Assisted Problem Solving and Decompositions of Finite Automata

Assisted Problem Solv ing and Decomp os itions of Finite Automata ∗ P eter Ga ˇ zi Branisla v Ro v an Department of Computer Science, Comenius Universit y Mlynsk´ a d olina, 842 48, Bratisla v a, Slo v akia { gazi,rov an } @dcs.fmph.u niba.sk Abstract A study of assisted problem solving formalized via decompositions of deterministic finite aut omata is initiated. The land scap e of new types of decomp ositions of finite automata this s tud y uncov ered i s presented. Lan- guages with vari ous degrees of decomp osability b etw een und ecomp osable and p erfectly decomp osable are sho wn t o exist. 1 In tro duction In the present paper w e initiate the study of assiste d pr obl em solving . W e intend to mo del and study situations , where solution to the pr oblem ca n b e sought based o n some additional a prior i informa tion ab out the inputs. One can exp ect to o bta in simpler so lutio n in such case. There ar e similar a pproaches k nown in the literature , most notably the notions o f advice functions [1], where the additional information is base d o n the length of the input word and the notion of promise problems [2], wher e the set o f inputs is separa ted into three cla sses – those with “yes” answer, those with “no” a nswer a nd those wher e we do not care a bo ut the outcome. By considering the simples t case where the “proble m solving” machinery is the deterministic finite a utomaton (DF A) we obtain a new motiv atio n for studying new types of finite automata decomp ositions. In this pap er we shall th us consider the case wher e s olving a pro blem sha ll mean constructing an a uto maton for a given language L . The “assista nce” shall be given b y additional informa tio n ab out the input, e.g., that we can assume the inputs s hall b e restricted to words from a particular re g ular language L ′ . Thus, instead of lo oking for an automaton A such tha t L = L ( A ) we ca n lo ok for a (po ssibly simpler) automaton B such that L = L ( B ) ∩ L ′ . W e can then say that B accepts L with the assistance o f L ′ . W e shall ca ll L ′ (or the corres po nding automaton A ′ such that L ′ = L ( A ′ )) an advisor to B . In this case the advisor A ′ provides a ssistance to the so lver B by guar anteeing that A ′ accepts the given input w ord. W e sha ll als o study a case where the a ssistance pr ovides more detailed informatio n ab out the o utcome of the computation of A ′ on the input word (e.g., the state r eached). Clearly the adviso r can be consider e d useful only if it enables B to b e simpler than A and at the same time A ′ is not more ∗ This w ork was supp orted in part b y the grant VEGA 1/3106/06. 1 complicated tha n A . The measure of complexity w e sha ll consider is the num b er of states of the deterministic finite automato n. This mea sure of complex ity was used quite o ften recen tly due to r enewed interest in finite automata prompted by applica tions such as mo del chec king (see e.g. [3] for a recent survey). (Note that results complementary to ours, namely r esults on complexity of automata for the intersection of regular sets were studied in [4].) The contribution of our paper is t wofold. First, we can interpret the ‘so lver’ and the ‘advisor’ as tw o par a llel pro c e sses each p erforming a differ ent task and joint ly solv ing a problem. Since our appro ach lends itself to a generalisation to k a dvisors it may stimulate new parallel solutions to problems (the tra ditio nal ones usually us ing parallel pro cess e s to per form essentially the same tas k). Sec- ond, the choice of finite automata as the simplest problem solving mac hinery brought a bo ut new t yp es o f deco mpo sitions motiv ated b y the info r mation the ‘advisor’ can provide to the ‘so lver’. Our results provide a c omplete pictur e of the landscap e of these deco mpo sitions. The problem within this scenario we shall a ddress in this pap er is the exis- tence of a useful advis o r for a given automaton A . W e shall co mpare the p ow er of s e veral t yp es of advisor s, and inv estigate the effect of the advisor on the complexity of the assisted solver B . W e can for mu late this als o as a pro blem of decomp osition of deterministic finite state a utomata – g iven DF A A find DF A A 1 (a solver) and A 2 (an advisor) such that w ∈ L ( A ) ca n be determined from the computations of A 1 and A 2 . W e shall study several new types of decomp o- sitions of DF A, one of them is ana logous to the s tate b ehavior decomp osition of finite state tra nsducers studied in [5]. In Sect. 3 we prov e rela tions among these decomp ositions. F o r each type of deco mpo s ition there are automata which are undecomp osable and automata for which there is a decomp os ition that is the bes t p o ssible. In Sect. 4 we consider the space b etw een these extreme po int s and study the degr ee of decomp o sability . 2 Definitions and Notation W e shall use standard no tions o f the theo ry of forma l lang uages (see e.g. [6]). Our no tation shall be as follows. Σ ∗ denotes the set of all words ov er the alphab et Σ, the length o f a word w is deno ted by | w | , ε denotes the empty word, a nd for a languag e L w e shall denote b y Σ L the minimal alphabet suc h that L ⊆ Σ ∗ L . The num b er of o ccurrenc e s o f a g iven letter a in a word w is denoted b y # a ( w ). Thr oughout this pap er we sha ll consider deter minis tic finite automata only . A deterministic finite automaton (DF A) is a quintuple ( K, Σ , δ, q 0 , F ), such that K is a finite set of s tates, Σ is a finite input alphabet, q 0 ∈ K is the initial state, F ⊆ K is the set of accepting states and δ : K × Σ → K is a transition function. As usual, we shall denote by δ als o the s tandard extension of δ to words, i.e., δ : K × Σ ∗ → K . W e shall denote b y | K | the n umber o f sta tes in K . F o r malizing the notions of assisted pro blem so lv ing from the Introduction we shall now define sev eral types of decomp ositions of DF A A into tw o (sim- pler) DF As A 1 and A 2 (a solver and a n advisor) so that the member ship of an input word w in L ( A ) can b e determined based o n the information on the computations of A 1 and A 2 on w . W e first in tro duce an ac c eptanc e- identifying decomp osition of deterministic 2 finite automata. Definition 2.1 . A p air of DF A s ( A 1 , A 2 ) , wher e A 1 = ( K 1 , Σ , δ 1 , q 1 , F 1 ) and A 2 = ( K 2 , Σ , δ 2 , q 2 , F 2 ) , forms an acceptance- ide ntifying decomp osition (A I- de c omp osition) of a DF A A = ( K, Σ , δ, q 0 , F ) , if L ( A ) = L ( A 1 ) ∩ L ( A 2 ) . This de c omp osition is no n trivia l if | K 1 | < | K | and | K 2 | < | K | . By decomp osing A in this manner, o ne of the decomp osed automata (say A 2 ) can act as an advisor and narrow down the set of input words for the other one (say A 1 ), whose task to recognize the words of L ( A ) may b ecome ea sier. Another r equirement we could p ose on a decomp osition is to identify the final state of any computation of the orig inal automaton by only knowing the final states o f b oth corresp onding computations o f the automata forming the decomp osition. This r equirement ca n be formalized as follows. Definition 2.2 . A p air of DF A s ( A 1 , A 2 ) , wher e A 1 = ( K 1 , Σ , δ 1 , q 1 , F 1 ) and A 2 = ( K 2 , Σ , δ 2 , q 2 , F 2 ) , forms a state-identifying decomp osition (SI- de c omp osition) of a D F A A = ( K , Σ , δ, q 0 , F ) , if ther e exists a mapping β : K 1 × K 2 → K , such that it holds β ( δ 1 ( q 1 , w ) , δ 2 ( q 2 , w )) = δ ( q 0 , w ) for al l w ∈ Σ ∗ . This de c omp osi- tion is nontrivial if | K 1 | < | K | and | K 2 | < | K | . The third – and the w eakest – requir ement w e p os e on a decomp osition of a DF A is to r e q uire that there m ust exist a wa y to determine whether the origina l automaton would accept some given input word based on knowing the states in which the computations of both decomp osition automata have finished. Definition 2.3 . A p air of DF A s ( A 1 , A 2 ) , wher e A 1 = ( K 1 , Σ , δ 1 , q 1 , F 1 ) and A 2 = ( K 2 , Σ , δ 2 , q 2 , F 2 ) , forms a we ak acceptance-identifying decomp os ition (wAI-de c omp osition) of a DF A A = ( K, Σ , δ, q 0 , F ) , if ther e exists a r elation R ⊆ K 1 × K 2 such that it holds R ( δ 1 ( q 1 , w ) , δ 2 ( q 2 , w )) ⇔ w ∈ L ( A ) for al l w ∈ Σ ∗ . This de c omp osition is non trivial if | K 1 | < | K | and | K 2 | < | K | . Note that in the last tw o definitions, the sets of accepting sta tes of A 1 and A 2 are irrele v ant. By a decompo sability of a regular lang uage L in some wa y , we shall mean the decomp osability of the corresp onding minimal auto maton o ver Σ L . T o be able to co mpa re these new t ype s of dec omp o sition to the p ar al lel de c omp ositions of state b ehavior in tro duced for sequen tial mac hines in [5 ], we shall redefine them fo r DF As. Definition 2 .4. A DF A A ′ = ( K ′ , Σ , δ ′ , q ′ 0 , F ′ ) is sa id to r ealize the state behavior of a DF A A = ( K , Σ , δ, q 0 , F ) if ther e exists an inje ctive mapping α : K → K ′ such that (i) ( ∀ a ∈ Σ)( ∀ q ∈ K ); δ ′ ( α ( q ) , a ) = α ( δ ( q , a )) , (ii) α ( q 0 ) = q ′ 0 . Mor e over, A ′ is said to r ealize the state and ac c eptance b ehavior of A , if in addition the fol lowing pr op erty holds: (iii) ( ∀ q ∈ K ); α ( q ) ∈ F ′ ⇔ q ∈ F . 3 Definition 2 .5. The parallel connection of t wo DF A A 1 = ( K 1 , Σ , δ 1 , q 1 , F 1 ) and A 2 = ( K 2 , Σ , δ 2 , q 2 , F 2 ) is the DF A A = A 1 || A 2 = ( K 1 × K 2 , Σ , δ, ( q 1 , q 2 ) , F 1 × F 2 ) such that δ (( p 1 , p 2 ) , a ) = ( δ 1 ( p 1 , a ) , δ 2 ( p 2 , a )) . Definition 2.6. A p air of DF As ( A 1 , A 2 ) is a state b ehavior (SB- ) decomp osi- tion of a DF A A if A 1 || A 2 r e alizes t he state b ehavior of A . The p air ( A 1 , A 2 ) is an acceptance a nd state behavior (ASB-) decompos ition of A if A 1 || A 2 r e alizes the state and ac c eptanc e b ehavior of A . This de c omp osition is nontrivial if b oth A 1 and A 2 have fewer st ates than A . W e hav e modified the definitions to fit the for malism and purp ose of deter- ministic finite automata (i.e., to accept for mal lang uages) without loo sing the connection to the str ongly related and use ful concept of S.P.p artitions , exhibited below. W e shall use the following notation and proper ties of S.P . partitions from [5]. A pa rtition π o n a set of states of a DF A A = ( K, Σ , δ, q 0 , F ) has substitution pr op erty (S.P .), if it ho lds ∀ p, q ∈ K ; p ≡ π q ⇒ ( ∀ a ∈ Σ; δ ( p, a ) ≡ π δ ( q , a )). If π 1 and π 2 are partitions on a g iven set M , then (i) π 1 · π 2 is a partition on M such that a ≡ π 1 · π 2 b ⇔ a ≡ π 1 b ∧ a ≡ π 2 b , (ii) π 1 + π 2 is a partitio n on M such that a ≡ π 1 + π 2 b iff there exists a seq uenc e a = a 0 , a 1 , a 2 , . . . , a n = b , such that a i ≡ π 1 a i +1 ∨ a i ≡ π 2 a i +1 for all i ∈ { 0 , . . . , n − 1 } , (iii) π 1  π 2 if it holds ( ∀ x, y ∈ M ); x ≡ π 1 y ⇒ x ≡ π 2 y . The set of all par titions on a given set (with the partial order  , join rea lized by + and meet rea lized by . ) forms a lattice. The set of all S.P . partitions on the set of states of a g iven DF A forms a sublattice o f the lattice of all partitions on this set. The trivial partitions {{ q 0 } , { q 1 } , . . . , { q n }} and {{ q 0 , q 1 , . . . , q n }} shall b e denoted by symbols 0 and 1, res p ectively . The blo ck of a partition π containing the sta te q shall b e denoted b y [ q ] π . In addition, we sha ll use the following separation notion. Definition 2.7. T he p artitions π 1 = { R 1 , . . . , R k } and π 2 = { S 1 , . . . , S l } on a set of states of a DF A A = ( K, Σ , δ, q 0 , F ) ar e said to sepa rate the final states of A if ther e exist indic es i 1 , . . . , i r and j 1 , . . . , j s such that it holds ( R i 1 ∪ . . . ∪ R i r ) ∩ ( S j 1 ∪ . . . ∪ S j s ) = F . 3 Relations Bet w een T yp es of Decomp ositions The concept of partitions separ ating the final s tates allows us to derive a nec es- sary and sufficient co ndition for the existence of SB- and ASB-deco mpos itions similar to the one stated in [5]. Theorem 3.1. A DF A A = ( K, Σ , δ, q 0 , F ) has a nontrivial SB-de c omp osition iff ther e exist two nontr ivial S.P. p artitions π 1 and π 2 on the set of states of A such that π 1 · π 2 = 0 . This de c omp osition is an ASB-de c omp osition if and only if these p artitions sep ar ate the fi nal states of A . Pr o of. The pro of is analogo us to that in [5] but had to be e xtended for the ASB-decomp osition. W e omit it due to space constraints. 4 F o r the o ther decompositio ns, w e can derive the following sufficient condi- tions that exploit the concept of S.P . partitions. Theorem 3.2. L et A = ( K , Σ , δ, q 0 , F ) b e a deterministic fin ite automaton, let π 1 and π 2 b e nontrivial S.P. p art itions on t he set of states of A , su ch that they sep ar ate the final states of A . Then A has a nontrivial AI-de c omp osition. Pr o of. Since π 1 and π 2 separate the final states of A , there e xist blo cks B 1 , . . . , B k and C 1 , . . . , C l of the partitions π 1 and π 2 resp ectively , such that ( B 1 ∪ . . . ∪ B k ) ∩ ( C 1 ∪ . . . ∪ C l ) = F . W e shall cons truct t w o automata A 1 and A 2 having states corresp onding to blo cks of these partitions and show that ( A 1 , A 2 ) is a nontrivial AI-deco mpo sition of A . Let A 1 = ( π 1 , Σ , δ 1 , [ q 0 ] π 1 , { B 1 , . . . , B k } ) and A 2 = ( π 2 , Σ , δ 2 , [ q 0 ] π 2 , { C 1 , . . . , C l } ) be DF As with δ i defined by δ i ([ q ] π i , a ) = [ δ ( q , a )] π i , i ∈ { 1 , 2 } (this definition do es not dep end on the choice o f q since π i is an S.P . partition). W e now need to prov e that L ( A ) = L ( A 1 ) ∩ L ( A 2 ). Let w ∈ L ( A ). Suppose that the c o mputation of A on the word w ends in some a ccepting sta te q f ∈ F . Then, from the construction o f A 1 and A 2 it follows that the computation of A i on the word w ends in the state corresp onding to the blo ck [ q f ] π i of the par tition π i . Since q f ∈ F , it must hold [ q f ] π 1 ∈ { B 1 , . . . , B k } and [ q f ] π 2 ∈ { C 1 , . . . , C l } , hence from the constructio n of A i , these blocks corres po nd to the accepting sta tes in the resp ective automata. Thu s w ∈ L ( A i ) for i ∈ { 1 , 2 } , therefore L ( A ) ⊆ L ( A 1 ) ∩ L ( A 2 ). Now s uppo s e w ∈ L ( A 1 ) ∩ L ( A 2 ), Thus the computatio n of A 1 on w ends in one o f the states B 1 , . . . , B k , which means that the c o mputation of A o n w would end in a state fro m the union of blo cks B 1 ∪ . . . ∪ B k . Using the s a me argument for A 2 , we get that the computation of A on w would end in a state from C 1 ∪ . . . ∪ C l . Since ( B 1 ∪ . . . ∪ B k ) ∩ ( C 1 ∪ . . . ∪ C l ) = F w e o btain that the computation o f A ends in an accepting state, hence w ∈ L ( A ) and L ( A 1 ) ∩ L ( A 2 ) ⊆ L ( A ). Since both partitions are nontrivial, so is the AI-decomp osition obtained. Theorem 3.3. L et A = ( K , Σ , δ, q 0 , F ) b e a deterministic finite automaton, let π 1 and π 2 b e n ontrivial S.P. p artitions on the set of states of A , such that π 1 · π 2  { F, K − F } . Then A has a nontrivial wAI-de c omp osition. Pr o of. W e shall construct A 1 and A 2 corres p o nding to the S.P . partitions π 1 and π 2 as follows: A i = ( π i , Σ , δ i , [ q 0 ] π i , ∅ ), where δ i ([ q ] π i , a ) = [ δ ( q , a )] π i and i ∈ { 1 , 2 } . T o show that ( A 1 , A 2 ) is a wAI-decomp osition of A , w e define the relation R ⊆ π 1 × π 2 by the equiv alence R ( D 1 , D 2 ) ⇔ ( D 1 ∩ D 2 ⊆ F ),where D i is some blo ck of the par tition π i . No w w e need to prove that ∀ w ∈ Σ ∗ ; w ∈ L ( A ) ⇔ R ( δ 1 ([ q 0 ] π 1 , w ) , δ 2 ([ q 0 ] π 2 , w )). Let the computation of A o n w end in some state p ∈ K . It follows that the computation of A i on the word w ends in the s ta te corr esp onding to the blo ck [ p ] π i , i ∈ { 1 , 2 } . Thus R ( δ 1 ([ q 0 ] π 1 , w ) , δ 2 ([ q 0 ] π 2 , w )) ⇔ R ([ p ] π 1 , [ p ] π 2 ) and by the definition of R , we have R ( δ 1 ([ q 0 ] π 1 , w ) , δ 2 ([ q 0 ] π 2 , w )) ⇔ [ p ] π 1 ∩ [ p ] π 2 ⊆ F . Since p ∈ [ p ] π 1 ∩ [ p ] π 2 , [ p ] π 1 ∩ [ p ] π 2 is a block of the partition π 1 · π 2 and π 1 · π 2  { F , K − F } , it must hold that either [ p ] π 1 ∩ [ p ] π 2 ⊆ F o r [ p ] π 1 ∩ [ p ] π 2 ⊆ K − F . Therefore R ( δ 1 ([ q 0 ] π 1 , w ) , δ 2 ([ q 0 ] π 2 , w )) ⇔ p ∈ F and the pro of is co mplete. It follows dire c tly from the definitions, that each SI- decomp osition is a ls o a wAI-decomp osition, and s o is each AI-decomp osition. Also, each ASB-decomp osition is an AI-decomp osition, which is a conseq uence o f the definition of acceptance 5 and state behavior realizatio n. F o r minimal automata, a relationship b et ween AI- and SI-decomp ositio ns ca n be obtained. Theorem 3 .4. L et A = ( K, Σ , δ, q 0 , F ) b e a minimal DF A, let ( A 1 , A 2 ) b e its AI-de c omp osition. Then ( A 1 , A 2 ) is also an SI- de c omp osition of A . Pr o of. Since ( A 1 , A 2 ) is an AI-decomp osition of A , L ( A ) = L ( A 1 ) ∩ L ( A 2 ). Therefore if w e use the well-kno wn Cartes ian pr o duct constr uction, we obtain the automa ton A 1 || A 2 such that L ( A 1 || A 2 ) = L ( A ). Since A is the minimal automaton accepting the language L ( A ), there exists a mapping β : K ′ → K such that it holds ( ∀ w ∈ Σ ∗ ); β ( δ ′ ( q ′ 0 , w )) = δ ( β ( q ′ 0 ) , w ), wher e δ ′ is th e transition function of A 1 || A 2 , K ′ is its set of s tates and q ′ 0 is its initial state. Since A 1 || A 2 is a parallel connection (i.e., K ′ = K 1 × K 2 , q ′ 0 is the pair of initial states of A 1 and A 2 ), it is easy to see that β is in fact exactly the mapping required by the definition of the SI-decomp osition. The ASB-decomp ositio n is a combination of the SB-de c o mpo sition and the AI-decomp osition, as the next theorem shows. Theorem 3. 5. L et A b e a D F A without unr e achable states. ( A 1 , A 2 ) is an ASB-de c omp osition of A iff ( A 1 , A 2 ) is b oth an SB-de c omp osition and an AI- de c omp osition of A . Pr o of. The first implication c le arly follows from the definitions, Theorem 3.1 and Theorem 3.2. No w let ( A 1 , A 2 ) be a n SB- and AI-decomp osition of A = ( K, Σ , δ, q 0 , F ). Let α be the mapping given by the definition of SB-decompo sition. W e need to prove that for all sta tes q of A , q ∈ F iff α ( q ) ∈ F 1 × F 2 , where F i is the set of accepting states of A i , i ∈ { 1 , 2 } . Let q ∈ K and let w b e a word such that δ ( q 0 , w ) = q . Then q ∈ F ⇔ w ∈ L ( A ) ⇔ w ∈ L ( A 1 ) ∩ L ( A 2 ) ⇔ α ( q ) ∈ F 1 × F 2 , where the first equiv alence is implied b y the choice o f w , the second holds b eca us e ( A 1 , A 2 ) is an AI-decomp os ition and the third is a consequence of the pro per ties of α guara n teed b y the SB-dec o mpo sition definition. There is also a relationship betw een SB- and SI-decomp ositions , in fact SB- is a stronger version of the state-identifying decompo sition, as the following tw o theorems show. W e need the notion of reachabilit y on pairs of states . Definition 3.1 . L et A 1 = ( K 1 , Σ , δ 1 , p 1 , F 1 ) and A 2 = ( K 2 , Σ , δ 2 , p 2 , F 2 ) b e DF As. We shal l c al l a p air of states ( q , r ) ∈ K 1 × K 2 reachable , if ther e exists a wor d w ∈ Σ ∗ such that δ 1 ( p 1 , w ) = q and δ 2 ( p 2 , w ) = r . Theorem 3.6 . L et A = ( K, Σ , δ, q 0 , F ) b e a DF A and let ( A 1 , A 2 ) b e its SB- de c omp osition. Then ( A 1 , A 2 ) also forms an SI-de c omp osition of A . Pr o of. Let A i = ( K i , Σ , δ i , q i , F i ), i ∈ { 1 , 2 } . Since ( A 1 , A 2 ) is an SB-decomp os ition of A , there exists an injective mapping α : K → K 1 × K 2 such that it holds α ( q 0 ) = ( q 1 , q 2 ) and ( ∀ a ∈ Σ)( ∀ p ∈ K ); α ( δ ( p, a )) = ( δ 1 ( p 1 , a ) , δ 2 ( p 2 , a )), where α ( p ) = ( p 1 , p 2 ). Let us define a new mapping β : K 1 × K 2 → K b y β ( p 1 , p 2 ) =  p if ∃ p ∈ K , α ( p ) = ( p 1 , p 2 ) q 0 otherwise. (1) Since α is injectiv e, there exists at most one suc h p and this definition is corr ect. 6 W e now need to prov e that β satisfies the condition from the definition of SI-decomp osition, i.e., that ( ∀ w ∈ Σ ∗ ); β ( δ 1 ( q 1 , w ) , δ 2 ( q 2 , w )) = δ ( q 0 , w ). Since α ( q 0 ) = ( q 1 , q 2 ) and all the pairs of states we enco unter in the computation o f A 1 || A 2 are th us reachable, this follo ws fr om the definition of α a nd (1) by an easy induction. Lemma 3 .7. L et A b e a DF A without u nr e achable st ates and let ( A 1 , A 2 ) b e its SI-de c omp osition, with β b eing the c orr esp onding mapping. Then ( A 1 , A 2 ) is an SB-de c omp osition of A if and only if β is inje ctive on al l r e achable p airs of states. Pr o of. Let ( A 1 , A 2 ) be an SB-decomp osition of A . It clearly follows from Def- inition 2.2, that the cor resp onding β satis fie s the equation (1) in the pro of o f Theorem 3.6 on all reachable pairs of s ta tes. Since the mapping α is a bijection betw een the s et of states of A and the s et of all reachable pairs o f states o f A 1 and A 2 , β defined as its inv ers e on the set o f r eachable pa ir s o f states w ill b e injectiv e o n this set. F o r the other implication, let ( A 1 , A 2 ) b e a n SI-decomp osition of A and let β b e injectiv e on the set of reachable pairs of states, let β r denote the ma pping β restricted onto the se t of all reachable pairs of states of A 1 , A 2 . Since A has no unr eachable states, β r is a lso sur jectiv e, thus we can define a new mapping α : K → K 1 × K 2 by the equation α ( q ) = β − 1 r ( q ). Since β maps the initia l state onto the initial state, so do es α , and since β satisfies the condition fro m the Definition 2.2, it implies that also α satis fie s the condition (i) from the definition of rea lization of state b ehavior. Therefo re ( A 1 , A 2 ) is an SB-deco mpo sition of A , with the cor resp onding mapping α . The con verse of Theore m 3.6 do es not hold. The minimal automaton for the language L = { a 4 k b 4 l | k ≥ 0 , l ≥ 1 } gives a counterexample. Insp ecting its S.P . partitions s hows that it has no nontrivial SB-decomp osition, but it ca n b e AI- decomp osed in to minimal automata for languages L 1 = { a 4 k b l | k ≥ 0 , l ≥ 1 } and L 2 = { w | # b ( w ) = 4 l ; l ≥ 0 } . According to Theo rem 3.4, this AI-decomp os ition is also state-identifying. Each ASB-decomp osition is obviously a lso an SB-decomp osition. On the other ha nd, there exist SB-decomposa ble automata, that a re ASB-undeco mpo sable. F o r example, the minimal automa ton for the languag e L 1 = { w ∈ { a, b, c } ∗ | # a ( w ) mo d 3 = 0 ∧ # b ( w ) mo d 5 = 0 } ∪ ∪ { w ∈ { a, b, c } ∗ | # a ( w ) mo d 3 = 2 ∧ # b ( w ) mo d 5 = 4 } has this pr op erty , be c ause the co rresp onding S.P . partitions o n the set of its states do not sepa r ate the final states in the sense of Definition 2.7. It is also not s o difficult to see that for an y non- minima l automaton A without unreachable states, there exists a no n trivia l AI- and wAI-de c o mpo sition ( A 1 , A 2 ) such that A 1 is the minimal automa ton equiv alent to A and A 2 has only one state. This decomp ositio n is obviously no t state-identifying. Figure 1 summarizes all the relationships among the decomp osition types that we have shown so far. Now w e show that for the cas e of so -called p erfect decomp ositions, so me of the types of decomp os itio n men tioned co incide. 7 AS B { { v v v v v v v v v # # G G G G G G G G AI × v v v v ; ; v v v v min H H H H # # H H H H   × # # S B { { w w w w w w w w × G G G G c c G G G G S I { { v v v v v v v v v × w w w w ; ; w w w w wAI × ; ; Description: A / / B : ev ery A-decompo sition is also a B- decomp osition A × / / B : not every A- de c o mpo sition is also a B-decomp osition A × / / B : there exists a DF A that has a non- trivial A-decompo sition but do es not ha ve a nontrivial B- de c omp o sition Figure 1: Rela tionships b e tw een decomp ositio n t yp es of DF A Definition 3.2. L et t b e a typ e of de c omp osition, t ∈ { AS B , S B , AI , S I , w AI } . L et A b e a DF A having n states, let A 1 and A 2 b e DF As having k and l states, r esp e ctively. We shal l c al l the p air ( A 1 , A 2 ) a p erfect t -dec o mpo sition of A , if it forms a t - de c omp osition of A and n = k · l . Theorem 3. 8. L et A b e a DF A with n o unr e achable states and let ( A 1 , A 2 ) b e a p air of DF As. Then ( A 1 , A 2 ) forms a p erfe ct SI-de c omp osition of A iff ( A 1 , A 2 ) forms a p erfe ct SB-de c omp osition of A . Pr o of. One of the implications is a consequence of T heo rem 3.6. As to the second one, since ( A 1 , A 2 ) forms a p erfect SI-decomp osition of A , ea ch of the pairs of states of A 1 and A 2 is r eachable and each pair has to cor resp ond to a different state of A in the mapping β , therefore β is bijectiv e and the theorem follows from Theorem 3.7. Corollary 3.9. L et A b e a minimal DF A and let ( A 1 , A 2 ) b e a p air of DF A s . Then ( A 1 , A 2 ) forms a p erfe ct AI-de c omp osition of A iff ( A 1 , A 2 ) forms a p erfe ct ASB-de c omp osition of A . Pr o of. The claim follo ws from Theor em 3.5, Theorem 3.4 and Theorem 3.8. As a consequence of these facts, we can use the neces sary and sufficien t conditions stated in Theo r em 3.1 to lo ok for p erfect AI- and SI-decomp ositions. Now, let us inspect the relationship b etw een decompo sitions of an automaton and the decomp ositions of the corres po nding minimal automaton. Theorem 3. 10. L et A = ( K , Σ , δ, q 0 , F ) b e a D F A and let A min b e a min- imal DF A such that L ( A ) = L ( A min ) . L et ( A 1 , A 2 ) b e an SI-de c omp osition (AI-de c omp osition, wAI-de c omp osition) of A , then ( A 1 , A 2 ) also forms a de- c omp osition of A min of t he same typ e. Pr o of. First, note that this theorem doe s no t state that any of the decomp osi- tions is nontrivial. T o prov e the statement for SI- de c o mpo sitions, suppo s e that ( A 1 , A 2 ) is an SI-decompo sition of A , th us there exists a mapping α : K 1 × K 2 → K such that it holds ( ∀ w ∈ Σ ∗ ); α ( δ 1 ( q 1 , w ) , δ 2 ( q 2 , w )) = δ ( q 0 , w ), where δ i and q i are the tra ns ition function and the initial state of the automaton A i . Since A min is the minimal automaton co rresp onding to A , there exists some mapping β : K → K min such that ( ∀ w ∈ Σ ∗ ); β ( δ ( q 0 , w )) = δ min ( β ( q 0 ) , w ), wher e δ min is 8 a 1 b / / a   R a 0 b / / a H H b 1 b ) ) a O O b 0 b i i a m m a 1 b / / a   R 1 b * * R 0 b j j a 0 b / / a H H b 1 b ) ) a O O b 0 b i i a O O Figure 2: T ransitio n functions of A min and A ′ . the transition function of A min and K min is the set of states of A min . By the com- po sition of thes e mappings we obtain the mapping β ◦ α : K 1 × K 2 → K min , which combines A 1 and A 2 int o A min in the w ay that the definition of SI-decompositio n requires. F or b oth the AI- and the wAI-deco mpo sition, this statement is trivial, since L ( A ) = L ( A min ). Based on the ab ov e theor em it thus suffices to inspect the SI- (AI-, w AI- ) decomp osability o f the minimal automa ton acc epting a g iven la nguage, and if we show its undecomp osa bility , w e know that the r ecognition of this language cannot be simplified using an advis o r of the res pective type. Ho wev er, this do es not hold for SB- a nd ASB-decompo sitions, as e x hibited b y the following example. Example 3.1. L et us c onsider the language L = { a 2 k b 2 l | k ≥ 0 , l ≥ 1 } . The minimal aut omaton A min = ( K, Σ L , δ, a 0 , { a 0 , b 0 } ) has its tr ansition function define d by the firs t tr ansition diagr am in Fig.2. We c an e asily show t hat this automaton do es not have any nont rivial SB- (and thus neither A S B-) de c omp o- sition by en u mer ating its S.P. p artitions. Now let us examine the automaton A ′ = ( K ′ , Σ L , δ ′ , a 0 , { a 0 , b 0 } ) with the tr ansition function δ ′ define d by the se c ond tr ansition diagr am in Fig.2. Cle arly, L ( A ′ ) = L ( A min ) , but by insp e cting the lattic e of S.P. p artitions of A ′ , we c an find the p air π 1 = {{ a 0 } , { a 1 } , { b 0 , b 1 } , { R 0 , R 1 }} and π 2 = { { a 0 , a 1 , b 0 , R 0 } , { b 1 , R 1 }} such that π 1 · π 2 = 0 and they sep ar ate the final states of A ′ . By The or em 3.1 we c an use these p artitions to c onstruct a n ont rivial ASB- (and thus also SB-) de c omp osition of A ′ forme d by the automata A 1 and A 2 having two and four states, r esp e ctively. Note that b oth A 1 and A 2 have less states than A min . In the following theo rem (inspired by a similar theorem in [5]) we state a condition, under which the situation from the last example cannot occur , i.e., under which any SB-decomp osition of a DF A implies a (maybe simpler) SB- decomp osition of the e q uiv alent minimal DF A. Theorem 3. 11. L et A = ( K, Σ , δ, q 0 , F ) b e a deterministic finite automa- ton and let A min = ( K min , Σ , δ min , q min , F min ) b e the minimal DF A such that L ( A ) = L ( A min ) . L et ( A 1 , A 2 ) b e a nontrivial SB-de c omp osition of A c onsisting of automata having k and l states. If the lattic e of S.P. p artitions of A is dis- tributive, then t her e exists an SB-de c omp osition of A min c onsisting of automata having k ′ and l ′ states, su ch t hat k ′ ≤ k and l ′ ≤ l . Pr o of. Since A min is the minimal DF A such that L ( A ) = L ( A min ), there exists a ma pping f : K → K min such that ( ∀ w ∈ Σ ∗ ); f ( δ ( q 0 , w )) = δ min ( q min , w ). Using the mapping f , let us define a par titio n ρ o n the set of states of A by p ≡ ρ q ⇔ f ( p ) = f ( q ). Clearly , ρ is an S.P . par tition. 9 Since ( A 1 , A 2 ) is a no nt rivia l SB-deco mpo sition o f A , we can use it to obtain S.P . partitions π 1 and π 2 on the set of states of A such that π 1 · π 2 = 0. Let us define new partitions π ′ 1 and π ′ 2 on the set of states of A min by f ( p ) ≡ π ′ i f ( q ) ⇔ p ≡ ρ + π i q . Since it holds tha t ρ + π i  ρ , this definition do es not depend on the choice of the states p and q . It holds that | π ′ i | = | ρ + π i | ≤ | π i | , therefore if we prov e that π ′ 1 and π ′ 2 are S.P . pa rtitions and π ′ 1 · π ′ 2 = 0 , we can use them to construct the desire d deco mpo sition. The fact that π ′ i is a n S.P . pa r tition on the set of s tates of A min is a trivial consequence of the fact that ρ + π i is an S.P . pa rtition on the set of states of A . W e need to prove that π ′ 1 · π ′ 2 = 0. Let us assume that p ′ and q ′ are states of A min such that p ′ ≡ π ′ 1 · π ′ 2 q ′ and p, q a re some states of A suc h that f ( p ) = p ′ and f ( q ) = q ′ . The n p ′ ≡ π ′ 1 q ′ and p ′ ≡ π ′ 2 q ′ , and b y definition o f π ′ i we get p ≡ ρ + π 1 q and p ≡ ρ + π 2 q , which is equiv alent to p ≡ ( ρ + π 1 ) · ( ρ + π 2 ) q . Since the lattice of all S.P . partitions o f A is distributive, w e have ( ρ + π 1 ) · ( ρ + π 2 ) = ρ + ( π 1 · π 2 ) = ρ + 0 = ρ , therefore p ≡ ρ q , which by definition of ρ implies that f ( p ) = f ( q ), in other words p ′ = q ′ . Hence π ′ 1 · π ′ 2 = 0 . 4 Degrees of Decomp osabilit y It is easy to see that for e ach t ype of deco mpo s ition, there exist undecomp osable regular languag es (e.g. L ( n ) = { a k | k ≥ n − 1 } is wAI-undecomp osable for each n ∈ N ). Ther e also exist reg ular langua g es, that are pe rfectly dec omp o sable in each wa y (e.g. L ( k,l ) = { w ∈ { a, b } ∗ | # a ( w ) mo d k = 0 ∧ # b ( w ) mo d l = 0 } has a p erfect ASB-decomp osition for a ll k, l ≥ 2). W e shall now investigate whether all v alues b et ween these t wo limits can be achiev ed. Definition 4.1. L et A b e a DF A, let ( A 1 , A 2 ) b e its nontrivial SB- (AS B-) de c omp osition with the c orr esp onding S.P. p artitions π 1 and π 2 . We shal l c al l this de c omp osition redundan t , if ther e ex ist S.P. p artitions π ′ 1  π 1 and π ′ 2  π 2 such that at le ast one of these ine qualities is strict, but it stil l holds π ′ 1 · π ′ 2 = 0 (and π ′ 1 and π ′ 2 sep ar ate the final states of A ). Lemma 4.1. F or e ach r, s ∈ N , r , s ≥ 2 , t her e exists a minimal DF A A c onsist- ing of r.s states and ha ving only one nontrivial nonr e dundant SB-de c omp osition (ASB-de c omp osition) up to the or der of automata, c onsisting of automata hav- ing r and s states. Pr o of. Let us study the minimal automa ton A r,s = ( K, Σ , δ, q 0 , 0 , F ) defined by K = { q i,j | i ∈ { 0 , . . . , r − 1 } , j ∈ { 0 , . . . , s − 1 }} , F = { q r − 1 ,s − 1 } a nd the transition function δ defined b y δ ( q i,j , a ) = q i +1 ,j for i ∈ { 0 , . . . , r − 2 } , j ∈ { 0 , . . . , s − 1 } δ ( q r − 1 ,j , a ) = q r − 1 ,j for j ∈ { 0 , . . . , s − 1 } δ ( q i,j , b ) = q i,j +1 for i ∈ { 0 , . . . , r − 1 } , j ∈ { 0 , . . . , s − 2 } δ ( q i,s − 1 , b ) = q i,s − 1 for i ∈ { 0 , . . . , r − 1 } . T o insp ect the SB-decomp ositio ns o f A r,s , let us study the S.P . partitions o n the s et o f its states . F ro m the metho d for gener ating a ll S.P . partitions of an automaton that is describ ed in [5], we know tha t each non trivial S.P . pa rtition can b e obtained as a sum of some pa rtitions π m p,t , where π m p,t denotes the minimal 10 S.P . partition such that it do es not distinguish b etw een states p and t , i.e., they belo ng in to the s ame blo ck. Let us determine π m p,t for v arious states p and t o f A r,s . First, let us consider the case of π m p,t such that p = q i,j , t = q i ′ ,j ′ and b oth inequalities i < i ′ and j < j ′ hold. Since q i,j ≡ π q i ′ ,j ′ , δ ( q i,j , a i ′ − i b j ′ − j ) = q i ′ ,j ′ and δ ( q i ′ ,j ′ , a i ′ − i b j ′ − j ) = q 2 i ′ − i, 2 j ′ − j (if 2 i ′ − i < r and 2 j ′ − j < s ), as a consequence of the substitution pr op erty of π , we obtain q i,j ≡ π q 2 i ′ − i, 2 j ′ − j . By applying this argument a finite num b er of times (keeping in mind the con- struction of A r,s ), we obtain q i,j ≡ π q r − 1 ,s − 1 . Now let k ∈ { i, . . . , r − 1 } and let l ∈ { i, . . . , s − 1 } . Then δ ( q i,j , a k − i b l − j ) = q k,l and δ ( q i ′ ,j ′ , a k − i b l − j ) = q k + i ′ − i,l + j ′ − j (if such states exist), therefore q k,l ≡ π q k + i ′ − i,l + j ′ − j . Again, w e can use the same arg ument to show that q k,l ≡ π q r − 1 ,s − 1 . Therefore, for this t yp e of π = π m p,t , w e hav e q k,l ≡ π q k ′ ,l ′ for all k, l , k ′ , l ′ such tha t i ≤ k , k ′ < r and j ≤ l , l ′ < s . Now let us conside r the case o f π m p,t such that p = q i,j , t = q i ′ ,j ′ and it ho lds i > i ′ and j < j ′ . Since q i,j ≡ π q i ′ ,j ′ , δ ( q i,j , a r − 1 − i b s − 1 − j ′ ) = q r − 1 ,s − 1 − ( j ′ − j ) and δ ( q i ′ ,j ′ , a r − 1 − i b s − 1 − j ′ ) = q r − 1 − ( i − i ′ ) ,s − 1 , as a consequence of the substitu- tion prop erty of π , w e have q r − 1 ,s − 1 − ( j ′ − j ) ≡ π q r − 1 − ( i − i ′ ) ,s − 1 . B y exploiting the substitution prop erty again on this equiv alence, using the words a i − i ′ − 1 , b j ′ − j − 1 and b j ′ − j , w e o bta in q r − 2 ,s − 1 ≡ π q r − 1 ,s − 1 ≡ π q r − 2 ,s − 2 . Therefore in this case, no suc h π m p,t partition can distinguish b etw een states q r − 2 ,s − 1 , q r − 1 ,s − 1 and q r − 2 ,s − 2 . The last case to consider is the case o f π m p,t such that p = q i,j , t = q i ′ ,j ′ and it holds i = i ′ (the case j = j ′ is analog ous). Without loss o f generality , w e can assume that j < j ′ . No w, using the same arguments as in the fir st case, we can show that q i,l ≡ π q i,l ′ for all l , l ′ such that j ≤ l , l ′ < s . Ther efore for ea ch given k such that i ≤ k < r , it holds that q k,l ≡ π q k,l ′ and all o f the states no t men tioned in this equiv alence form separa te blo cks of π m p,t . It is ea sy to verify that one non trivia l ASB-decomp os ition of A r,s is giv en by the S.P . pa rtitions π 1 = {{ q 0 , 0 , . . . , q 0 ,s − 1 } , { q 1 , 0 , . . . , q 1 ,s − 1 } , . . . , { q r − 1 , 0 , . . . , q r − 1 ,s − 1 }} and π 2 = {{ q 0 , 0 , . . . , q r − 1 , 0 } , { q 0 , 1 , . . . , q r − 1 , 1 } , . . . , { q 0 ,s − 1 , . . . , q r − 1 ,s − 1 }} Now we sho w that any o ther SB-decomp osition of A r,s is given by S.P . par titio ns preceding to π 1 and π 2 in the partia l order  and therefore is r edundant. Indeed, notice that none of the π m p,t partitions o f the firs t and the seco nd discussed t yp e can distinguish betw een any of the states q r − 2 ,s − 1 , q r − 1 ,s − 1 and q r − 2 ,s − 2 , therefor e no sum of them can, either. F or the par titions o f the third t yp e, it ho lds either q r − 2 ,s − 1 ≡ π q r − 1 ,s − 1 or q r − 1 ,s − 1 ≡ π q r − 2 ,s − 2 , therefor e it will take t w o partitions to distinguish betw een these thr ee states. Hence a n y nontrivial SB-decomp osition is determined by tw o S.P . par titions, b oth of which m ust b e of the third t yp e. But it is easy to see that for any partition π of this t yp e it holds either π  π 1 or π  π 2 . Definition 4.2. L et A = ( K, Σ , δ, q 0 , F ) b e a deterministic fin ite automaton, let K ∩ { p 0 , p 1 , . . . , p k − 1 } = ∅ and let c b e a new symb ol not include d in Σ . We shal l define a k -extension A ′ of the automaton A by the fol lowing c onst ruction: A ′ = ( K ∪ { p 0 , p 1 , . . . , p k − 1 } , Σ ∪ { c } , δ ′ , p 0 , F ) , wher e the tr ansition fun ction δ ′ 11 is define d as fol lows: ( ∀ q ∈ K ) ( ∀ a ∈ Σ); δ ′ ( q , a ) = δ ( q , a ) ( ∀ q ∈ K ); δ ′ ( q , c ) = q ( ∀ p ∈ { p 0 , p 1 , . . . , p k − 1 } ) ( ∀ a ∈ Σ); δ ′ ( p, a ) = p ( ∀ i ∈ { 0 , 1 , . . . , k − 2 } ); δ ′ ( p i , c ) = p i +1 δ ′ ( p k − 1 , c ) = q 0 . Lemma 4. 2 . L et A b e a DF A c onsisting of n states, al l of which ar e r e ach- able. L et A ′ b e its k -ext ension. Then A has a nontrivial nonr e dun dant SB- de c omp osition (ASB-de c omp osition) c onsisting of automata having r and s states iff A ′ has a nontrivial nonr e dundant de c omp osition of the same typ e, c onsisting of automata having k + r and k + s states. Pr o of. W e will try to insp ect S.P . partitions o n the set o f states of A ′ , using the notation from Definition 4.2. Let us assume that π ′ is an S.P . partition on the set of states o f A ′ such that p i and p j are in the sa me blo ck of π ′ ; i, j ∈ { 0 , 1 , . . . , k − 1 } . As a consequence o f the S.P . pro p e rty , if i, j < k − 1 then also p i +1 and p j +1 are in the same blo ck of π ′ , b ecause δ ′ ( p i , c ) = p i +1 and δ ′ ( p j , c ) = p j +1 . By applying this argument a finite n um b er of times, we can show that there exists some l ∈ { 0 , 1 , . . . , k − 2 } such that p l ≡ π ′ p k − 1 , and using the argument once more, w e obtain p l +1 ≡ π ′ q 0 . Howev er, it holds δ ′ ( p l , a ) = p l for a ll a ∈ Σ, hence p l ≡ π ′ δ ′ ( q 0 , w ) for a ll w ∈ Σ ∗ . Since a ll of the states of A ′ are reachable, we have p l ≡ π ′ q for all q ∈ K . Th us suc h a par tition cannot distinguish b etw een the orig inal states of the automaton A . Now let us suppo se that π ′ is an S.P . par tition on the set of states of A ′ such that for so me i ∈ { 0 , 1 , . . . , k − 1 } , p i ≡ π ′ q for some q in K . Then it a lso holds that p i ≡ π ′ p i +1 , b ecause δ ( p i , c ) = p i +1 and δ ( q , c ) = q . But we hav e alr eady shown that p i ≡ π ′ p i +1 implies that all of the states in K are equiv ale nt mo dulo π ′ , th us this S.P . par tition cannot distinguish b etw een the states o f A , either. ¿F ro m these observ atio ns it follows that if π ′ is any S.P . pa r tition on the set of sta tes of A ′ such that the s tates o f A a re not all equiv alent modulo π ′ , then π ′ m ust als o co n tain k blo cks, each of which contains only one state p i , where i ∈ { 0 , 1 , . . . , k − 1 } . Now we can prov e the equiv a lence stated in the theorem. Let A have an SB-decompo sition consisting of r and s states. Then there exist S.P . partitions π 1 and π 2 on the set o f states of A having r and s blo cks, such tha t π 1 · π 2 = 0. Let us no w co nstruct new partitions π ′ 1 and π ′ 2 on the set of s tates o f A ′ by π ′ 1 = π 1 ∪ {{ p 0 } , { p 1 } , . . . , { p k − 1 }} and π ′ 2 = π 2 ∪ {{ p 0 } , { p 1 } , . . . , { p k − 1 }} . Obviously , π ′ 1 and π ′ 2 hav e substitution pro pe rty , b e- cause for the sta tes in K this prop er t y is inher ited from π 1 and π 2 , and the new states p 0 , p 1 , . . . , p k − 1 cannot vio late this prop erty either, b ecause each of these states b elongs to a separ ate blo ck in π ′ 1 and π ′ 2 , mak ing the substitution pro p- erty hold trivially . Neither do the new c -mov es defined on the states from K violate the substitution proper ty . Finally , it holds that π ′ 1 · π ′ 2 = 0 . T o see this, note that for a state q ∈ K , it holds [ q ] π ′ 1 · π ′ 2 = [ q ] π 1 · π 2 = { q } , since π 1 · π 2 = 0. F o r a state q ∈ K ′ − K , [ q ] π ′ i = { q } for i ∈ { 1 , 2 } th us [ q ] π ′ 1 · π ′ 2 = { q } , to o. Hence each state of A ′ belo ngs to a sepa rate blo ck of π ′ 1 · π ′ 2 , whic h implies π ′ 1 · π ′ 2 = 0. Therefore π ′ 1 and π ′ 2 induce an SB-deco mpos ition of A ′ . It is also eas y to see that if π 1 and π 2 separate the final states of A , then also π ′ 1 and π ′ 2 separate the final states of A ′ , making the induced decomp os ition an ASB-decomp osition. 12 On the other hand, let us now assume that A ′ has an SB-decomp osition a nd π ′ 1 and π ′ 2 are the S.P . par titions on K ′ that induce this decomp osition, thus π ′ 1 · π ′ 2 = 0. F r om the obser v ations made in the b eginning of this pro of, w e kno w that any S.P . partition that ca n dis tinguish b et ween the states in K in any wa y , m ust contain each of the sta tes p 0 , p 1 . . . p k − 1 in a separate blo ck con taining only this state. As π ′ 1 · π ′ 2 = 0, for all q 1 , q 2 ∈ K , at least o ne of these partitions must distinguish b etw een these states, i.e., [ q 1 ] π ′ i 6 = [ q 2 ] π ′ i . If o ne of the par titions distinguished b etw een all such pairs, it would imply that this partition must contain a se pa rate blo ck for each one of the states in K ′ , th us b ecoming a trivial partition 0, resulting in a trivial deco mpo s ition. Ther efore b oth π ′ 1 and π ′ 2 hav e to distinguis h b etw een so me pair of states from K , whic h implies that they both contain a separate blo ck for each of the states p 0 , p 1 . . . p k − 1 containing no other state. By removing these k blo cks from π ′ 1 and π ′ 2 , w e obtain new partitions π 1 and π 2 on the set K , such that π 1 = π ′ 1 − {{ p 0 } , { p 1 } , . . . , { p k − 1 }} and π 2 = π ′ 2 − {{ p 0 } , { p 1 } , . . . , { p k − 1 }} . The s e partitions preserve the substitution prope rty , since ( ∀ a ∈ Σ)( ∀ q ∈ K ): δ ( q , a ) ∈ K and π ′ 1 and π ′ 2 were S.P . partitions. It also holds π 1 · π 2 = 0 , as for all q 1 , q 2 ∈ K , q 1 ≡ π 1 · π 2 q 2 implies q 1 ≡ π ′ 1 · π ′ 2 q 2 and that implies q 1 = q 2 . So π 1 and π 2 induce an SB- decomp osition of A . As π ′ 1 and π ′ 2 were nontrivial, s o are π 1 and π 2 and the obtained decomp osition. It is a gain easy to see that if π ′ 1 and π ′ 2 separate the final states of A ′ , then als o π 1 and π 2 m ust s e pa rate the final states of A . The descr ibed relationship betw een the S.P . par titions on the set of states of A and the corr esp onding S.P . par titions o n A ′ also implies, that each de- comp osition o f A is nonr edundant iff the co r resp onding decomp ositio n of A ′ is nonredundant, to o . Since a k -extensio n of a minimal DF A is again a minim al DF A, we can combine the le mma s to obtain the following theorem. Theorem 4.3. L et n ∈ N b e su ch that n = k + r.s , wher e r , s, k ∈ N , r , s ≥ 2 . Then ther e exists a minimal DF A A c onsisting of n states, such that it has only one n ontrivial n onre dundant SB-de c omp osition (ASB-de c omp osition) up to the or der of the automata in the de c omp osition, and t his de c omp osition c onsists of automata with k + r and k + s states. References [1] J. L. Balcaza r , J. Diaz , J. Gabarro, Structur al Complexity I. , Springer- V erla g New Y ork , 1988 [2] S. Even, A. L. Selma n, Y. Y acobi, The Complexity o f Pr omise Pr oblems with Applic ations to Public-Key Crypto gr aphy , Information and Control 61(2): 159-1 73 (198 4) [3] S. Y u, State Complexity: R e c ent R esults and Op en Pr oblems , F undament a Informaticae 64: 471 -480 (2005) [4] J. C. Birg et, Interse ction and Union of R e gular L anguages and S tate Com- plexity , Information Pro cessing Letters 43: 185 -190 (1 9 92) [5] J. Hartmanis and R. E. Stear ns, Algebr aic Structu re The ory of S e qu en tial Machines , Prentice-Hall, 1 966 13 [6] J. E. Hopcro ft and J. 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