I F I G R e s e a r c h R e p o r t. Minimal and Hyper-Minimal Biautomata. IFIG Research Report 1401 March Institut für Informatik

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1 I F I G R e s e a r h R e p o r t Institut für Informatik Minimal and Hyper-Minimal Biautomata Markus Holzer Seastian Jakoi IFIG Researh Report 1401 Marh 2014 Institut für Informatik JLU Gießen Arndtstraße Giessen, Germany Tel: Fax: mail@informatik.uni-giessen.de

2 IFIG Researh Report IFIG Researh Report 1401, Marh 2014 Minimal and Hyper-Minimal Biautomata Markus Holzer 1 and Seastian Jakoi 2 Institut für Informatik, Universität Giessen Arndtstraße 2, Giessen, Germany Astrat. We ompare deterministi finite automata (DFAs) and iautomata under the following two aspets: strutural similarities etween minimal and hyper-minimal automata, and omputational omplexity of the minimization and hyper-minimization prolem. Conerning lassial minimality, the known results suh as isomorphism etween minimal DFAs, and NL-ompleteness of the DFA minimization prolem arry over to the iautomaton ase. But surprisingly this is not the ase for hyper-minimization: the similarity etween almostequivalent hyper-minimal iautomata is not as strong as it is etween almost-equivalent hyper-minimal DFAs. Moreover, while hyper-minimization is NL-omplete for DFAs, we prove that this prolem turns out to e omputationally intratale, i.e., NP-omplete, for iautomata. Categories and Sujet Desriptors: F.1.1 [Computation y Astrat Devies]: Models of Computation Automata; F.1.3 [Computation y Astrat Devies]: Complexity Measures and Classes Reduiility and ompleteness; F.2.2 [Analysis of Algorithms and Prolem Complexity]: Nonnumerial Algorithms and Prolems Computations on disrete strutures; F.4.3 [Mathematial Logi and Formal Languages]: Formal Languages Deision prolems; Additional Key Words and Phrases: iautomata, almost-equivalene, hyper-minimization, omputational omplexity 1 holzer@informatik.uni-giessen.de 2 seastian.jakoi@informatik.uni-giessen.de Copyright 2014 y the authors

3 1 Introdution The minimization prolem for finite automata is well studied in the literature, see, e.g., [9] for a reent overview on some automata related prolems. The prolem asks for the smallest possile finite automaton that is equivalent to a given one. Beause regular languages are used in many appliations and one may like to represent the languages suintly, this prolem is also of pratial relevane. It is well known that for a given n-state deterministi finite automaton (DFA) one an effiiently ompute an equivalent minimal automaton in O(n log n) time [11]. More preisely, the DFA minimization prolem is omplete for NL, even for DFAs without inaessile states [4]. On the other hand, minimization of nondeterministi finite automata (NFAs) is highly intratale, namely PSPACE-omplete [14]. These results go along with the strutural properties of minimal finite automata. While minimal DFAs are unique up to isomorphism, this is not the ase for minimal nondeterministi state devies anymore [1]. In fat, the haraterization of minimal DFAs is one of the asi uilding loks for effiient DFA minimization algorithms. When hanging from minimization to hyper-minimization a quite similar piture as mentioned aove emerges. Hyper-minimization asks for the smallest automaton that is equivalent to a given one up to a finite numer of exeptions this form of equivalene is referred to as almost-equivalene in the literature. Let us disuss the situation for hyper-minimal DFAs and NFAs in more detail. First, all of the aove mentioned omputational omplexity results remain valid for hyper-minimization. Thus, omputing a hyper-minimal DFA an e done in O(n log n) time [10] and the hyper-minimization prolem is NL-omplete [6]. In fat it is known that minimization for DFAs linearly redues to hyper-minimization [10]. Moreover, the intrataility result for NFAs remains, that is, hyper-minimization for NFAs is PSPACE-omplete [6], just as it is for ordinary NFA minimization. What an e said aout the strutural properties of hyperminimal finite state mahines? Neither hyper-minimal DFAs nor hyper-minimal NFAs are unique up to isomorphism. Nevertheless, hyper-minimal DFAs oey a strutural haraterization as shown in [2]. Almost-equivalent hyper-minimal DFAs have isomorphi kernels and isomorphi preamles up to state aeptane. Here the kernel of an automaton onsists of the states that are reahale from the start state y an infinite numer of inputs; all other states elong to the preamle of the automaton. Reently, an alternative automaton model to deterministi finite automata, the so alled iautomaton (DBiA) [19] was introdued. Roughly speaking, a iautomaton onsists of a deterministi finite ontrol, a read-only input tape, and two reading heads, one reading the input from left to right (forward transitions), and the other head reading the input from the opposite diretion, i.e., from right to left (akward transitions). An input word is aepted y a iautomaton, if there is an aepting omputation starting the heads on the two ends of the word meeting somewhere in an aepting state. Although the hoie of reading a symol y either head is nondeterministi, a deterministi outome of the omputation of the iautomaton is enfored y two properties: (i) The heads read input symols independently, i.e., if one head reads a symol and 2

4 the other reads another, the resulting state does not depend on the order in whih the heads read these single letters. (ii) If in a state of the finite ontrol one head aepts a symol, then this letter is aepted in this state y the other head as well. Later we all the former property the -property and the latter one the F-property. In [19] and a series of forthoming papers [7,8, 15, 18] it was shown that iautomata share a lot of properties with ordinary finite automata. For instane, as minimal DFAs, also minimal DBiAs are unique up to isomorphism [19]. Moreover, in [7] it was shown that lassial DFA minimization algorithms an e adapted to iautomata as well. As a first result we show that iautomaton minimization is NL-omplete as for ordinary DFAs. Now the question arises, whih of the strutural similarities etween almostequivalent or hyper-minimal DFAs similarly hold for iautomata as well? Moreover, what an e said aout the omputational omplexity of iautomaton hyper-minimization? We give answers to oth questions in the forthoming. Some of the strutural similarities found for almost-equivalent and hyper-minimal DFAs arry over to the ase of iautomata, ut there are sutle differenes. On the one hand we show that the kernel isomorphism for almost-equivalent DFAs arries over to almost-equivalent iautomata, ut on the other hand, the isomorphism for the preamle for almost-equivalent hyper-minimal DFAs does not transfer to the iautomaton ase. In fat, we present an example, of two almost-equivalent hyper-minimal iautomata the preamles of whih are not isomorphi at all oserve, that the size of oth preamles must e the same due to the hyper-minimality of the devies, and the kernel isomorphism. The oserved phenomenon is related to the struture of the almost-equivalene lasses of hyper-minimal iautomata. In ontrast to hyper-minimal DFAs, where two different ut almost-equivalent states an only appear in the kernel, the indued almost-equivalene lasses in ase of hyper-minimal iautomata may in addition also span etween preamle and kernel states, or even etween two preamle states. Later we use this fat in order to prove the main result of this paper, namely that hyper-minimizing iautomata is not as easy as for DFAs. More preisely, we show that hyper-minimization for iautomata is NP-omplete. This is in sharp ontrast to the ase of hyper-minimal DFAs. 2 Preliminaries A deterministi finite automaton (DFA) is a quintuple A = (Q, Σ, δ, q 0, F), where Q is the finite set of states, Σ is the finite set of input symols, q 0 Q is the initial state, F Q is the set of aepting states, and δ: Q Σ Q is the transition funtion. The language aepted y the deterministi finite automaton A is L(A) = { w Σ δ(q 0, w) F }, where the transition funtion is reursively extended to δ: Q Σ Q as usual. A deterministi iautomaton (DBiA) is a sixtuple A = (Q, Σ,,, q 0, F), where Q, Σ, q 0, and F are defined as for DFAs, and where and are mappings from Q Σ to Q, alled the forward and akward transition funtion, respetively. It is ommon in the literature on iautomata to use an infix notation for these funtions, i.e., writing q a and q a instead of (q, a) and (q, a). Similar as for the transition funtion of a DFA, the forward transition funtion 3

5 an e extended to : Q Σ Q y q λ = q, and q av = (q a) v, for all states q Q, symols a Σ, and words v Σ. The extension of the akward transition funtion to : Q Σ Q is defined as follows: q λ = q and q va = (q a) v, for all states q Q, symols a Σ, and words v Σ. Notie that onsumes the input from right to left, hene the name akward transition funtion. The DBiA A aepts a word w Σ if there are words u i, v i Σ, for 1 i k, suh that w an e written as w = u 1 u 2...u k v k...v 2 v 1, and ((...((((q 0 u 1 ) v 1 ) u 2 ) v 2 )...) u k ) v k F. The language aepted y A is L(A) = { w Σ A aepts w }. The DBiA A has the -property, if (q a) = (q ) a, for all q Q and a, Σ, and it has the F-property, if we have q a F if and only if q a F, for all q Q and a Σ. The iautomata as introdued in [19] always had to satisfy oth these properties, while in [7, 8] also iautomata that lak one or oth of these properties, as well as nondeterministi iautomata were studied. Throughout the urrent paper, when writing of iautomata, or DBiAs, we always mean deterministi iautomata that satisfy oth the -property, and the F-property, i.e., the model as introdued in [19]. For suh iautomata the following is known from the literature [7,19]: (q u) v = (q v) u, for all states q Q and words u, v Σ, (q u) vw F if and only if (q uv) w F, for all states q Q and words u, v, w Σ. From this one an onlude that for all words u i, v i Σ, with 1 i k, we have ((...((((q 0 u 1 ) v 1 ) u 2 ) v 2 )...) u k ) v k F if and only if q 0 u 1 u 2...u k v k... v 2 v 1 F. Therefore, the language aepted y a iautomaton A an as well e defined as L(A) = { w Σ q 0 w F }. In the following we define the two DFAs ontained in a DBiA, whih aept the language, and the reversal of the language aepted y the iautomaton. Let A = (Q, Σ,,, q 0, F) e a DBiA. We denote y Q fwd (Q wd, respetively) the set of all states reahale from q 0 y only using forward (akward, respetively) transitions, i.e., and Q fwd = { q Q u Σ : q 0 u = q }, Q wd = { q Q v Σ : q 0 v = q }. Now we define the DFA A fwd = (Q fwd, Σ, δ fwd, q 0, F fwd ), with F fwd = Q fwd F, and where δ fwd (q, a) = q a, for all states q Q fwd and symols a Σ. Similarly, we define the DFA A wd = (Q wd, Σ, δ wd, q 0, F wd ), with F wd = Q wd F, and δ wd (q, a) = q a, for all q Q and a Σ. One readily sees that L(A fwd ) = L(A). Moreover, sine q uv = (q v) u, one an also see L(A wd ) = L(A) R. 4

6 For a state q of an automaton A (DFA or DBiA), the right language of q is the language L A (q) aepted y the automaton that is otained from A y making q its initial state. Notie that the right language of the initial state q 0 of A is L A (q 0 ) = L(A). We say that two automata A and A are equivalent, denoted y A A, if L(A) = L(A ). Similarly, if q is a state of A and q a state of A, then q and q are equivalent, for short q q, if L A (q) = L A (q ). An automaton A is minimal if there is no automaton B of the same type, that has fewer states than A and satisfies A B. Let L e a language over Σ and let u, v Σ. The left derivative of L y u is the language u 1 L = { w Σ uw L }, and the right derivative of L y v is Lv 1 = { w Σ wv L }. Notie that u 1 (Lv 1 ) = (u 1 L)v 1, so we may denote oth-sided derivatives y u 1 Lv 1 = { w Σ uwv L }. Derivatives are used in [19] for the definition of the anonial iautomaton of a regular language, whih is similar to the anonial DFA as desried in [3]. The set of states of the anonial iautomaton for a regular language L onsists of all derivatives of L this is a finite set eause L is regular and the right language of a state u 1 Lv 1 is the language u 1 Lv 1. We often use regular expressions to desrie languages see, e.g., [12]. As usual we identify an expression with the language it desries, and y ause of notation we also use regular expressions as names for states. Reently, the notions of almost-equivalene and hyper-minimality were introdued [2]. Two languages L and L are almost-equivalent, denoted y L L, if their symmetri differene L L := (L\L ) (L \L) is finite. This notion naturally arries over to automata and states: two automata A and A are almostequivalent, for short A A, if L(A) L(A ), and two states q and q of A and, respetively, A are almost-equivalent, for short q q, if L A (q) L A (q ). An automaton A is hyper-minimal if there is no automaton B of the same type, that has fewer states than A and satisfies A B. A useful onept for the study of almost-equivalent automata is the partitioning of the state set into preamle and kernel states. A state q of an automaton A is a kernel state if it is reahale from the initial state of A y an infinite numer of inputs, otherwise q is a preamle state. In ase A is a iautomaton over alphaet Σ, this means that q is a kernel state if and only if there are infinitely many pairs of words u, v Σ suh that (q 0 u) v = q here q 0 is the initial state, and and are the transition funtions of A. The set of all preamle states of A is denoted y Pre(A), and the set of kernel states is Ker(A). We assume familiarity with the asi onepts of omplexity theory [12, 20] suh as redutions, ompleteness, and the inlusion hain NL P NP. Here NL is the set of prolems aepted y nondeterministi logarithmi spae ounded Turing mahines. Moreover, let P (NP, respetively) denote the set of prolems aepted y deterministi (nondeterministi, respetively) polynomial time ounded Turing mahines. 5

7 3 Strutural Similarity Between Minimal Automata The well-known fat that two equivalent minimal DFAs are isomorphi an e formulated as follows. Theorem 1. Let A = (Q, Σ, δ, q 0, F) and A = (Q, Σ, δ, q 0, F ) e two minimal deterministi finite automata with A A. Then there exists a mapping h: Q Q that is ijetive, and that satisfies the following onditions: 1. q h(q), for all q Q (in partiular q F if and only if h(q) F ). 2. h(q 0 ) = q h(δ(q, a)) = δ (h(q), a), for all q Q and a Σ. Further, the following haraterization of minimal DFAs is well known. Theorem 2. A deterministi finite automaton is minimal if and only if all its states are reahale, and there is no pair of distint, ut equivalent states. An isomorphism as in Theorem 1 an also e found etween equivalent minimal iautomata, whih follows from results from [19]. Theorem 3. Let A = (Q, Σ,,, q 0, F) and A = (Q, Σ,,, q 0, F ) e two minimal iautomata 1 with A A. Then there exists a mapping h: Q Q that is ijetive, and that satisfies the following onditions: 1. q h(q), for all q Q (in partiular q F if and only if h(q) F ). 2. h(q 0 ) = q h(q a) = h(q) a, and h(q a) = h(q) a, for all q Q and a Σ. Also the following haraterization of minimal DBiAs, whih is similar to Theorem 2 for DFAs, was shown in [7]. Theorem 4. A iautomaton is minimal if and only if all its states are reahale, and there is no pair of distint, ut equivalent states. We an draw another onnetion etween iautomata and finite automata. Reall that any DBiA A ontains the two DFAs A fwd and A wd, aepting the languages L(A fwd ) = L(A) and L(A wd ) = L(A) R. In fat, if A is a minimal iautomaton, then the two ontained DFAs are minimal, too, as the following result shows. Lemma 5. Let A = (Q, Σ,,, q 0, F) e a minimal iautomaton. Then A fwd is a minimal deterministi finite automaton for L(A) and A wd is a minimal deterministi finite automaton for L(A) R. Proof. Notie that if q is a state of A fwd, then L Afwd (q) = L A (q), and if q is a state of A wd, then L Awd (q) = L A (q) R. Therefore, if A fwd or A wd ontains a pair of equivalent states, then these states are also equivalent in the iautomaton A. Now if A is a minimal iautomaton, then Theorem 4 implies that it does not ontain a pair of distint, ut equivalent states. Therefore also the DFAs A fwd, and A wd ontain no suh pair. Sine y definition all states of A fwd and A wd are reahale, oth DFAs must e minimal, due to Theorem 2. 1 Rememer that throughout this paper a iautomaton is always a deterministi iautomaton whih satisfies oth the -property and the F-property. 6

8 The following example shows that the onverse of Lemma 5 is not true, whih means that a DBiA A where oth DFAs A fwd and A wd are minimal needs not to e minimal itself. Example 6. Consider the iautomaton A = (Q, Σ,,, q 0, F) with the state set Q = {q 0, q 1,...,q 6 }, initial state q 0, final states F = {q 2, q 4, q 5 } and the transition funtions and of whih an e read from Figure 1 solid arrows denote forward transitions y, and dashed arrows denote akward transitions y. Oviously the three aepting states q 2, q 4, and q 5 are equivalent, so we q a 0 q 1 q 2 q a 3 q 4 a q 5 Fig. 1. A non-minimal iautomaton A where oth ontained DFAs A fwd and A wd are minimal. The sink state q 6 and all transitions to it are not shown. know y Theorem 4 that this is not a minimal iautomaton. However, one an easily see that oth ontained DFAs A fwd and A wd are minimal. This shows that the onverse of Lemma 5 does not hold. 4 Strutural Similarity Between Hyper-Minimal Automata The notions of almost-equivalene and hyper-minimality were introdued in [2]. There it was shown that two almost-equivalent hyper-minimal DFAs are isomorphi in their kernels, and isomorphi in their preamles (up to aeptane values of preamle states). The following theorem, whih summarizes results from [2], should e ompared to the orresponding Theorem 1 for equivalent minimal DFAs. Theorem 7. Let A = (Q, Σ, δ, q 0, F) and A = (Q, Σ, δ, q 0, F ) e two minimal deterministi finite automata with A A. Then there exists a mapping h: Q Q satisfying the following onditions. 1. If q Pre(A) then q h(q), and if q Ker(A) then q h(q). 2. If q 0 Pre(A) then h(q 0 ) = q 0, and if q 0 Ker(A) then h(q 0 ) q The restrition of h to Ker(A) is a ijetion etween the kernels of A and A, that is ompatile with taking transitions: 3.a We have h(ker(a)) = Ker(A ), and if q 1, q 2 Ker(A) with h(q 1 ) = h(q 2 ) then q 1 = q We have h(δ(q, a)) = δ (h(q), a), for all q Ker(A) and all a Σ. 7

9 Further, if A and A are hyper-minimal then also the following ondition holds. 4. The restrition of h to Pre(A) is a ijetion etween the preamles of A and A, that is ompatile with taking transitions, exept for transitions from preamle to kernel: 4.a We have h(pre(a)) = Pre(A ), and if q 1, q 2 Pre(A) with h(q 1 ) = h(q 2 ) then q 1 = q We have h(δ(q, a)) = δ (h(q), a), for all q Pre(A) and all a Σ, that satisfy δ(q, a) Pre(A). Notie that the ijetion etween the preamle states does not preserve finality of states. Further, the mapping h does not neessarily respet the transitions from preamle states to kernel states see Condition 4. of Theorem 7. Thus, two almost-equivalent hyper-minimal DFAs an differ in the following: aeptane values of preamle states, transitions leading from preamle to kernel states, the initial state, if the preamle is empty. However, the transitions onneting preamle and kernel of almost-equivalent DFAs annot differ aritrarily. Assume that we have a state q Pre(Q), and some symol a Σ, suh that δ(q, a) Ker(Q). Then it ould e that the two states h(δ(q, a)) and δ (h(q), a) are different, ut they must at least e almost-equivalent. This follows from the following result from [2]. Lemma 8. Let A = (Q, Σ, δ, q 0, F) and A = (Q, Σ, δ, q 0, F ) e two (not neessarily distint) deterministi finite automata, with q Q and q Q. Then q q if and only if δ(q, w) δ (q, w), for all w Σ. Moreover, q q implies δ(q, w) δ (q, w), for all words w Σ with w k = Q Q. Also a haraterization of hyper-minimal DFAs, whih is similar to Theorem 2, was shown in [2]: Theorem 9. A deterministi finite automaton is hyper-minimal if and only if it is minimal, and there is no pair of distint ut almost-equivalent states suh that one of them is in the preamle. Now let us investigate, whih of the strutural similarity results for almostequivalent hyper-minimal DFAs arry over to iautomata. We first show that a result similar to Lemma 8 also holds for iautomata. Lemma 10. Let A = (Q, Σ,,, q 0, F) and A = (Q, Σ,,, q 0, F ) e two iautomata. Let q Q and q Q, then q q if and only if (q u) v (q u) v, for all words u, v Σ. Moreover, q q implies (q u) v (q u) v, for all words u, v Σ with uv k = Q Q. Proof. Assume there are words u, v Σ suh that the states p = (q u) v and p = (q u) v are not almost-equivalent. This means that there are infinitely many words w L A (p) L A (p ), whih implies that there are infinitely many words uwv L A (q) L A (q ). Thus states q and q are not 8

10 almost-equivalent. Therefore, if (q u) v (q u) v, for all u, v Σ, then states q and q are almost-equivalent. The reverse impliation is trivial: if (q u) v (q u) v, for every u, v Σ, then we otain q q y hoosing u = v = λ. This proves the first part of the Lemma. For the seond part assume q q, and onsider two words u = a 1 a 2... a l and v = a m... a l+2 a l+1, with a 1, a 2,...,a m Σ suh that uv = m k. Consider the sequene of state pairs (q i, q i ), for 0 i m, that the automata pass through in their omputations (q u) v and (q u) v: (q, q ), for i = 0, (q i, q i) = (q i 1 a i, q i 1 a i ) for 1 i l, (q i 1 a i, q i 1 a i ) for l + 1 i m. Beause m Q Q, there must e integers i, j with 0 i < j m, for whih we have (q i, q i ) = (q j, q j ). If j l then the word u an e written as u = u 1 u 2 u 3 suh that q u 1 = q i, q i u 2 = q i, (q i u 3 ) v = p, q u 1 = q i, q i u 2 = q i, (q i u 3 ) v = p. If the states p = (q u) v and p = (q u) v are not equivalent, then there is a word w L A (p) L A (p ), and it follows that u 1 u n 2 u 3wv L A (q) L A (q ), for all n 0. This is a ontradition to q q. If l + 1 i then we an find a similar partition of the word v = v 3 v 2 v 1, suh that a word w in the symmetri differene of the two states p = (q u) v and p = (q u) v indues infinitely many words uwv 3 v n 2 v 1 L A (q) L A (q ), for all n 0. It remains to disuss the ase i l < j. Now the words u and v an e written as u = u 1 u 2 and v = v 2 v 1 suh that q u 1 = q i, (q i u 2 ) v 1 = q i, q i v 2 = p, q u 1 = q i, (q i u 2 ) v 1 = q i, q i v 2 = p. Now, if the states p = (q u) v and p = (q u) v are not equivalent, then there is a word w L A (p) L A (p ), and it follows that u 1 u n 2 wv 2v n 1 L A(q) L A (q ), for all n 0. This is again a ontradition to q q, hene the two states (q u) v and (q u) v must e equivalent. Now we ome to a mapping etween the states of two almost-equivalent iautomata. As in the ase of finite automata, we an find an isomorphism etween the kernels of the two automata. However, we annot find a similar isomorphism etween their preamles. Of ourse, two almost-equivalent hyperminimal iautomata must have the same numer of states, and if their kernels are isomorphi, then also their preamles must e of same size. But still we annot always find a ijetive mapping that preserves almost-equivalene, as in the ase of finite automata. We will later see an example for this phenomenon, ut first we present our result on the strutural similarity etween almostequivalent minimal iautomata. 9

11 Theorem 11. Let A = (Q, Σ,,, q 0, F) and A = (Q, Σ,,, q 0, F ) e two minimal iautomata with A A. There exists a mapping h: Q Q that satisfies the following onditions. 1. If q Pre(A) then q h(q), and if q Ker(A) then q h(q). 2. If q 0 Pre(A) then h(q 0 ) = q 0, and if q 0 Ker(A) then h(q 0 ) q The restrition of h to Ker(A) is a ijetion etween the kernels of A and A, that is ompatile with taking transitions: 3.a We have h(ker(a)) = Ker(A ), and if q 1, q 2 Ker(A) with h(q 1 ) = h(q 2 ) then q 1 = q We have h(q a) = h(q) a and h(q a) = h(q) a, for all q Ker(A) and all a Σ. Proof. In order to define the mapping h we hoose for every state q Q tow words u q and v q as follows: If q is a kernel state of A then there are infinitely many pairs u, v Σ suh that (q 0 u) v = q. Hene we an hoose for every kernel state q Ker(A) two words u q and v q with u q v q k = Q Q suh that q = (q 0 u q ) v q. Moreover, for every preamle state q Pre(A), we fix some shortest words u q and v q, i.e., where u q v q is shortest possile, suh that also (q 0 u q ) v q = q. The mapping h: Q Q is then defined y h(q) = (q 0 u q ) v q. In the following we show that this mapping satisfies the statements of the theorem. Sine A A it must e q 0 q 0. If q 0 Pre(Q), then we have u q0 = v q0 = λ, and so h(q 0 ) = q 0, whih proves one part of Statement 2. From Lemma 10 we otain (q 0 u q ) v q (q 0 u q ) v q, for all preamle states q Pre(A), and further even (q 0 u q ) v q (q 0 u q ) v q, for all kernel states q Ker(A). This proves Statement 1. Now, if q 0 Ker(A), then q 0 q 0 h(q 0 ), whih proves the other part of Statement 2. Moreover, for all kernel states q Ker(A) we have q h(q), whih implies h(q a) q a h(q) a and h(q a) q a h(q) a, for all symols a Σ. Sine A is a minimal iautomaton, it does not ontain a pair of different, ut equivalent states. Therefore it must e h(q a) = h(q) a and h(q a) = h(q) a, for all a Σ, whih proves Statement 3.. It remains to prove Statement 3.a, namely that the mapping is ijetive etween the kernels of the two DBiAs. If q Ker(A) then u q v q Q, so the omputation path in A that leads from q 0 to h(q) must ontain a yle. Hene state h(q) must e a kernel state of A, and we get h(ker(a)) Ker(A ). Moreover, the mapping h is injetive, whih an e seen as follows: if h(p) = h(q), then we know p h(p) = h(q) q, ut sine A is a minimal iautomaton, this implies p = q. Therefore it is Ker(A) Ker(A ). By exhanging the roles of the automata A and A we an also find an injetive mapping h : Q Q that satisfies h (Ker(A )) Ker(A), whih in turn shows that Ker(A ) Ker(A). Altogether we otain Ker(A) = Ker(A ), so the mapping h must also e surjetive on Ker(A ). This onludes our proof. Notie that Theorem 11 requires the almost-equivalent DBiAs A and A to e minimal, ut not neessarily hyper-minimal. Of ourse, the theorem also 10

12 holds for hyper-minimal automata, sine these are always minimal. However, the question is whether we an find more strutural similarities like Statement 4. from Theorem 7 on DFAs if oth DBiAs are hyper-minimal. Unfortunately the answer is no, as the following example demonstrates. Example 12. Consider the iautomaton A whih is depited on top in Figure 2 as usual, transitions whih are not shown lead to a non-aepting sink state, whih is also not shown. The state laels of the eight states in the lower two rows of the automaton denote the right languages of the respetive states. The kernel of A onsists of those states, and the sink state. The right languages of the states q 0, q 1, and q 2, whih onstitute the preamle of A, are as follows: L A (q 0 ) = L(A) = (a+)a + a, L A (q 1 ) = a +λ, and L A (q 2 ) = a. One an verify that A satisfies the - and the F-property. Let us first show that A is hyper-minimal. Claim. The iautomaton A depited on top in Figure 2 is hyper-minimal. Proof. Assume B is a minimal iautomaton that is almost-equivalent to A. We have to show that B has at least as many states as A. We know from Theorem 11, that the kernels of A and B are isomorphi, hene it suffies to show that B has at least three states in its preamle. Let us denote the initial state of B y q0 B, and its forward and akward transition funtions y B and B, respetively. We have q0 B q 0 eause B A, and y Lemma 10 follows (q0 B B u) B v (q 0 u) v, for all u, v Σ. Sine state q 0 is not almost-equivalent to any kernel state of A, also state q0 B is not a kernel state of B either rememer that the kernels of A and B are isomorphi. Further, also the state q 1 = q 0 a is not almost-equivalent to any kernel state, and not almost-equivalent to q 0, therefore the state q0 B B a must e a another preamle state in B, too. Let us denote this state y q1 B. If we an show that B has another preamle state, then we know that A is hyper-minimal. Therefore assume for the sake of ontradition, that q0 B and qb 1 are the only preamle states of B. Beause no kernel state of A is almostequivalent to q 1, and due to the isomorphism etween the kernels of A and B, state q1 B is the only state of B that is almost-equivalent to state q 1 of automaton A. Beause q 1 q 2, state q1 B is also the only state of B almost-equivalent to q 2. By Lemma 10, the state q 2 = q 0 of A must e almost-equivalent to state q0 B B of B, so we onlude q0 B B = q1 B = qb 0 B a. Now we onsider two ases, namely whether q1 B is an aepting state, or not. If q1 B = q0 B B a is not aepting, then also the state q0 B B a must not e aepting due to the F-property of B. However, state q0 B B a must e almost-equivalent to state q 0 a, whih is the aepting kernel state in A. By Theorem 11, also the orresponding kernel state of B must e aepting, therefore this annot e the target of the transition q0 B B a. Sine there is no other kernel state whih is almost-equivalent to, we onlude that the automaton B must have yet another preamle state q0 B B a, different from q0 B and q1 B a ontradition. The other ase is similar: if q1 B = qb 0 B is an aepting state, then also state q0 B B must e aepting. Moreover, this state must e almost-equivalent 11

13 to the kernel state q 0, i.e., the state (a+)a of A. The orresponding kernel state of B is also non-aepting, so it annot e the target of the transition q0 B B. Again, there is no other kernel state in B that is almost-equivalent to the state (a+)a, so B must possess another preamle state, different from q0 B and qb 1. This onludes our proof. Now onsider the DBiA A, depited on the ottom of Figure 2. This iautomaton aepts the language L(A ) = (a + )a + a, so it is almostequivalent to A. Sine A and A have the same numer of states and A is hyper-minimal, the automaton A is hyper-minimal, too. Consider a mapping h from the states of A to the states of A, that satisfies the onditions of Theorem 11. Between the kernels of the automata, the mapping is lear. Moreover, sine q 0 and q 0 are preamle states, it must e h(q 0) = q 0. This an even e onluded if q 0 and q 0 were not the initial states, eause state q 0 of A is the only state that is equivalent to q 0, and h must satisfy q h(q) for all states q. With the same argumentation we otain h(q 1 ) = h(q 2 ) = q 2. The mapping h is now fully defined, so in this example, there is no other possile mapping from the states of A to the states of A that preserves almost-equivalene. Notie that mapping h is not a ijetion etween the preamles: eause we have h(q 1 ) = h(q 2 ) = q 2, it is not injetive, and it neither is surjetive, sine no state of A is mapped to state q 1 of B. This shows that the ijetion Condition 4.a of Theorem 7 for preamles of deterministi finite automata does not hold for iautomata. Similarly, also Condition 4. of Theorem 7 annot e satisfied here, whih is witnessed y the following. We have h(q 0 a) = h( ) = here in h( ) denotes the kernel state of A, and after the equation symol denotes the kernel state of A ut it is h(q 0 ) B a = q 0 B a = q 1, so h(q 0 a) h(q 0 ) B a. Of ourse there exist ijetive mappings etween the state sets of the two automata A and A, ut none of these an preserve almost-equivalene eause the orresponding almost-equivalene lasses in the state sets are not always of same size. For example, there are two states in A that are almost-equivalent to q 1, namely q 1 itself and q 2, ut in A there is only state q 2 in its equivalene lass. In the previous example we have seen two hyper-minimal iautomata, where one iautomaton (the lower automaton from Figure 2) has an almost-equivalene lass of states that is ut y the preamle-kernel order: the preamle state q 1 is almost-equivalent to the kernel state. In the other iautomaton (the upper automaton from Figure 2) all almost-equivalene lasses lie entirely in either the preamle or the kernel. Now one may ask, whether for a given iautomaton one an always find an almost-equivalent hyper-minimal iautomaton where no almost-equivalene lass ontains oth a kernel and a preamle state. But even this is not possile: in the proof of the forthoming Theorem 14 we will see a iautomaton where every almost-equivalent iautomaton must ontain a preamle state that is almost-equivalent to some kernel state f. Figure 4. Another question is whether two almost-equivalent states, from whih one is a preamle state, an only differ in aeptane, i.e., whether their symmetri 12

14 q 0 a a preamle kernel a q 1 q 2 (a+)a a, a a a a a a+ a, λ a a a a a, a q 0 a, a preamle kernel a q 2 q 1 (a+)a a, a a a a a a+ a, λ a a a a a, a Fig. 2. Two hyper-minimal iautomata that are almost-equivalent. Biautomaton A (top) aepts the language L(A) = (a + )a + a, and A (ottom) aepts the language L(A ) = (a+)a + a. The preamles are Pre(A) = {q 0, q 1, q 2} and Pre(A ) = {q 0, q 1, q 2}. The states q 1, q 2, q 1, and q 2 have the following right languages: L A(q 1) = a + λ, L A(q 2) = L A (q 2) = a, and L A (q 1) =. The gray shading of state pairs denotes almostequivalene, i.e., we have q 1 q 2, a + λ, and q 1. 13

15 differene is at most {λ}. Also this is not the ase, whih an also e seen in the automaton from Figure 4, namely for the states q 1 and q 1. We have seen in Lemma 5 that the DFAs A fwd, and A wd ontained in a minimal DBiA A are minimal DFAs for the language L(A), and L(A) R, respetively. Notie that this relation does not hold if we onsider hyper-minimal automata: the iautomaton A from Example 12 is hyper-minimal. But the ontained DFA A fwd is not hyper-minimal eause the two preamle states q 1 and q 2 almost-equivalent, whih ontradits the haraterization of hyper-minimal DFAs y Theorem 9. In fat, one an hek that for every hyper-minimal iautomaton B that is almost-equivalent to A, either B fwd or B wd is not hyper-minimal. Due to the lak of strutural similarity in the preamles of almost-equivalent hyper-minimal iautomata, we do not hope for a nie haraterization of hyperminimal iautomaton, as we have seen in Theorems 2, 4, and 9. Another effet related to these unsatisfying strutural properties of hyper-minimal iautomata will show up in the following setion, where we show that hyper-minimizing iautomata is omputationally hard. 5 Computational Complexity of (Hyper)-Minimization Given a deterministi finite automaton, it is an easy task to onstrut an equivalent minimal automaton. A lot of minimization algorithms for DFAs are known, the most effetive of them eing Hoproft s algorithm [11] with a running time of O(n log n), where n is the numer of states of the input DFA. In fat, the deision version of the DFA minimization prolem given a DFA A and an integer n, deide whether there exists an n-state DFA B with A B is NLomplete [4]. Conerning minimization of iautomata, it was disussed in [7] how lassial DFA minimization tehniques an also e applied to DBiAs. In the following we investigate the omputational omplexity of the minimization prolem for iautomata, and show that it is NL-omplete, too. For proving NL-hardness we give a redution from the following variant of the graph reahaility prolem [16, 17] whih is NL-omplete, too. Reahaility: given a direted graph G = (V, E) with V = {v 1, v 2,...,v n }, where every vertex has at most two suessors, and at most two predeessors, deide whether v n is reahale from v 1. 2 The next theorem reads as follows: Theorem 13 (DBiA Minimization Prolem). The prolem of deiding for a given iautomaton A, and an integer n, whether there exists an n-state iautomaton B with A B, is NL-omplete. 2 The general graph reahaility prolem an e redued to the ase where every vertex has at most two suessors y appending after eah vertex that has more than two suessors a small tree-like sugraph to simulate the multiple outgoing edges. A similar onstrution an e used to redue the numer of predeessors, to otain a graph where also the numer of predeessors of a vertex is at most two. 14

16 Proof. For the NL upper ound we use the following algorithm for omputing the numer k of equivalene lasses of the state set of A. Let q 1, q 2,...q m e some fixed order of the states of A, and initially set k = 0. For all states q i (in asending order) do the following: if q i q j, for all j < i, then inrement k. Finally, if k n then the answer is yes, otherwise it is no. Beause A has the -property, and the F-property, it suffies to onsider only forward transitions to deide whether q i q j. Therefore, in order to hek whether q i q j holds, we an hek equivalene of the two DFAs otained from A y making q i, and q j, respetively, the initial state, and onsidering only forward transitions. Sine (in-)equivalene of DFAs an e deide in NL, the whole algorithm an e implemented on a non-deterministi logarithmi spae-ounded Turing mahine. For NL-hardness we give a redution from the aove desried variant of the graph reahaility prolem. The idea is to transform the graph into a DFA A 1 that aepts the empty language if the instane of the graph reahaility prolem is a no instane, and it aepts a non-empty, and non-universal language otherwise. Then an equivalent DBiA A is uilt y a ross-produt onstrution of A 1 and the reverse of A 1. The iautomaton A is equivalent to a single-state iautomaton if and only if the graph reahaility prolem is a no instane. The only prolem in this approah is to make sure that the onstrution of the reverse of A 1 an e done y a logarithmi spae-ounded Turing mahine, eause in general this onstrution indues an exponential low-up in the numer of states of the finite automaton. Therefore we onstrut the DFA A 1 suh that its reversal is also a deterministi automaton. Let G = (V, E), with V = {v 1, v 2,...,v n } e a direted graph where every vertex has at most two suessors, and at most two predeessors. We onstrut the partial DFA A 1 = (Q, Σ, δ, q 0, F) here partial means that some transitions of A 1 may e undefined over the alphaet Σ = {a, } as follows: The states set onsists of the verties and edges of G, i.e., Q = V E, the initial state is q 0 = v 1, and set of final states is F = {v n }. The transitions in states v i V are defined as follows: if v i has two suessors v j1 and v j2, i.e., if (v i, v j1 ), (v i, v j2 ) E, and j 1 < j 2, then δ(v i, a) = (v i, v j1 ) and δ(v i, ) = (v i, v j2 ), if v i has one suessors v j, i.e., if (v i, v j ) is the only edge in E with v i as on the left-hand side, then δ(v i, a) = (v i, v j ), Finally, the transitions in states (v i, v j ) E are defined as follows: if (v i, v j ) is the only edge in E with vertex v j on the right-hand side, then δ((v i, v j ), a) = v j, if there are two edges (v i1, v j ) and (v i2, v j ) in E with vertex v j on the righthand side, then δ((v i1, v j ), a) = v j and δ((v i2, v j ), ) = v j. All other transitions are undefined. Note that every state (v i, v j ) E has exatly one outgoing and one ingoing transition, and every state v i V has at most one outgoing and at most one ingoing transition for eah alphaet symol. Therefore, the reverse automaton A R 1 = (Q, Σ, δr, v n, {v 1 }), where all transitions are reversed, and the initial and (single) final state are interhanged is also a partial 15

17 DFA. The iautomaton A an now e onstruted y a ross-produt onstrution, simulating A 1 in the first omponent using its forward transitions, and simulating A R 1 in the seond omponent using akward transitions see [19] for details of this onstrution. Clearly this onstrution an e realized y a logarithmi spae-ounded deterministi Turing mahine. It remains to prove the orretness of the redution. First assume that in the graph G the vertex v n is not reahale from v 1. Then learly the language L(A 1 ) = L(A) is empty, so there exists a single-state iautomaton B that is equivalent to A. Next assume v n is reahale from v 1 in G. Then learly the language L(A) = L(A 1 ) is not empty, and eause v 1 v n it is also not Σ. Therefore there exists no single-state iautomaton B that is equivalent to A. Sine NL = onl [13, 21], the theorem is proven. Now we turn to hyper-minimization. For deterministi finite automata the situation is similar as in the ase of lassial minimization: effiient hyperminimization algorithms with running time O(n log n) are known [5,10], and it was shown in [6] that the hyper-minimization prolem for DFAs is NL-omplete. On the one hand, sine lassial DFA minimization methods also work well for DBiAs, one ould expet that hyper-minimization of DBiAs is as easy as for ordinary DFAs. On the other hand, the prolems related to the struture of hyper-minimal iautomata, whih we disussed in Setion 4, already give hints that hyper-minimization of DBiAs may not e so easy. In fat, we show in the following that the hyper-minimization prolem for iautomata is NP-omplete. To prove NP-hardness we give a redution from the NP-omplete MAX-2-SAT prolem [20] whih is defined as follows. MAX-2-SAT: given a Boolean formula ϕ in onjuntive normal form, where eah lause has exatly two literals, and an integer k, deide whether there exists an assignment that satisfies at least k lauses of ϕ. Before we give a detailed proof of NP-hardness, we want to desrie the key idea of the redution. Given as instane of MAX-2-SAT a formula ϕ and numer k, we onstrut a DBiA A ϕ suh that for every lause that an e satisfied in ϕ, we an save one state of A ϕ, otaining an almost-equivalent DBiA. Every lause of ϕ will e translated to a part of the iautomaton using a separate alphaet, so that the lause gadgets in A ϕ are mostly independent from eah other. Assume that ϕ i = (l i1 l i2 ) is a lause of ϕ, and the first literal is l i1 = x u, and the seond is l i2 = x v, for some variales x u and x v. Then the DBiA A ϕ ontains the struture whih is depited in Figure 3. The states q 1 and q 1 orrespond to literal l i 1, and states q 2 and q 2 to the literal l i2. States p u and p v orrespond to the variales x u and x v, respetively, and are shared y all lause gadgets related to these variales. Now assume that there is a truth assignment ξ to the variales that satisfies lause ϕ i, say y ξ(x u ) = 1. Then we make the preamle state p u aepting, and merge the preamle state q 1 to the almost-equivalent state q 1. To preserve the - property of the automaton, we further re-route the akward transition of the initial state, making state s 1 the target of the transition. The ase where the lause ϕ i is satisfied y the seond literal orresponds to the similar situation, 16

18 preamle kernel q 1 1 q 1 a 1, 2 t 1 p u a 1 a 1, 1, 2 1 q 0 s s 1 a 2 2 s 2 a 1 p v a 2 a 2, 1, 2 2 q 2 a 2, 1 t 2 q 2 Fig. 3. Simplified struture of A ϕ orresponding to the lause ϕ i = (x u x v). The gray shading denotes almost-equivalene of states. where state p v stays non-aepting, state q 2 is merged to q 2, and the target of the akward transition from q 0 is state s 2. The hanging of aeptane of preamle states only introdues a finite numer of errors. Further, the merging of preamle states to almost-equivalent kernel states also yields an almostequivalent automaton. 3 Therefore, if k lauses of ϕ an e satisfied, then k states of A ϕ an e saved. The other diretion, i.e., the dedution of a truth assignment ξ from a given automaton B that is almost-equivalent to A ϕ, is similar: let ξ(x u ) = 1 if and only if state p u of automaton B is aepting. Now assume that state B has k states less than A ϕ. The redution will make sure that only the states q 1 and q 2 an e saved. If for example state q 1 is not present in B, then the initial state of B must enter state q 1 on reading symol 1 with a forward transition. Due to the F-property of B, the state p u reahed from the initial state y taking a 3 In general, this needs some more argumentation. Here the desried hanges in the automaton preserved the -property, and the F-property. Therefore the languages aepted y the original and the modified iautomata are the same as the languages aepted y the ontained DFAs (using only forward transitions). Now almost-equivalene of these DFAs, and thus, of the iautomata, follows from the fat that merging preamle states to almostequivalent kernel states in a DFA preserves almost-equivalene. 17

19 akward 1 transition must e aepting. Sine the variale states are shared y all lause gadgets, the information that p u is aepting i.e., that variale x u should e assigned truth value 1 is transported to all other lause gadgets that use variale x u. Therefore, no state orresponding to the negative literal x u an e saved, i.e., no lause an e satisfied y a literal x u. 4 It may e the ase that oth states q 1 and q 2 are merged to their almost-equivalent kernel states q 1 and q 2, respetively. But then, due to the -property, the initial state must go to some state s on reading a symol with a akward transition, and this state s must go to state t 1 on a forward 1 transition, and it must go to state t 2 on a forward 2 transition. Suh a state is not present in the automaton A ϕ, so this state s is an additional state in the preamle of B. Hene, even if oth states q 1 and q 2 are merged into the kernel, the lause gadget in B annot save more than one state ompared to the lause gadget in A ϕ. Altogether, for every state that B has less than A, there is a lause of ϕ that is satisfied y ξ. We now present our result on the NP-hardness of the hyper-minimization prolem with a detailed proof. Theorem 14. The prolem of deiding for a given iautomaton A, and an integer n, whether there exists an n-state iautomaton B, with A B, is NPhard. Proof. Let ϕ and k form an instane of the MAX-2-SAT prolem where k is an integer, and ϕ = ϕ 1 ϕ 2 ϕ m is a Boolean formula in onjuntive normalform over the set of variales X = {x 1, x 2,...,x n }, and where eah lause ϕ i ontains exatly two literals l i1 and l i2. Instead of diretly onstruting the DBiA A ϕ for the instane of the hyper-minimization prolem, we desrie the language L ϕ aepted y this iautomaton. The integer k for the instane of the hyper-minimization prolem will e the numer of states of A ϕ minus k, i.e., for eah lause that has to e satisfied in ϕ a state of A ϕ is to e saved. From the definition of L ϕ one an see that A ϕ an indeed e onstruted from ϕ in polynomial time. After the definition of L ϕ we will analyze the struture of A ϕ and prove the orretness of the desried redution. Let us define the language L ϕ. The alphaet Σ over whih L ϕ is defined is m n m+1 Σ = {$} Σ (i) {# j,h }, i=1 where for 1 i m the alphaet Σ (i) is with j=1 h=1 Σ (i) = {a (i) 1, a(i) 2 } B(i) 1 B (i) 2 { (i), d (i), e (i) 1, e(i) 2, f(i) 1, f(i) 2, f(i) 3, f(i) 4, f(i) 5 }, B (i) m+1 1 = h=1 { (i) 1,h m+1 } and B(i) 2 = h=1 { (i) 2,h }. 4 The reader may have notied that there is still a possiility to heat: one ould use aepting and non-aepting opies of variale states in the preamle in order to satisfy a lot more lauses than possile. We take are of this prolem in the detailed proof. (The prolem an e solved y using many opies 1,j and 2,j of the 1 and 2 symols, eah onneted to a different opy p u,j of variale states. If the numer of these opies is larger than the numer of lauses, then the heat turns out to e a ad trade-off.) 18

20 The language L ϕ onsists of lause languages L ϕi and variale languages L xj : L ϕ = m L ϕi i=1 n L xj. The variale languages L xj, for 1 j n, are defined as follows: L xj = m+1 h=1 j=1 ( {#j,h } {$} ) {# j,h } B xj,h, where the set B xj,h ontains the symol (i) 1,h ((i) 2,h, respetively) if and only if the first (seond, respetively) literal in lause ϕ i orresponds to the variale x j. More formally, B xj,h = { (i) 1,h ϕ i = (l i1 l i2 ) and l i1 {x j, x j } } { (i) 2,h ϕ i = (l i1 l i2 ) and l i2 {x j, x j } }. For 1 i m, the lause language L ϕi is defined over the alphaet ({$} Σ (i) ) as follows for a etter readaility we omit the upper index (i) for symols from Σ (i) : If ϕ i = (x i1 x i2 ) then L ϕi = a 1 (e 1 + d + ) + B 1 (e d ) + a 2 (e 2 + d ) + B 2 (e d+ ) ( + $ + f 1 (e 1 + d + ) f 1 + f 2 (e 2 + d ) f 2 + f 3 ((a 1 + B 2 ) d + + (a 2 + B 1 ) d ) f 3 + f 4 ((a 1 + B 1 + B 2 ) d + + a 2 d ) f 4 + f 5 (a 1 d + + (a 2 + B 1 + B 2 ) d ) f 5 ) $ +. If ϕ i = (x i1 x i2 ) then L ϕi is defined similar as in the first ase, ut we use (e d ) instead of (e 2 + d ): (parts whih are the same as in the first ase are shown grayed out) L ϕi = a 1 (e 1 + d + ) + B 1 (e d ) + a 2 (e d ) + B 2 (e d+ ) ( + $ + f 1 (e 1 + d + ) f 1 + f 2 (e d ) f 2 + f 3 ((a 1 + B 2 ) d + + (a 2 + B 1 ) d ) f 3 + f 4 ((a 1 + B 1 + B 2 ) d + + a 2 d ) f 4 + f 5 (a 1 d + + (a 2 + B 1 + B 2 ) d ) f 5 ) $ +. If ϕ i = (x i1 x i2 ) then L ϕi is defined as in the first ase, ut we use (e + 1 +d+ ) instead of (e 1 + d+ ). If ϕ i = (x i1 x i2 ) then L ϕi is defined as in the first ase, ut we use (e + 1 +d+ ) instead of (e 1 + d+ ), and (e d ) instead of (e 2 + d ). 19

21 This onludes the definition of the language L ϕ. Let A ϕ = (Q, Σ,,, q 0, F) e the anonial iautomaton [19] for L ϕ, with state set Q = { u 1 L ϕ v 1 u, v Σ }, initial state q 0 = L ϕ, set of final states F = { q Q λ q }, and where q a = a 1 q and q a = qa 1 for all q Q and a Σ. Further let k = Q k. That the instane (A ϕ, k ) an e onstruted in polynomial time from the given instane (ϕ, k) of the MAX-2-SAT prolem an e seen as follows. Every lause language indues a fixed numer of states in A ϕ, and exept for the $ symol the lause languages are defined over disjoint alphaets. Hene the numer of states orresponding to lause languages is linear in the numer of lauses. Further, the numer of states indued y the variale languages is O(mn), so the automaton A ϕ an e onstruted in polynomial time. Before we show the orretness of the redution, let us analyze the struture of A ϕ. The preamle of A ϕ onsists of the states Pre(A ϕ ) = {q 0 } {q 0 (i) x,h, q 0 (i) x,h 1 i m, 1 h m + 1, x {1, 2} }, whih an e seen as follows. Let 1 i m, 1 h m + 1, and assume ϕ i = (l i1 l i2 ) with l i1 {x u, x u } and l i1 {x v, x v }, then q 0 (i) 1,h = + e(i) 1 + d (i) (i), q 0 (i) 1,h = # u,h$ # u,h, q 0 (i) 2,h = + e(i) 2 + d (i) + (i), q 0 (i) 2,h = # v,h$ # v,h. One an see from the desriptions of the languages L ϕi and L xj, that these states are only reahale y reading a single symol from m i=1 (B(i) 1 B (i) 2 ), so they are preamle states. By examining all other states that an e reahed from the initial state y reading a symol whih is not in m i=1 (B(i) 1 B (i) 2 ), one an see that there are no further preamle states. For example onsider the state q 0 a (i) 1. This state an also e reahed from q 0 y reading words from $ + f 1 with forward transitions and words from f 1 $ + with akward transitions. Therefore, state q 0 a, and all states reahale from it are kernel states. The reader is invited to verify that all other states of A ϕ are kernel states, too. Figure 4, whih is similar to Figure 3 shown efore the proof, exemplarily shows a part of the struture of A ϕ that orresponds to a lause language L ϕi, for ϕ i = (x u x v ). The states are renamed as follows: 20