Finitely generated ideal languages and synchronizing automata
Vladimir V. Gusev, Marina I. Maslennikova, Elena V. Pribavkina
aa r X i v : . [ c s . F L ] M a y Finitely generated ideal languagesand synchronizing automata
Vladimir V. Gusev, Marina I. Maslennikova, Elena V. Pribavkina
Ural Federal University, Ekaterinburg, Russia [email protected], [email protected],[email protected]
Abstract.
We study representations of ideal languages by means ofstrongly connected synchronizing automata. For every finitely generatedideal language L we construct such an automaton with at most 2 n states,where n is the maximal length of words in L . Our constructions are basedon the De Bruijn graph. Keywords: ideal language, synchronizing automaton, synchronizing word,reset complexity.
Let A = h Q, Σ, δ i be a deterministic finite automaton (DFA for short), where Q is the state set , Σ stands for the input alphabet , and δ : Q × Σ → Q is the transition function defining an action of the letters in Σ on Q . When δ is clearfrom the context, we will write q . w instead of δ ( q, w ) for q ∈ Q and w ∈ Σ ∗ .A DFA A = h Q, Σ, δ i is called synchronizing if there exists a word w ∈ Σ ∗ which leaves the automaton in unique state no matter at which state in Q it isapplied: q . w = q ′ . w for all q, q ′ ∈ Q . Any word w with such property is said to be synchronizing (or reset ) word for the DFA A . For the last 50 years synchronizingautomata received a great deal of attention. For a brief introduction to the theoryof synchronizing automata we refer the reader to the recent surveys [7, 8].In the present paper we focus on language theoretic aspects of the theory ofsynchronizing automata. We denote by Syn( A ) the language of synchronizingwords for a given automaton A . It is well known that Syn( A ) is regular [8]. Fur-thermore, it is an ideal in Σ ∗ , i.e. Syn( A ) = Σ ∗ Syn( A ) Σ ∗ . On the other hand,every ideal language L serves as a language of synchronizing words for some au-tomaton. For instance, the minimal automaton of the language L is synchronizedby L [4]. Thus, synchronizing automata can be considered as a special represen-tation of ideal languages. Effectiveness of such representation was addressedin [4]. The reset complexity rc ( L ) of an ideal language L is the minimal possiblenumber of states in a synchronizing automaton A such that Syn( A ) = L . Ev-ery such automaton A is called minimal synchronizing automaton (for brevity,MSA). Let sc ( L ) be the number of states in the minimal automaton recognizing L . Then for every ideal language L we have rc ( L ) ≤ sc ( L ) ⋆ . Moreover, there are ⋆ since the minimal automaton is synchronized by L Vladimir V. Gusev, Marina I. Maslennikova, Elena V. Pribavkina languages L n for every n ≥ rc ( L n ) = n and sc ( L n ) = 2 n − n , see [4].Thus, representation of an ideal language by means of a synchronizing automa-ton can be exponentially smaller than “traditional” representation via minimalautomaton. However, no reasonable algorithm is known for computing MSA ofa given language. One of the obstacles is that MSA is not uniquely defined. Forinstance, there is a language with at least two different MSA’s: one of them isstrongly connected, another one has a sink state [4]. Therefore, some refinementof the notion of MSA seems to be necessary. Another important observation isthe following: minimal synchronizing automata for the aforementioned languages L n are strongly connected. Thus, one may expect that there is always a stronglyconnected MSA for an ideal language. In the present paper we show that it is notthe case. Moreover, the smallest strongly connected automaton with a language L as the language of synchronizing words may be exponentially larger than aminimal synchronizing automaton of L .Another source of motivation for studying representations of ideal languagesby means of synchronizing automata comes from the famous ˇCern´y conjecture.ˇCern´y already in 1964 conjectured that every synchronizing automaton possessesa synchronizing word of length at most ( n − . Despite intensive efforts of re-searchers this conjecture is still widely open. We can restate the ˇCern´y conjecturein terms of reset complexity as follows: if ℓ is the minimal length of words inan ideal language L then rc ( L ) ≥ √ ℓ + 1. Thus, we hope that deeper under-standing of reset complexity will bring us new ideas to resolve this long standingconjecture. It is well known that the ˇCern´y conjecture holds true whenever itholds true for strongly connected automata. In this regard an interesting relatedquestion was posed in [2]: does every ideal language serve as the language of syn-chronizing words for some strongly connected automaton? For instance, if theanswer is negative then there is a way to simplify formal language statement ofthe ˇCern´y conjecture. Unfortunately, it is not the case. Recently Reiss and Ro-daro [6] for every ideal language ⋆⋆ L presented a strongly connected automaton A such that Syn ( A ) = L . Their proof is non-trivial and technical. In the presentpaper we give simple constructive proof of the fact that every finitely generatedideal language L , i.e. L = Σ ∗ U Σ ∗ for some finite set U , serves as the languageof synchronizing words of some strongly connected automaton. Our construc-tions reveal interesting connections with classical objects from combinatorics onwords. Let Σ be a finite alphabet with | Σ | >
1. Let L be a finitely generated ideallanguage over Σ , i.e. L = Σ ∗ SΣ ∗ , where S is a finite set of words. In thissection we construct a strongly connected synchronizing automaton for which L = Σ ∗ SΣ ∗ .First recall some standard definitions and fix notation. A word u is a factor ( prefix , suffix ) of a word w , if w = xuy ( w = uy , w = xu respectively) for some ⋆⋆ over an alphabet with at least two lettersinitely generated ideal languages and synchronizing automata 3 x, y ∈ Σ ∗ . By Fact ( w ) we denote the set of all factors of w . The i th letter of theword w is denoted by w [ i ]. The factor w [ i ] w [ i + 1] · · · w [ j ] is denoted by w [ i..j ].By Σ n ( Σ ≤ n , Σ ≥ n ) we denote the set of all words over Σ of length n (at most n , at least n respectively).Note, that if a word s ∈ S is a factor of some other word t ∈ S , then the word t may be deleted from the set S without affecting the ideal language, generatedby S . Thus, we may assume, that the set S is anti-factorial , i.e. no word in S isa factor of another word in S . Σ n Theorem 1.
Let Σ = { a, b } . There is unique up to isomorphism strongly con-nected synchronizing automaton B such that Syn( B ) = Σ ≥ n .Proof. Consider De Bruijn graph for the words of length n . Recall that thevertices of this graph are the words of length n , and there is a directed edgefrom the vertex u to the vertex v , if u = xs and v = sy for some s ∈ Σ n − , x, y ∈ Σ . By labeling each edge e = ( u, v ) by the last letter of v we obtain DeBruijn automaton. Its state set is Q = Σ n , and transition function is defined inthe following way: xs . y = sy for s ∈ Σ n − , x, y ∈ Σ . De Bruijn automaton isknown to be strongly connected. Thus it remains to verify that Syn( B ) = L .It is easy to see that for an arbitrary word u of length at most n we have Q . u = Σ n −| u | u . Hence for any word w of length n we have | Q . w | = 1, and forany word u of length less than n we have | Q . u | >
1. So, Syn( B ) = L. Let C = h Q, Σ, δ i be a strongly connected synchronizing DFA such thatSyn( C ) = L . Let us prove that | Q | ≤ n . Strong connectivity implies Q . a ∪ Q . b = Q . By induction it is easy to see that Q = S | w | = k Q . w . In particular, wehave Q = S | w | = n Q . w . Thus, | Q | = | S | w | = n Q . w | ≤ P | w | = n | Q . w | = 2 n . Thelast equality follows from the fact that every word of length n synchronizes C , soeach Q . w is a singleton. For the converse inequality 2 n ≤ | Q | consider the DFA C a , obtained from C by removing all transitions corresponding to the action of b in C . The word a n synchronizes C , so C a contains no cycles but unique loop.So the automaton C a has a tree-like structure as it is shown on Fig.1. Denoteby s the state of C such that s . a = s . The state s is called root of the tree, andthe states p , p , . . . , p k having no incoming transitions labeled by a are called leaves of the tree. The height h ( p i ) of a vertex p i is the length of the path from p i to the root s . The height of the tree h ( C a ) is the maximal height of its leaves.We have h ( C a ) = n . Indeed, if h ( C a ) = h < n , then we would have Q . a h = { s } ,meaning that a h ∈ Syn( C ), which is impossible.Consider the set of leaves H = Q \ Q . a = { p , p , ..., p k } . Since the DFA C is strongly connected, for each state p ℓ in H there exists a state q ℓ such that q ℓ . b = p ℓ . Thus H ⊆ Q . b . We show that H is exactly Q . b , meaning that
Q . a ∩ Q . b = ∅ . Take a leaf of height n . Without loss of generality supposeit is p . Let q be such that q . b = p . The word ba n − is synchronizing, so Q . ba n − = { q } for some q ∈ Q . We have q . ba n − = q , and q . a = s (seeFig.1). Suppose there is p ∈ Q . a ∩ Q . b.
Then there is a state q such that q . b = p . Vladimir V. Gusev, Marina I. Maslennikova, Elena V. Pribavkina p p p p k ...sq aa a a aa aa a aa Fig. 1.
The action of a in C Since p is not a leaf, we have h ( p ) < n . Then q . ba n − = p . a n − = s = q . Acontradiction. Hence H = Q . b . Furthermore, the height of any leaf of C a isexactly n . To see this assume that there exists a state p m such that h ( p m ) < n ,i.e. p m . a ℓ = s , for some ℓ < n . Then the word ba n − is not synchronizing. Indeed,take a state q m such that q m . b = p m . We have q m . ba n − = p m . a n − = s = q .Consider an arbitrary state p ∈ Q . a . Let δ − ( p, u ) = { p ′ ∈ Q | p ′ . u = p } . We prove that | δ − ( p, a ) | ≥ p ∈ Q . a . For the root s we have { s, q } ⊆ δ − ( s, a ), thus, | δ − ( s, a ) | ≥
2. Let p be an arbitrary state in Q . a .Strong connectivity of C implies that there exists a state p and a word w ∈ Σ n such that p . w = p . Since w is synchronizing, we have Q . w = { p } . Consider theword w [1 ..n −
1] that does not synchronize C . Then | Q . w [1 ..n − | ≥
2. However,(
Q . w [1 ..n − . w [ n ] = p . And we obtain the inequality | δ − ( p, a ) | ≥
2. Denote H = { q } and construct sets H i = δ − ( H i − , a ) for 1 ≤ i ≤ n −
1. We have | H i | ≥ i for all 1 ≤ i ≤ n −
1. Then C possesses at least 1 + 1 + 2 + 4 + ... + 2 n − = 2 n states.Thus we have | Q | = 2 n . Moreover, Q = ∪ | w | = n Q . w . It means that with eachstate q of Q we can associate the word w of length n such that Q . w = { q } . Itis clear that it gives us the desired isomorphism between C and B . ⊓⊔ Remark 1.
In case Σ = { a, b }∪ ∆ , where ∆ = ∅ , we consider De Bruijn automa-ton constructed for the binary alphabet { a, b } and put the action of each letterin ∆ to be the same as the action of the letter a . It is clear that the language ofsynchronizing words of the modified De Bruijn automaton coincides with Σ ≥ n .The Proposition implies that the minimal DFA recognizing an ideal lan-guage L can be exponentially smaller than a strongly connected MSA B with Syn ( B ) = L . initely generated ideal languages and synchronizing automata 5 Let U ( Σ n . There is a strongly connected synchronizing automa-ton B U with n states such that Syn( B U ) = Σ ∗ U Σ ∗ .Proof. We modify the De Bruijn automaton B from the section 2.1 to obtainthe desired automaton B U . First of all it is convenient to view the states of theautomaton B not as the words of length n , but as pairs ( x, u ), where x ∈ Σ and u ∈ Σ n − . Then by the definition of the transitions in B we have( x, u ) y −→ ( z, v ) ⇔ uy = zv (1)For a word uy which is not in U , we modify the corresponding transition givenby (1) in the following way. If uy / ∈ U ∪ { a n , b n } we put( x, u ) y −→ ( x, v ) , (2)where v is defined by (1).If uy = a n / ∈ U ( uy = b n / ∈ U respectively) we put( a, a n − ) a −→ ( b, a n − ) , (( b, b n − ) b −→ ( a, b n − ) respectively) . (3)The other transitions remain unchanged. The obtained automaton is denoted ( a, aa ) ( a, ab )( a, ba ) ( a, bb )( b, aa ) ( b, ab )( b, ba ) ( b, bb ) a bb a ba ba b a ba b aa b Fig. 2.
De Bruijn automaton for n = 3 by B U . The examples of the automaton B and the corresponding modifiedautomaton B U for U = { aaa, abb, bab } are shown on Fig.2 and Fig.3 respectively.We prove that the automaton B U satisfies the statement of the proposition. Firstwe show that B U is strongly connected. For this purpose we prove that all thestates are reachable from the state ( a, a n − ), and the state ( a, a n − ) is reachablefrom all states. Vladimir V. Gusev, Marina I. Maslennikova, Elena V. Pribavkina( a, aa ) ( a, ab )( a, ba ) ( a, bb )( b, aa ) ( b, ab )( b, ba ) ( b, bb ) a b a ba ba b a bba b aa b Fig. 3.
Automaton B U for U = { aaa, abb, bab } First we show that a state ( a, u ) is reachable from ( a, a n − ) for any u ∈ Σ n − .If u = a n − , the claim obviously holds. Hence we may assume u = a k b ˆ u , where k ≥
0, ˆ u ∈ Σ n − k − . By the definition of transitions in B U we have( a, a n − ) b −→ ( a, a n − b ) ˆ u [1] −−→ ( a, a n − b ˆ u [1]) ˆ u [2] −−→ · · · ˆ u [ n − k − −−−−−−→ ( a, a k +1 b ˆ u [1 ..n − k − ˆ u [ n − k − −−−−−−→ ( a, a k b ˆ u [1 ..n − k − a, u ) . Symmetrically any state ( b, u ) is reachable from the state ( b, b n − ). The latterstate is reachable from ( a, b n − ). Thus the state ( b, u ) is reachable also from( a, a n − ): ( a, a n − ) ( a, b n − ) b −→ ( b, b n − ) ( b, u ) . Now we show that the state ( a, a n − ) is reachable from any other state.Apply the word a n − to an arbitrary state ( x, u ). By the definition of transitionswe have ( x, u ) . a n − ∈ { ( a, a n − ) , ( b, a n − ) } . If ( x, u ) . a n − = ( a, a n − ) we aredone. If ( x, u ) . a n − = ( b, a n − ), then we apply once more the letter a and obtain( x, u ) . a n = ( a, a n − ).Thus the constructed automaton B U is strongly connected. Next we showthat Syn( B U ) = Σ ∗ U Σ ∗ . It is easy to see that for any word u ∈ Σ n − we have Q . u ⊆ { ( a, u ) , ( b, u ) } , and Q . u ∩ Q . v = ∅ for u, v ∈ Σ n − such that u = v .Thus Q ⊇ S | u | = n − Q . u . Next we check that Q = S | u | = n − Q . u . Indeed, if a n ∈ U we have ( a, a n − ) u −→ ( a, u ) for all u ∈ Σ n − . If a n U take any word u ∈ Σ n − .If u = a n − then u maps the state ( a, a n − ) or the state ( b, a n − ) to ( a, u ).Let us assume now that u = a k b ˆ u . If k is even (odd, respectively) then u maps( a, a n − ) (( b, a n − ), respectively) to ( a, u ). So any states ( a, u ) belongs to theset S | u | = n − Q . u . Symmetrically any states ( b, u ) belongs to the latter set. Hence Q = S | u | = n − Q . u . Since | Q | = 2 n , if there is a synchronizing word u of length initely generated ideal languages and synchronizing automata 7 n −
1, we would have 2 n = | Q | = | S | u | = n − Q . u | < n , which is a contradiction.Thus, none of the words of length n − w of length n and factorize it as w = uy with u ∈ Σ n − and y ∈ Σ . Wehave Q . u = { ( a, u ) , ( b, u ) } . If w ∈ U , then the corresponding transitions fromthe states ( a, u ) and ( b, u ) were not changed, and we have Q . uy = { ( z, v ) } ,where uy = zv , so w is synchronizing. If w / ∈ U , then Q . uy = { ( a, v ) , ( b, v ) } ,where v is such that uy = zv for some z ∈ Σ , so w / ∈ Syn( B U ). ⊓⊔ Let S be finite and anti-factorial set of words in Σ + . There is astrongly connected synchronizing automaton C S such that Syn( C S ) = Σ ∗ SΣ ∗ . This automaton has at most n states, where n = max {| s | | s ∈ S } .Proof. Let T = { w ∈ Σ n | ∃ s ∈ S, s ∈ Fact ( w ) } . First we construct the au-tomaton B T as described in the previous proposition. In that proposition thestates of B T were viewed as pairs ( x, u ) with x ∈ Σ , u ∈ Σ n − . Here it will beconvenient to view the states as the words xu of length n (as it was in the initialDe Bruijn automaton). Note, that since S is anti-factorial, every state in T canbe uniquely factorized as usv such that s ∈ S , u, v ∈ Σ ∗ and sv does not containfactors in S except s . In what follows we will use this unique representationwithout stating it explicitly.Next we define an equivalence relation ≃ on the set of states of this automaton(i.e. on words of length n ) in the following way. Let w, w ′ ∈ T . We have w ≃ w ′ iff w = usv and w ′ = u ′ sv , where s ∈ S , u, u ′ , v ∈ Σ ∗ . On the set Σ n \ T therelation ≃ is defined trivially, i.e. for w, w ′ ∈ Σ n \ T we have w ≃ w ′ iff w = w ′ .It is easy to see that ≃ is indeed an equivalence relation on Σ n . In fact, ≃ isa congruence on the set of states of the automaton B T . Let us check that forany x ∈ Σ and any w, w ′ ∈ Σ n w ≃ w ′ implies w . x ≃ w ′ . x . If w, w ′ ∈ Σ n \ T ,then w = w ′ and we are done. if w, w ′ ∈ T , then w = usv , w ′ = u ′ sv . If u = u ′ = ε , then w = w ′ , and there is nothing to prove. So we may assume,that u, u ′ = ε . Then usv . x = tsvx and u ′ sv . x = t ′ svx for some t, t ′ ∈ Σ ∗ .Since the obtained two words have the same suffixes, containing a word in S ,they are equivalent. So we can consider the factor automaton B T / ≃ , whosestates are the equivalence classes of ≃ , and the transition function is inducedfrom the initial automaton. Let us denote by [ sv ] the equivalence class of a word usv ∈ T , and by [ u ] the equivalence class of a word u / ∈ T . We claim, that C S = B T / ≃ . In other words, the constructed automaton is strongly connected,and Syn( B T / ≃ ) = Σ ∗ SΣ ∗ . The first property holds trivially, since a factorautomaton of a strongly connected automaton is strongly connected.For any w ∈ Σ ∗ and s ∈ S previously in B T we had w . s = us , where u ∈ Σ ∗ . Since S is anti-factorial, in B T / ≃ we have [ w ] . s = [ s ], so any s ∈ S issynchronizing for the automaton B T / ≃ . Now let t be a synchronizing word, sothere is a state [ w ] such that for any state [ w ′ ] we have [ w ′ ] . t = [ w ]. If [ w ] is a Vladimir V. Gusev, Marina I. Maslennikova, Elena V. Pribavkina one-element class, then the word t was synchronizing for the initial automaton B T , so t contains some word in S as a factor, i.e. t ∈ Σ ∗ SΣ ∗ . Consider the casewhere [ w ] is a class consisting of elements u sv, u sv, . . . , u k sv , k >
1. Notethat in this case u i = ε for each i = 1 , . . . , k . This means that t = usv for some u ∈ Σ ∗ , thus, also in this case t ∈ Σ ∗ SΣ ∗ . ⊓⊔ Complete this section with an example. Let S = { a , aba } and Σ = { a, b } .Construct the corresponding set T = { a , a b, ba , aba } . Next build the DFA B T . The resulting automaton is shown on the left side of Fig.4. aaa aababa abbbaa babbba bbba b a bbaa b a bba baa b [ aa ] [ aab ][ aba ] [ abb ][ bab ][ bba ] [ bbb ] a b a bba a bba baa b Fig. 4.
Automata B T and B T / ≃ for T = { aaa, aab, baa, aba } By the definition of ≃ the class [ aa ] includes states aaa and baa . The restclasses are one-element. The resulting automaton B T / ≃ is shown on the rightside of Fig.4 Let S = { u, v } ⊆ Σ + and let | u | = n , | v | = m . Again we suppose that S is anti-factorial. In this case we can construct a strongly connected automaton D u,v such that Syn( D u,v ) = Σ ∗ ( u + v ) Σ ∗ with n + m states, thus improving construc-tion from the previous section. For simplicity we state and prove the followingtheorem only for the case of binary alphabet, although the same argument worksin general. Theorem 4.
Let Σ = { a, b } , and let u ∈ Σ n \ { ab n − , a n − b, ba n − , b n − a } , v ∈ Σ m \{ ab m − , a m − b, ba m − , b m − a } . There is a strongly connected synchronizingautomaton D u,v having n + m states such that Syn( D u,v ) = Σ ∗ ( u + v ) Σ ∗ .Proof. In order to obtain D u,v we combine minimal automata for the languages Σ ∗ uΣ ∗ and Σ ∗ vΣ ∗ . For a letter x ∈ { a, b } by x we denote its complementary initely generated ideal languages and synchronizing automata 9 letter, i.e. a = b , and b = a . Recall the construction of the minimal automatonrecognizing the language Σ ∗ wΣ ∗ , where w ∈ Σ + . It is well-known that thisautomaton has | w | + 1 states. We enumerate the states of this automaton by theprefixes of the word w so that the state w [1 ..i ] maps to the state w [1 ..i + 1] underthe action of the letter w [ i + 1] for all i , 0 ≤ i < k . The other letter w [ i + 1]sends the state w [1 ..i ] to state p such that p is the maximal prefix of w thatappears in w [1 ..i + 1] as a suffix. The state w is the sink state of the automaton.The initial state is ε and the unique final state is w , see Fig.5 (the transitionslabeled by complementary letters w [ i ] are not shown). ε w [1] w [1 .. w... a, bb w [1] w [2] w [ n ] w [3] Fig. 5.
The minimal DFA A w . Construct minimal automata A u and A v . Denote by A ′ u the automaton ob-tained from A u by deleting the sink state and the transition from u [1 ..n − u [ n ]. Denote by A ′ v the corresponding automaton for v . Define theaction of letters u [ n ] and v [ m ] on states u [1 ..n −
1] and v [1 ..m −
1] as follows.Denote by p the state in A ′ u corresponding to the maximal prefix of u thatappears in v as a suffix. Denote by s the state in A ′ v corresponding to the max-imal prefix of v that appears in u as a suffix. We put u [1 ..n − . u [ n ] = s and v [1 ..m − . v [ m ] = p . Denote the resulting automaton by D u,v and prove thatit satisfies the desired properties. Figures 6,7,8 illustrate the construction for u = abaab and v = babab . ε a ab aba abaa abaaba b a a bb a a, bb ab Fig. 6.
The minimal DFA recognizing Σ ∗ abaabΣ ∗ . ε b ba bab baba bababb a b a ba b a, ba ba Fig. 7.
The minimal DFA recognizing Σ ∗ bababΣ ∗ .0 Vladimir V. Gusev, Marina I. Maslennikova, Elena V. Pribavkina ε b ba bab babab a b aa ba baε a ab aba abaaa b a ab ab abb b Fig. 8.
The DFA D u,v . The following claim is rather easy to see. The explicit proof can be found in [3].
Claim. If w ∈ Σ n \ { a n − b, ab n − } , then the automaton A ′ w is strongly con-nected.By the Claim automata A ′ u and A ′ v are strongly connected. By the definitionof the action of letters u [ n ] and v [ m ] on states u [1 ..n −
1] and v [1 ..m − D u,v is also strongly connected.Now we are going to verify that u, v ∈ Syn( D u,v ). The state set of D u,v is theunion of the state set of A ′ u (denoted by Q u ), and the state set of A ′ v (denotedby Q v ). To avoid confusion when necessary we will use the upper indices u and v for the states in Q u and Q v respectively. Let w be an arbitrary word. We claimthat ε u . w = r , where r is the maximal prefix of either u or v which is a suffixof w . Let us consider the path from ε u to r . As long as we do not use modifiedtransitions, i.e. the ones that lead from Q u to Q v or vice versa, the claim holdstrue by the definition of A ′ u and A ′ v . Suppose now that the path contains atransition u [1 ..n − u [ n ] −−→ s and ε u . w ′ = u [1 ..n − w ′ is a prefix of w . Let s ′ be the maximal prefix of the word v which is a suffix of w ′ u [ n ]. Notethat w ′ u [ n ] has u as a suffix. Therefore | s ′ | < | u | , otherwise u is a factor of v .Since | s ′ | < | u | we have s ′ = s . Similar reasoning applies in case of transition v [1 ..m − v [ m ] −−−→ p . Therefore, the claim holds true. It is not hard to see thatalso ε v . w = r ′ , where r ′ is the maximal prefix of either u or v which is a suffixof w .Now we are ready to show that u is a synchronizing word for D u,v . By thedefinition of D u,v we have ε u . u = s . Let us consider an arbitrary state t ∈ Q u .Let ε u . tu = r . Note that the maximal prefix of the word u which is a suffix of tu is equal to u . Then by the claim r is a prefix of v . Since u is not a factor of v wehave | r | < | u | . Thus, r = s due to maximality. Since ε u . t = t we have t . u = s .Thus, Q u . u = { s } . Arguing in the same way for the state ε v we get Q v . u = { s } .So, u is synchronizing. Analogously, one can show that v is synchronizing. initely generated ideal languages and synchronizing automata 11 To complete the proof it remains to verify that each word from the setSyn( D u,v ) contains u or v as a factor. Take w ∈ Syn( D u,v ) and a state r ∈ Q u .If r . w ∈ Q u , then w maps all states in the component Q v into the samestate. In particular, ε v . w ∈ Q u . Thus v appears in w as a factor. Analo-gously if r . w ∈ Q v , the word u appears as a factor in w . So we proved thatSyn( D u,v ) = Σ ∗ ( u + v ) Σ ∗ . ⊓⊔ Acknowledgement . The authors acknowledge support from the Presiden-tial Programm for young researchers, grant MK-266.2012.1.
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