necessrily column sis, the σ-predecessor of stte like q 1 ove, could e either or, which re not in Q. Extending to q 1 does not do ny hrm, s L(q 1 )=L( ). Ignoring stte does not do ny hrm either, s is covered y nd, possily on some other pth. These completion rules re necessry when computing the inverse of Q, i.e. representing AQ in Q to otin A. Theorem 4. (Row reduction) Given FA M with ρ r (O)=k<n,letR e row sis for O. Define Q =[q 1,...,q k ] s follows: for every i [1,k] nd j [1,n], if row O j is linerly dependent on R i then q ij =1;otherwiseq ij =0. Then chnge of sis x T = x T Q oeying ove completion rules results in FA M tht: (1) hs sme output; (2) hs sttes with independent lnguges. Proof. (1) Sttes q stisfy L(q)=L(x), where x is the independent stte in q. Trnsitions A σ = Q 1 [A σ q 1...A σ q n ], hve A σ q i s the σ-predecessors of sttes in q i.theroleofq 1 is to represent A σ q i in Q. If this fils, it is corrected s discussed ove. (2) Dependent rows hve een identified with their summnds. For exmple, consider FA M 7 in Figure 4(). The oservility mtrix O of M 7 is given elow: 6 6 We show only prt of the columns in O.

9 202 R. Grosu C A C A C A C A CA CA C q 1 q 2 q O(M 7 )= x Q(M 7 )= Row = +. This determines the construction of Q s shown ove. Using Q in x T = x T Q,resultsinFAM 8 shown in Figure 4(). Note tht A q 1 = hs een removed when representing A q 1 in Q. Using ρ r (O) typiclly results in n NFA, even when strting with DFA. This is ecuse vectors in Q my hve overlpping rows, due to liner dependencies in O. The use of ρ d r (O) ensures resulting DFA, s columns do not overlp. Identity equivlence lso simplifies the trnsformtion. First, Theorem 3 nd the computtion of ρ d r cn e performed on-the-fly s prtition-refinement: [C], prtitions sttes, sed on oservtions of length 0; [CAC], further distinguishes the sttes in previous prtition, sed on oservtions of length 1; nd so on. Second, no completion is ever necessry, s Aq is lwys representle in Q. Theorem 5. (Deterministic row reduction) Given n FA M with ρ d r (O)=k<n proceed s in Theorem 4 ut using ρ d r(o). ThenifM isdfa,thensoism. Proof. (1) Theorem 4 ensures correctness. (2) Sttes in Q re disjoint. Hence, no row in A = Q 1 AQ hs two entries for the sme input symol. For exmple, let us pply Theorem 5 to the DFA M 2 in Figure 2(). The corresponding oservility mtrix is shown elow: C A C A C A C q 1 q 2 q O(M 2 )= Q(M 2 )= Rows = nd =. This determines the construction of the sis Q s shown ove. Using this sis in the equivlence trnsformtion, results in DFA M 3, which is shown grphiclly in Figure 2(c). Corollry 1 (Myhill-Nerode theorem). Theorem 5 is equivlent to the DFA minimiztion lgorithm of the Myhill-Nerode theorem [2]. Column-sis trnsformtions. These trnsformtions pick Q s column sis for O. The definition of sis depends on the notion of liner independence used, nd this lso impcts the column rnk computtion vi Theorem 3. Using ρ r (O), one fully exploits liner dependencies, nd chooses miniml column sis Q s the sttes of M. The trnsitions of M re then determined y representing ll the predecessors AQ of the sttes Q =[q 1...q k ]ofm in Q. In contrst to the generl row trnsformtion, Aq i,fori [1,k], is representle in Q, sq is column sis for O. Hence, no completion is ever necessry. Like in vector spces, the resulting mtrices A re in compnion form.

10 FA s Time-Inv LS Oservility, Rechility nd More 203 4, c () () 4,c,c c, x,c 3 (c),c (d),c x,c 1, x x x c Fig. 5. () NFA M 9.()DFAM 10. (c)dfam 11. (d)nfam 12. Theorem 6. (Column reduction) Given n FA M with ρ c (O)=k<n. Define Q s column sis of O. Then chnge of sis x T = x T Q results in FA M with: (1) sme output; (2) sttes with distinguishing ccepting word. Proof. (1) Trnsitions A σ = Q 1 [A σ q 1...A σ q n ], hve A σ q i s the σ-predecessors of sttes in q i.theroleofq 1 is to represent A σ q i in Q, nd this never fils. (2) Dependent columns in O hve een identified with their summnds. Forexmple,considertheFAM 5 shown in Figure 3() nd its ssocited oservility mtrix, shown elow of Figure 3(). No row-rnk reduction pplies, s ρ r (O) = 4. However, s ρ c (O) = 3, one cn pply column-sis reduction, with Q s the first three columns of O. The resulting FA is shown in Figure 3(c). The column-sis trnsformtion for ρ d c (O) simplifies, s dependence reduces to identity. Moreover, in this cse M cn e constructed on-the-fly, s follows: Strt with Q, Q n =[C]. Then repetedly remove the first stte q Q n,nddd the trnsition p σ q to A for ech p Aq. Ifp Q, then lso dd p t the end of Q nd Q n.stopwhenq n is empty. The resulting M T is deterministic. Theorem 7. (Deterministic column trnsformtion) Given FA M proceed s in Theorem 6 ut using ρ d c(o). ThenM T is DFA with Q 2 n 1. Proof. Ech row of A T σ hs single 1 for ech input symol σ Σ. Forexmple,considertheFAM 11 shown in Figure 5(c). Construct the sis Q y selecting ll columns in O. Using this sis in the equivlence trnsformtion x T = x T Q, results in the FA M 12 showninfigure5(d). 5 Rechility Trnsformtions of FA The oolen semiring B is commuttive, thtis = holds. When viewed s semimodule, left linerity is therefore equivlent to right linerity, tht is i I x i i = i I i x i. This in turn mens tht (AB) T = B T A T. Consequently, in B the rechility of n FA M =[I,A,C]is reducile to the oservility of the FA M T =[C T,A T,I T ], nd ll the results nd trnsformtions in Section 4, cn e directly pplied without ny further proof! For illustrtion, consider the FA M 9 shown in Figure 5(). The rechility mtrix R T (M 9 ) is given elow. It is identicl to O(M T 9 ).

11 204 R. Grosu () 5 3, () Fig. 6. () NFA M 13. ()NFAM 14. I A T I AT I AT I AT I AT c I AT ci q1 q2 q3 q R T (M 9 )= Q(M 9 )= Ech row i of R T corresponds to stte x i.acolumna T w I of RT is 1 in row i iff x i is rechle from I with w R, or dully, if stte x i ccepts w in M T. 7 Row-sis trnsformtions. These trnsformtions utilize sets of rechilityequivlent sttes in M to uild the independent sttes q Q of M. These sttes re, s discussed efore, the oservility equivlent sttes of M T. Theorem 8. (Row reduction) Given n FA M, Theorem 4 pplied to M T resultsinnfam with: (1) sme output; (2) sttes with independent sets of reching words. For exmple, in R T (M 9 )ove,row = +. This determines the construction of the sis Q, lso shown ove. Using this sis in the equivlence trnsformtion, results in the DFA M 10 shown in Figure 5(). Identifying linerly dependent sttes with their genertors nd repiring lone σ-successors might preclude M T to e DFA, even if M T ws DFA. Identifying only sttes with identicl rechility however, ensures it. Theorem 9. (Deterministic row reduction) If M T is DFA, then Theorem 5 pplied to M T ensures tht M T is lso DFA. For exmple, let us pply Theorem 9 to the NFA M 13 showninfigure6(), the dul of the DFA M 2 shown in Figure 2(). Hence, M T 11 = M 2 is DFA. The rechility mtrix R T (M 13 ) is shown elow. It is identicl to O(M 2 ). I A T I AT I AT I q1 q2 q R T (M 13 )= Q(M 13 )= We write w R for the reversed form of w.

12 FA s Time-Inv LS Oservility, Rechility nd More 205 () 2,,c,c x3 () x3 2 (c),, 2 (d) Fig. 7. () FA M 15. ()FAM 16. (c)fam 17. (d)fam 18. Rows = nd =. This determines the construction of the sis Q s shown ove. Using this sis in the equivlence trnsformtion, results in NFA M 14, shown grphiclly in Figure 6(c). The FA M T 14 is DFA, nd M T 14 = M 3. Column-sis trnsformtions. Given n FA M, these trnsformtions construct FA M y choosing column sis of R T s the sttes Q of M. The generl form of the trnsformtions uses the full concept of liner dependency, in order to look for column sis in R T. Hence, this trnsformtion computes the smllest possile column sis. Theorem 10. (Column reduction) Given FA M, Theorem 6 used on M T results in FA M with: (1) sme output; (2) sttes reched with distinguishing word. Consider the NFA M 4 shown in Figure 3(). Neither row nor column-sis oservility reduction is pplicle to M 4. However, one cn pply columnsis rechility reduction to M 4.ThemtrixR T (M 4 ) is given elow. I A T I AT I AT c I q1 q2 R T (M 4 )= x1 Q(M 4 )= Columns 1 nd 2 form sis for R T. This determines the construction of Q s shown ove. Using Q in x T = x T Q results in NFA M 15, shown in Figure 7(). In this cse, the column-sis rechility trnsformtion is identicl to row-sis rechility trnsformtion. Consequently, the ltter trnsformtion would not require ny utomtic completion of the σ-successors q T A σ of q Q. Given n FA M, the deterministic column-sis trnsformtion, with column rnk ρ d c (RT ), lwys constructs DFA M. This construction is dul to the deterministic column-sis oservility trnsformtion. Theorem 11. (Deterministic column trnsformtion) Given n FA M, Theorem 7 pplied to M T results in the DFA M. Consider for exmple the NFA M 17 shown in Figure 7(c). Its rechility mtrix R T (M 17 ) is given elow, where only the interesting columns re shown. R T (M 17 )= I A T I AT I q1 q2 q3 [ ] Q(M 17 )= [ As columns one nd two form sis for R T, the generl column-sis trnsformtion is the identity. The deterministic one is not, s it includes ll distinct ]

13 206 R. Grosu columns of R T in Q, sshownove.usingq in x T = x T Q results in the DFA M 18 shown in Figure 7(d). This DFA hs one more stte, compred to M 17. Applying deterministic column-sis trnsformtion to n FA M, doesnot necessrily increse the numer of sttes of M. For exmple, pplying such trnsformtion to NFA M 6 shown in Figure 3(c), results in the DFA M 16 shown in Figure 7(), which hs the sme numer of sttes s M 6. Moreover, in this cse, the generl nd the deterministic column-sis trnsformtions coincide. Corollry 2 (NFA determiniztion lgorithm). Theorem 11 is equivlent to the NFA determiniztion lgorithm [2]. 6 The Pumping Lemm nd FA Minimlity In previous sections we hve shown tht control-theoretic pproch to FA complements, nd lso llows to extend the rech of, the grph-theoretic pproch. In this section we give two dditionl exmples: An lterntive proof of the pumping lemm [2]; A lterntive pproch to FA minimiztion. Both tke dvntge of the oservility nd rechility mtrices. Theorem 12 (Pumping Lemm). If L is regulr set then there exists constnt p such tht every word w L of length w p cnewrittensxyz, where: (1) 0 < y, (2) xz p, nd(2)xy i z L for ll i 0 Proof. ConsiderDFA M ccepting L. SinceM is deterministic, ech column of R T is stndrd sis vector n i,ndthereretmostn such distinct columns in R T. Hence, for every word w of length greter thn n, there re words xyz = w stisfying (1) nd (2) such tht I T A x = I T A xy.sincei T A xy i = I T A xy A y i 1, it follows tht I T A xy i zc = I T A w C, for ll i 0. Cnonicl Forms. Row- nd column-sis trnsformtions re relted to ech other. Let Q c M i j (B), Q r M i k (B) e the oservility column nd row sis for n FA M. LetA c = Q 1 c AQ c nd A r = Q 1 r AQ r. Theorem 13 (Row nd column sis). There is mtrix R M k j (B) such tht: (1) Q c = Q r R;(2)A c = R 1 A r R;(3)A r = RA c R 1. Proof. (1) Let B(m) e the index in O of the independent row of q m Q r nd C(n) e the index in O of the independent column q n Q c, nd define R mn = O B(m)C(n),form [1,k], n [1,j]. Then Q c = Q r R; (2) As consequence A c =(Q r R) 1 A(Q r R)=R 1 A r R; (3) This implies tht A r = RA c R 1. Hence, A r is otined through rechility trnsformtion with column sis R fter n oservility trnsformtion with column sis Q c.leto nd R e the column sis oservility nd rechility trnsformtions, respectively. We cll M o = O(R(M)) nd M r = R(O(M)) the cnonicl oservle nd rechle FAs of M, respectively.

14 FA s Time-Inv LS Oservility, Rechility nd More 207 () x 6 x 7 () x 6, (c) x 6, Fig. 8. () DFA M 19. ()NFAM 20. (c)nfam 21. Theorem 14 (Cnonicl FA). For ny FA M, R(M o )=M r nd O(M r )=M o. Miniml FA. Cnonicl FAs re often miniml wrt. to the numer of sttes. For exmple, M 11 nd M 12 in Figure 5 re oth miniml FAs. Moreover, FA M 11 is cnonicl rechle nd FA M 12 is cnonicl oservle. For certin FAs however, the cnonicl FAs re not miniml. A necessry condition for the lck of minimlity, is the existence of weker form of liner dependence mong the sis vectors of the oservility/rechility mtrices: A set of vectors Q = {q i i I} in R is clled wekly linerly dependent if there re two disjoint susets I 1,I 2 I, such tht i I 1 q i = i I 2 q i [8]. For exmple, the DFA M 19 in Figure 8() hs the cnonicl rechle FA M 20 shown in Figure 8(), which is miniml. The oservility mtrix of M 20 shown elow, hs 7 independent columns. The cnonicl oservle FA of M 19 nd M 20 hs therefore 7 sttes! As consequence, it is not miniml. Note however, tht A C + A C = A C + A C. Hence, the 7 columns re wekly dependent. CA CA CA C A CA CA C q 1 q 2 q 3 q 4 q 5 q O(M 20 )= x Q(M 20 )= Theorem 15 (Miniml FA). Given the oservility mtrix O of n FA M, choose Q s set sis [13] of O, such tht AQ is representle in Q. Then the equivlence trnsformtion x T = x T Q results in miniml utomton. Alterntively, minimiztion cn e reduced to computing the miniml oolen reltion corresponding to O. For exmple, the Krnugh locks [9] in O(M 20 ) provide severl wys of constructing Q. OnesuchwyisQ(M 20 )shownove, where one lock is the first column in O(M 20 ), nd the other locks correspond to its rows. The resulting FA is M 21.Bothlterntivesled to NP-complete lgorithms. Rechility is treted in dul wy, y mnipulting R. Since ll equivlent FAs dmit n equivlence trnsformtion resulting in the sme DFA, nd since from this DFA one cn otin ll other FAs through n equivlence trnsformtion, ll FAs re relted through n equivlence trnsformtion! This provides clener wy of deling with the miniml FAs, when compred to the terminl FA (incorporting ll other FA), discussed in [1].

15 208 R. Grosu 7 Conclusions We hve shown tht regrding finite utomt (FA) s discrete, time-invrint liner systems over semimodules, llows to unify DFA minimiztion, NFA determiniztion, DFA pumping nd NFA minimlity s vrious properties of oservility nd rechility trnsformtions of FA. Our tretment of oservility nd rechility my lso llow us to generlize the Cyley-Hmilton theorem to FA nd derive chrcteristic polynomil. In future work, we would therefore like to investigte this polynomil nd its ssocited eigenvlues. Acknowledgments. I would like to thnk Gene Strk nd Gheorghe Stefnescu for their insightful comments to the vrious drfts of this pper. This reserch ws supported in prt y NSF Fculty Erly Creer Awrd CCR References 1. Nivt, M., Arnold, A., Dicky, A.: A note out miniml nondeterministic utomt. EATCS 45, (1992) 2. Aho, A.V., Ullmn, J.D.: The Theory of Prsing, Trnsltion nd Compiling. Prentice-Hll, Englewood Cliffs (1972) 3. Alur, R., Coucouetis, C., Hlwchs, N., Henzinger, T.A., Ho, P.H., Nicolin, X., Olivero, A., Sifkis, J., Yovine, S.: The lgorithmic nlysis of hyrid systems. Theoreticl Computer Sciences 138, 3 34 (1995) 4. Besley, L.B., Gutermn, A.E., Jun, Y.B., Song, S.Z.: Liner preservers of extremes of rnk inequlities over semirings: Row nd column rnks. Liner Alger nd its Applictions 413, (2006) 5. Chen, C.T.: Liner System Theory Design. Holt, Reinhrt nd Winston (1984) 6. Conwy, J.H.: Regulr Alger nd Finite Mchines. Chpmn nd Hll, Boc Rton (1971) 7. Friederg, S.H., Insel, A.J., Spence, L.E.: Liner Alger. Prentice-Hll, Englewood Cliffs (1989) 8. Gondrn, M., Minoux, M.: Grphs, Dioids nd Semirings: New Models nd Algorithms. Springer, Heidelerg (2008) 9. Krnugh, M.: The mp method for synthesis of comintionl logic circuits. Trnsctions of Americn Institute of Electricl Engineers 72(9), (1953) 10. Kuich, W., Slom, A.: Semirings, Automt, Lnguges. Springer, Heidelerg (1986) 11. Lynch, N., Segl, R., Vndrger, F.: Hyrid I/O utomt. Informtion nd Computtion 185(1), (2003) 12. Slom, A.: Theory of Automt. Pergmon Press, Oxford (1969) 13. Stockmeyer, L.J.: The set sis prolem is NP-complete. Technicl Report RC- 5431, IBM (1975)