On Nonpermutational Transformation Semigroups with an Application to Syntactic Complexity

Transcription

1 Acta Cybernetica 22 (2016) On Nonpermutational Transormation Semiroups with an Application to Syntactic Complexity Szabolcs Iván and Judit Nay-Györy Abstract We ive an upper bound o n((n 1)! (n 3)!) or the possible larest size o a subsemiroup o the ull transormational semiroup over n elements consistin only o nonpermutational transormations. As an application we ain the same upper bound or the syntactic complexity o (eneralized) deinite lanuaes as well. 1 Introduction A lanuae is eneralized deinite i membership can be decided or a word by lookin at its preix and suix o a iven constant lenth. Generalized deinite lanuaes and automata were introduced by Ginzbur [6] in 1966 and urther studied in e.. [4, 5, 13, 15]. This lanuae class is strictly contained within the class o star-ree lanuaes, lyin on the irst level o the dot-depth hierarchy [1]. This class possess a characterization in terms o its syntactic semiroup [12]: a reular lanuae is eneralized deinite i and only i its syntactic semiroup is locally trivial i and only i it satisies a certain identity x ω yx ω = x ω. This characterization is hardly eicient by itsel when the lanuae is iven by its minimal automaton, since the syntactic semiroup can be much larer than the automaton (a construction or a deinite lanuae with state complexity that is, the number o states o its minimal automaton n and syntactic complexity that is, the size o the transition semiroup o its minimal automaton e(n 1)! is explicit in [2]). However, as stated in [14], Sec. 5.4, it is usually not necessary to compute the (ordered) syntactic semiroup but most o the time one can develop a more eicient alorithm by analyzin the minimal automaton. As an example or this line o research, recently, the authors o [9] ave a nice characterization o minimal automata o piecewise testable lanuaes, yieldin a quadratic-time decision alorithm, matchin an alternative (but o course equivalent) earlier (also quadratic) characterization o [17] which improved the O(n 5 ) bound o [16]. Both authors were supported by the European Union and the State o Hunary, co-inanced by the European Social Fund in the ramework o TÁMOP A/ National Excellence Proram. Szabolcs Iván was also supported by NKFI rant number K University o Szeed, Hunary, {szabivan@in,nyj@math}.u-szeed.hu DOI: /actacyb

2 688 Szabolcs Iván and Judit Nay-Györy There is an onoin line o research or syntactic complexity o reular lanuaes. In eneral, a reular lanuae with state complexity n can have a syntactic complexity o n n, already in the case when there are only three input letters. There are at least two possible modiications o the problem: one option is to consider the case when the input alphabet is binary (e.. as done in [7, 10]). The second option is to study a strict subclass o reular lanuaes. In this case, the syntactic complexity o a class C o lanuaes is a unction n (n), with (n) bein the maximal syntactic complexity a member o C can have whose state complexity is (at most) n. The syntactic complexity o several lanuae classes, e.. (co)inite, reverse deinite, biix, actor and subword-ree lanuaes etc. is precisely determined in [11]. However, the exact syntactic complexity o the (eneralized) deinite lanuaes and that o the star-ree lanuaes (as well as the locally testable or the locally threshold testable lanuaes) is not known yet. In this note we ive an upper bound or the maximal size o a subsemiroup o T n, the transormation semiroup o {1,..., n}, consistin o nonpermutational transormations only. These are exactly the (transormation) semiroups satisyin the identity yx ω = x ω. It is known that a lanuae is deinite i its syntactic semiroup satisies the same identity; thus as a corollary we et that the same bound is also an upper bound or the syntactic complexity o deinite lanuaes. We also ive a orbidden pattern characterization or the eneralized deinite lanuaes in terms o the minimal automaton, and analyze the complexity o the decision problem whether a iven automaton reconizes a eneralized deinite lanuae, yieldin an NL-completeness result (with respect to lospace reductions) as well as a deterministic decision procedure runnin in O(n 2 ) time (on a RAM machine). Analyzin the structure o their minimal automata we conclude that the syntactic complexity o eneralized deinite lanuaes coincide with that o deinite lanuaes. 2 Notation When n 0 is an inteer, [n] stands or the set {1,..., n}. For the sets A and B, A B denotes the set o all unctions : B A. When A B and C B, then C A C denotes the restriction o to C. When A 1,..., A n are disjoint sets, A is a set and or each i [n], i : A i A is a unction, then the source tuplin o 1,..., n is the unction [ 1,..., n ] : ( ) A i A with a[1,..., n ] = a i or i [n] the unique i with a A i. T n is the transormation semiroup o [n] (i.e. [n] [n] ), where composition is understood as p() := (p) or p [n] and, : [n] [n] (i.e., transormations o [n] act on [n] rom the riht to ease notation in the automata-related part o the paper). Elements o T n are oten written as n-ary vectors as usual, e.. = (1, 3, 3, 2) is the member o T 4 with 1 = 1, 2 = 3, 3 = 3 and 4 = 2. When : A A is a transormation o a set A, and X is a subset o A, then X denotes the subset {x : x X} o A.

3 Nonpermutational transormation semiroups and syntactic complexity 689 A transormation : A A o a (inite) set A is nonpermutational i X = X implies X = 1 or any nonempty X A. Otherwise it s permutational. NP n stands or the set o all nonpermutational transormations o [n]. Another class o unctions used in the paper is that o the elevatin unctions: or the inteers 0 < k n, a unction : [k] [n] is elevatin i i i or each i [k] with equality allowed only in the case when i = n (note that this also implies k = n as well). We assume the reader is amiliar with the standard notions o automata and lanuae theory, but still we ive a summary or the notation. An alphabet is a nonempty inite set Σ. The set o words over Σ is denoted Σ, while Σ + stands or the set o nonempty words. The empty word is denoted ε. A lanuae over Σ is an arbitrary set L Σ o Σ-words. A (inite) automaton (over Σ) is a system A = (Q, Σ, δ, q 0, F ) where Q is the inite set o states, q 0 Q is the start state, F Q is the set o inal (or acceptin) states, and δ : Q Σ Q is the transition unction. The transition unction δ extends in a unique way to a riht action o the monoid Σ on Q, also denoted δ or ease o notation. When δ is understood, we write q u, or simply qu or δ(q, u). Moreover, when C Q is a subset o states and u Σ is a word, let Cu stand or the set {pu : p C} and when L is a lanuae, CL = {pu : p C, u L}. The lanuae reconized by A is L(A) = {x Σ : q 0 x F }. A lanuae is reular i it can be reconized by some inite automaton. The state q Q is reachable rom a state p Q in A, denoted p A q, or just p q i there is no daner o conusion, i pu = q or some u Σ. An automaton is connected i its states are all reachable rom its start state. Two states p and q o A are distinuishable i there exists a word u Σ such that exactly one o pu and qu belons to F. In this case we say that u separates p and q. A connected automaton is called reduced i each pair o distinct states is distinuishable. It is known that or each reular lanuae L there exists a reduced automaton, unique up to isomorphism, reconizin L. This automaton A L can be computed rom any automaton reconizin L by an eicient alorithm called minimization and is called the minimal automaton o L. The classes o the equivalence relation p q p q and q p are called components o A. A component C is trivial i C = {p} or some state p such that pa p or any a Σ, and is a sink i CΣ C. It is clear that each automaton has at least one sink and sinks are never trivial. The component raph Γ(A) o A is an ede-labelled directed raph (V, E, l) alon with a mappin c : Q V where V is the set o the -classes o A, the mappin c associates to each state q its class q/ = {p : p q} and or two classes p/ and q/ there exists an ede rom p/ to q/ labelled by a Σ i and only i p a = q or some p p, q q. It is known that the component raph can be constructed rom A in linear time. Note that the mappin c is redundant but it ives a possibility or determinin whether p q holds in constant time on a RAM machine, provided Q = [n] or some n > 0 and c is stored as an array. When A = (Q, Σ, δ, q 0, F ) is an automaton, its transormation semiroup T (A)

4 690 Szabolcs Iván and Judit Nay-Györy consists o the set o transormations o Q induced by nonempty words, i.e. T (A) = {u A : u Σ + } where u A : Q Q is the transormation deined as q qu. The state complexity stc(l) o a reular lanuae L is the number o states o its minimal automaton A L while its syntactic complexity syc(l) is the cardinality o its transormation semiroup T (A L ). The syntactic complexity o a class o lanuaes C is a unction : N N deined as (n) = max{syc(l) : L C, stc(l) n}, i.e. (n) is the maximal size that the transormation semiroup o a minimal automaton o a lanuae belonin to C can have, provided the automaton has at most n states. 3 Semiroups o nonpermutational transormations Observe that NP n is not a semiroup (i.e., not closed under composition) when n > 2. Indeed, i = (2, 3, 3) and = (1, 1, 2) (both bein nonpermutational), then their product = (1, 2, 2) is permutational with {1, 2} = {1, 2}. (See Fiure 1.) Fiure 1: and are nonpermutational, is permutational Thus, the ollowin question is nontrivial: how lare a subsemiroup o T n, which consists only o nonpermutational transormations can be? The obvious upper bound is n n, the size o T n. As a irst step we ive an upper bound o n n 2. Observe that the ollowin are equivalent or a unction : [n] [n]: i) is nonpermutational; ii) the raph o is a rooted tree with edes directed towards the root, and with a loop ede attached on the root; iii) ω, the unique idempotent power o is a constant unction. Here the raph o is o course the directed raph Γ on vertex set [n] and with (i, j) bein an ede i i = j. Indeed, assume is nonpermutational. Let X be the set o all nodes o Γ lyin on some closed path. (Since each node o the inite raph Γ has outderee 1, X is nonempty.) Then X = X, thus X = 1, i.e. has a unique ixed point Fix()

5 Nonpermutational transormation semiroups and syntactic complexity 691 and apart rom the loop ede on Fix(), Γ is a directed acyclic raph (DAG) with each node distinct rom Fix() havin outderee 1 that is, a tree rooted at Fix(), with edes directed towards the root, showin i) ii). Then n is a constant unction with value Fix(), showin ii) iii); inally, i X = X or some nonempty X [n], then X ω = X, showin X = 1 since the imae o ω is a sinleton. Now rom ii) we et that the members o NP n are exactly the rooted trees with edes directed towards the root on which a loop ede is attached we call such a raph an inverted looped arborescence, or ILA or short. By Cayley s theorem on the number o labeled rooted trees over n nodes, the number o all ILAs (i.e., NP n ) is n n 2, ivin a slihtly better upper bound. To achieve an upper bound o n!, suppose S NP n is a subsemiroup o T n. For i [n], let S i S be the subsemiroup { S : Fix() = i} o S. Note that S i is indeed a semiroup: by assumption, S is closed under composition and consists o nonpermutational transormations only, moreover, i i is the common (unique) ixed point o and, then it is also a ixed point o as well, thus S i is closed under composition. We ive an upper bound o (n 1)! or S i, i [n], yieldin S n!. To this end, let Γ i be the raph on vertex set [n] with (j, k) bein an ede i j = k or some S i. Then, apart rom the trivial case when S i =, (i, i) is an ede in Γ i, moreover i is a sink (since i = i or each S i ). Note that in the case when S i =, S i = 0 (n 1)! clearly holds. Observe that Γ i is transitive, since i (j, k) and (k, l) are edes o Γ i, then j = k and k = l or some, S i ; since S i is a semiroup, is also in S i thus (j, l) is also an ede in Γ i. Now assume some node j [n] is in a nontrivial stronly connected component (SCC) o Γ i, i.e. j lies on some closed path. By transitivity, (j, j) is an ede o Γ i, thus j = j or some S i, thus j = i since i = Fix() is the unique ixed point o S i. Hence by droppin the ede (i, i) we et a DAG aain, thus Γ i (viewed as a relation) is a strict partial orderin o [n] with larest element i. Let i stand or this partial orderin, i.e., let j i k i and only i j i and j = k or some S i. Let us also ix some arbitrary total orderin < i extendin i and write the members o [n] in the order a i,1 < i a i,2 < i... < i a i,n = i. Then or any S i and 1 j < n we have a i,j < i a i,j, and a i,n = a i,n. Since the number o unctions : [n] [n] satisyin this constraint is (n 1)! (a i,1 can et (n 1) dierent possible values, a i,2 can et (n 2) etc.), we immediately et S i (n 1)! as well, yieldin S n!. Via a somewhat cumbersome case analysis we can sharpen this upper bound to n((n 1)! (n 3)!). Without loss o enerality assume that S n is (one o) the larest o the semiroups S i and that < n is the usual orderin < o [n] (we can achieve this by a suitable bijection). Lemma 1. Suppose or each i < j and k < l with i k there exists a unction S n with i = j and k = l. Then the ollowin holds or each i, j [n] and S i : i) i j < i, then j < j;

6 692 Szabolcs Iván and Judit Nay-Györy ii) i i j, then j = i. Proo. By assumption, the statements clearly hold or i = n. Let i < n be arbitrary and S i a transormation. Clearly i = i by the deinition o S i. Also, n < n since i n is the unique ixed point o. Suppose j < j or some j. Then j = n has to hold: i j n, then by assumption j = j and n = n or some S n, thus both j and n are distinct ixed points o, a contradiction. (See Fiure 2.) This implies in particular that j j or each j < n. Also, i n < i, then n = i and i = n or some S n, in which case has two distinct ixed points n and i, a contradiction. (See Fiure 2.) Thus i n. j j n n i n n Fiure 2: Let: i j < j, j n, then has two ixed points. Riht: I n < i, then has two ixed points Assume i < n. Then (since n n = i < n) there is some k > 0 such that n k+1 < n. I k is chosen to be the smallest possible such k, then n n k, yieldin (n k ) < n n k, a contradiction (by (n k ) < n k, it should hold that (n k ) = n, see Fiure 3). Hence i = n is the unique ixed point o and or each j < i, j < j indeed has to hold, showin i). i n k+1 n n k n Fiure 3: I i < n, then has two distinct ixed points Finally, assume i < j < j. Then i = j and j = n or some S n (i j = n, then this latter case always ets satisied, otherwise it s by assumption on S n ), and has two distinct ixed points j and n. Thus we have indeed shown that n = i is the unique ixed point o, j < j or each i < j and j = i or each i j n.

7 Nonpermutational transormation semiroups and syntactic complexity 693 i j j n Fiure 4: I i < j < j, then has two distinct ixed points Lemma 1 has the ollowin corollary: Theorem 1. The cardinality o any subsemiroup S o T n consistin only o nonpermutational transormations is at most n((n 1)! (n 3)!). Proo. As beore, let S i stand or { S : Fix() = i} and without loss o enerality we assume that amonst them S n is one o the larest ones, moreover < n coincides with <. I or each i < j and i < j with i i there is some S n with i = j and i = j, then by Lemma 1 S i can consist o at most (n 1)(n 2)... (n i 1) = (n 1)! (n i)! elements (we have to choose or each j < i a larer inteer and that s all since the other elements have to be mapped to i). Also S n (n 1)! as well. Summin up we et an upper bound or these semiroups n i=1 (n 1)! (n i)! n 1 1 = (n 1)! j! j=0 = e(n 1)!, which comes rom the acts that e = j=0 1 j! and (n 1)! j=n 1 j! < 1. For the other case, suppose there exist an i < j and an i < j with i i such that i = j and i = j do not both hold or any S n. Still, i < i or each i < n and n = n, by deinition o S n and the assumption <=< n. The number o (n 1)! (n i)(n j) such unctions satisyin both i = j and i = j is (n 3)!, hence the size o S n is upper-bounded by (n 1)! (n 3)!. Since S n is the larest amonst the S i s and S is the disjoint union o them we et the claimed upper bound n((n 1)! (n 3)!). We note that the construction or the irst case, yieldin the upper bound e(n 1)! indeed constructs a semiroup B which is exactly the semiroup rom [2] conjectured there to be a candidate or the maximal-size such subsemiroup. Our proo can be viewed as a support or this conjecture and can be reormalized as ollows: i there exists some i such that many transormations share this ixed point i, then the size o S is upper-bounded by e(n 1)! and S is isomorphic to a subsemiroup o B. The question is, whether one can construct a larer semiroup by puttin not too many unctions sharin a common ixed point. We also conjecture that B is a ood candidate or a maximal-size subsemiroup o T n consistin o nonpermutational transormations only.

8 694 Szabolcs Iván and Judit Nay-Györy 4 Deinite and eneralized deinite lanuaes A lanuae L is deinite i there exists a constant k 0 such that or any x Σ, y Σ k we have xy L y L and is eneralized deinite i there exists a constant k 0 such that or any x 1, x 2 Σ k and y Σ we have x 1 yx 2 L x 1 x 2 L. These are both subclasses o the star-ree lanuaes, i.e. can be built rom the sinletons with repeated use o the concatenation, inite union and complementation operations. It is known that the ollowin decision problem is complete or PSPACE: iven a reular lanuae L with its minimal automaton, is L star-ree? In contrast, the question or these subclasses above are tractable. Minimal automata o these lanuaes possess a characterization in terms o orbidden patterns. In our settin, a pattern is an ede-labelled, directed raph P = (V, E, l), where V is the set o vertices, E V 2 is the set o edes, and l : E X is a labellin unction which assins to each ede a variable. An automaton A = (Q, Σ, δ, q 0, F ) admits a pattern P = (V, E, l) i there exists an injective mappin : V Q and a map h : X Σ + such that or each (u, v) E labelled x we have (u) h(x) = (v). Otherwise A avoids P. As an example, consider the pattern P d on Fiure 5. x x x x p q p y q (a) Pattern P d. (b) Pattern P. Fiure 5: Patterns or deinite and eneralized deinite lanuaes. 4.1 Syntactic complexity o deinite lanuaes A reduced automaton avoids P d i and only i it reconizes a deinite lanuae. Indeed, a lanuae L is deinite i its syntactic semiroup satisies the identity yx ω = x ω. Now assume L(A) admits P d with px = p and qx = q with p q and x Σ +. I q 0 x ω = p, then q 0 x ω q 0 yx ω or a (nonempty) word y with q 0 y = q. I q 0 x ω p, then q 0 x ω q 0 yx ω or a (nonempty) y with q 0 y = p, thus the identity is not satisied. For the other directon, i the transition semiroup o an automaton A does not satisy x ω = yx ω, then p 0 x ω 0 p 0 yx ω 0 or some p 0, x 0 and y; choosin p = p 0 x ω, q = p 0 y and x 0 = x ω witnesses admittance o P d. (For a more detailed discussion see e.. [2].) Observe that avoidin P d is equivalent to state that each nonempty word induces a transormation with at most one ixed point, which is urther equivalent to state that each nonempty word induces a non-permutational transormation: or each nonempty u, the word u Q! ixes each state belonin to a nontrivial component o the raph o u, hence u also can have only one state in a nontrivial component,

9 Nonpermutational transormation semiroups and syntactic complexity 695 i.e. u induces a nonpermutational transormation. (Aain, see [2] or a dierent ormulation. 1.) Thus Theorem 1 has the ollowin byproduct: Corollary 1. The syntactic complexity o the deinite lanuaes is at most n((n 1)! (n 3)!). 4.2 Syntactic complexity o eneralized deinite lanuaes In this subsection we show that the syntactic complexity o deinite and eneralized deinite lanuaes coincide. To this end we study the structure o the minimal automata o the members o the latter class. In the process we ive a (to our knowlede) new (but not too surprisin) characterization o the minimal automata o eneralized deinite lanuaes, leadin to an NL-completeness result o the correspondin decision problem, as well as a low-deree polynomial deterministic alorithm. Our irst observation is the ollowin characterization: Theorem 2. The ollowin are equivalent or a reduced automaton A: i) A avoids P. ii) Each nontrivial component o A is a sink, and or each nonempty word u and sink C o A, the transormation u C : C C is non-permutational. iii) A reconizes a eneralized deinite lanuae. Proo. Let A = (Q, Σ, δ, q 0, F ) be a reduced automaton. i) ii). Suppose A avoids P. Suppose that u C is permutational or some sink C and word u Σ +. Then there exists a set D C with D > 1 such that u induces a permutation on D. Then, x = u D! is the identity on D. Choosin arbitrary distinct states p, q D and a word y with py = q (such y exists since p and q are in the same component o A), we et that A admits P by the (p, q, x, y) deined above, a contradiction. Hence, u C is non-permutational or each sink C and word u Σ +. Now assume there exists a nontrivial component C which is not a sink. Then, pu = p or some p C and word u Σ +. Since C is not a sink, there exists a sink C C reachable rom p (i.e. all o its members are reachable rom p). Since u induces a non-permutational transormation on C, x = u C induces a constant unction on C. Let q be the unique state in the imae o x C. Since C is reachable rom p, there exists some nonempty word y such that py = q. Hence, px = p, qx = q, py = q and A admits P, a contradiction. ii) iii). Suppose the condition o ii) holds. We show that L = L(A) is eneralized deinite. By the assumption, q 0 u belons to a sink or any u with u Q. On 1 Since up to our knowlede [2] has not been published yet in a peer-reviewed journal or conerence proceedins, we include a proo o this act. Nevertheless, we do not claim this result to be ours, by any means.

10 696 Szabolcs Iván and Judit Nay-Györy the other hand, viewin a sink C as a (reduced) automaton C = (C, Σ, δ C, p, F C) with p bein an arbitrary state o C we et that the transition semiroup o C consists o nonpermutational transormations only, i.e. L(C) is k-deinite or some k = k C. Hence choosin n to be the maximum o Q and the values k C with C bein a sink we et that L is n-eneralized deinite (since the lenth-n preix o u determines the sink C to which q 0 u belons and the lenth-n suix o u, once we know C, determines the unique state in Cu). iii) i). Suppose L(A) is eneralized deinite. Then its syntactic semiroup satisies x ω yx ω = x ω (see e.. [14]). Now assume A L admits P with px = p, qx = q and py = q or the nonempty words x, y and dierent states p, q. Then px ω = p and px ω yx ω = q, and the identity is not satisied, thus L is not eneralized deinite. In [2] it has been shown that the class o deinite lanuaes has syntactic complexity e (n 1)!, thus the same lower bound also applies or the larer class o eneralized deinite lanuaes. Theorem 3. The syntactic complexity o the deinite and that o the eneralized deinite lanuaes coincide. Proo. It suices to construct or an arbitrary reduced automaton A = (Q, Σ, δ, q 0, F ) reconizin a eneralized deinite lanuae a reduced automaton B = (Q,, δ, q 0, F ) or some reconizin a deinite lanuae such that T (A) T (B). By Theorem 2, i L(A) is eneralized deinite and A is reduced, then Q can be partitioned as a disjoint union Q = Q 0 Q 1... Q c or some c > 0 such that each Q i with i [c] is a sink o A and Q 0 is the (possibly empty) set o those states that belon to a trivial component. Without loss o enerality we can assume that Q = [n] and Q 0 = [k] or some n and k, and that or each i [k] and a Σ, i < ia. The latter condition is due to the act that reachability restricted to the set Q 0 o states in trivial components is a partial orderin o Q 0 which can be extended to a linear orderin. Clearly, i Q 0 is nonempty, then by connectedness q 0 = 1 has to hold; otherwise c = 1 and we aain may assume q 0 = 1. Also, Q i Σ Q i or each i [c], and let Q 1 Q 2... Q c. Then, each transormation : Q Q can be uniquely written as the source tuplin [ 0,..., c ] o some unctions i : Q i Q with i : Q i Q i or 0 < i c. For any [ 0,..., c ] T = T (A) the ollowin hold: 0 (i) > i or each i [k], and j is non-permutational on Q j or each j [c]. For k = 0,..., c, let T k stand or the set { k : T } (i.e. the set o unctions Qk with T ). Then, T T k. 0 k c I Q c = 1, then all the sinks o A are sinleton sets. Thus there are at most two sinks, since i C and D are sinleton sinks whose members do not dier in their inality, then their members are not distinuishable, thus C = D since A is reduced. Such automata reconize reverse deinite lanuaes, havin a syntactic semiroup o size at most (n 1)! by [2], thus in that case B can be chosen to an arbitrary deinite automaton havin n state and a syntactic semiroup o size at least e(n 1)! (by the construction in [2], such an automaton exists). Thus we

11 Nonpermutational transormation semiroups and syntactic complexity 697 may assume that Q c > 1. (Note that in that case Q c contains at least one inal and at least one non-inal state.) Let us deine the sets T k o unctions Q i Q as T 0 is the set o all elevatin unctions rom [k] to [n], T c = T c and or each 0 < k < c, T k = QQ k c. Since T k Q Q k k and Q k Q c or each k [c], we have T k T k or each 0 k c. Thus deinin T = {[ 0,..., c ] : i T i } it holds that T T. We deine B as (Q, T, δ, q 0, F ) with δ (q, ) = (q) or each T. We show that B is a reduced automaton avoidin P d, concludin the proo. First, observe that B has exactly one sink, Q c, and all the other states belon to trivial components (since by each transition, each member o Q 0 ets elevated, and each member o Q i with 0 < i < c is taken into Q c ). Hence i B admits P d, then pt = p and qt = q or some distinct pair p, q Q c o states and t = [t 0,..., t c] T. This is urther equivalent to pt c = p and qt c = q or some p q in Q c and = T c, there exists a transormation o the orm t = [t 0,..., t c 1, t c] T induced by some word x, thus px = p and qx = q both hold in A, and since p, q are in the same sink, there also exists a word y with py = q. Hence A admits P, a contradiction. Second, B is connected. To see this, observe that each state p 1 is reachable rom 1 by any transormation o the orm t = [ p, t 1,..., t c ] where p : [k] [n] is the elevatin unction with 1 p = p and i p = n or each i > 1. O course 1 is also trivially reachable rom itsel, thus B is connected. Also, whenever p q are dierent states o B, then they are distinuishable by some word. To see this, we irst show this or p, q Q c. Indeed, since A is reduced, some transormation t = [t 0,..., t c ] T separates p and q (exactly one o t c T c. By deinition o T c pt = pt c and qt = qt c belon to F ). Since T c = T c, we et that p and q are also distinuishable by in B by any transormation o the orm t = [t 0,..., t c 1, t c ] T. Now suppose neither p nor q belon to Q c. Then, since {[t 0,..., t c 1] : t i T i } = Q Q\Qc c, and Q c > 1, there exists some t = [t 0,..., t c 1] with pt qt, thus any transormation o the orm [t 0,..., t c 1, t c ] T maps p and q to distinct elements o Q c, which are already known to be distinuishable, thus so are p and q. Finally, i p Q c and q / Q c, then let t c T c be arbitrary and t = [t 0,..., t c 1 ] Q Q\Qc c with qt pt c. Then [t, t c ] aain maps p and q to distinct states o Q c. Thus B is reduced, concludin the proo: B is a reduced automaton reconizin a deinite lanuae and havin a syntactic semiroup T with T T. 4.3 Complexity issues Usin the characterization iven in Theorem 2, we study the complexity o the ollowin decision problem GenDe: iven a inite automaton A, is L(A) a eneralized deinite lanuae? Theorem 4. Problem GenDe is NL-complete. Proo. First we show that GenDe belons to NL. By [3], minimizin a DFA can be done in nondeterministic lospace. Thus we can assume that the input is already

12 698 Szabolcs Iván and Judit Nay-Györy minimized, since the class o (nondeterministic) lospace computable unctions is closed under composition. Consider the ollowin alorithm: 1. Guess two dierent states p and q. 2. Let s := p. 3. Guess a letter a Σ. Let s := sa. 4. I s = q, proceed to Step 5. Otherwise o back to Step Let p := p and q := q. 6. Guess a letter a Σ. Let p := p a and q = q a. 7. I p = p and q = q, accept the input. Otherwise o back to Step 6. The above alorithm checks whether A admits P : irst it uesses p q, then in Steps 2 4 it checks whether q is accessible rom p, and i so, then in Steps 5 7 it checks whether there exists a word x Σ + with px = p and qx = q. Thus it decides 2 the complement o GenDe, in nondeterministic lospace; since NL = conl, we et that GenDe NL as well. For NL-completeness we recall rom [8] that the reachability problem or DAGs (DAG-Reach) is complete or NL: iven a directed acyclic raph G = (V, E) on V = [n] with (i, j) E only i i < j, is n accessible rom 1? We ive a lospace reduction rom DAG-Reach to GenDe as ollows. Let G = ([n], E) be an instance o DAG-Reach. For a vertex i [n], let N(i) = {j : (i, j) E} stand or the set o its neihbours and let d(i) = N(i) < n denote the outderee o i. When j [d(i)], then the jth neihbour o i, denoted n(i, j) is simply the jth element o N(i) (with respect to the usual orderin o inteers o course). Note that or any i [n] and j [d(i)] both d(i) and the n(i, j) (i exists) can be computed in lospace. We deine the automaton A = ([n + 1], [n], δ, 1, {n + 1}) where n + 1 i (i = n + 1) or (j = n) or (i < n and d(i) < j); δ(i, j) = 1 i i = n and j < n; n(i, j) otherwise. Note that A is indeed an automaton, i.e. δ(i, j) is well-deined or each i, j. We claim that A admits P i and only i n is reachable rom 1 in G. Observe that the underlyin raph o A is G, with a new ede (n, 1) and with a new vertex n + 1, which is a neihbour o each vertex. Hence, {n + 1} is a sink o A which is reachable rom all other states. Thus A admits P i and only i there exists 2 Note that in this orm, the alorithm can enter an ininite loop which its into the deinition o nondeterministic lospace. Introducin a counter and allowin at most n steps in the irst cycle and at most n 2 in the second we et a nondeterministic alorithm usin lospace and polytime, as usual.

13 Nonpermutational transormation semiroups and syntactic complexity 699 a nontrivial component o A which is dierent rom {n + 1}. Since in G there are no cycles, such component exists i and only i the addition o the ede (n, 1) introduces a cycle, which happens exactly in the case when n is reachable rom 1. Note that it is exactly the case when 1x = 1 or some word x Σ +. What remains is to show that the reduced orm B o A admits P i and only i A does. First, both 1 and n + 1 are in the connected part A o A, and are distinuishable by the empty word (since n + 1 is inal and 1 is not). Thus, i A admits P with 1x = 1 and (n + 1)x = n + 1 or some x Σ +, then B admits P with h(1)x = h(1) and h(n+1)x = h(n+1) (with h bein the homomorphism rom the connected part o A onto its reduced orm). For the other direction, assume h(p)x 0 = h(p) or some state p n + 1 (note that since n + 1 is the only inal state, p n + 1 i and only i h(p) h(n + 1)). Let us deine the sequence p 0, p 1,... o states o A as p 0 = p, p t+1 = p t x 0. Then, or each i 0, h(p i ) = h(p), thus p i [n]. Thus, there exist indices 0 i < j with p i = p j, yieldin p i x j i 0 = p i, thus A admits P with p = p i, q = n + 1, x = x j i 0 and y = n. Hence, the above construction is indeed a lospace reduction rom DAG-Reach to the complement o GenDe, showin NL-hardness o the latter; applyin NL = conl aain, we et NL-hardness o GenDe itsel. It is worth observin that the same construction also shows NL-hardness (thus completeness) o the problem whether the input automaton accepts a deinite lanuae. Thus, the complexity o the problem is characterized rom the theoretic point o view. However, nondeterministic alorithms are not that useul in practice. Since NL P, the problem is solvable in polynomial time now we ive an eicient (quadratic) deterministic decision alorithm: 1. Compute A = (Q, Σ, δ, q 0, F ), the reduced orm o the input automaton A. 2. Compute Γ(A ), the component raph o A. 3. I there exists a nontrivial, non-sink component, reject the input. 4. Compute B = A A and Γ(B). 5. Check whether there exists a state (p, q) o B in a nontrivial component (o B) or some p q with p bein in the same sink as q in A. I so, reject the input; otherwise accept it. The correctness o the alorithm is straihtorward by Theorem 2: ater minimization (which takes O(n lo n) time) one computes the component raph o the reduced automaton (takin linear time) and checks whether there exists a nontrivial component which is not a sink (takin linear time aain, since we already have the component raph). I so, then the answer is NO. Otherwise one has to check whether there is a (sink) component C and a word x Σ + such that x C has at least two dierent ixed points. Now it is equivalent to ask whether there is a state (p, q) in A A with p and q bein in the same component and a word x Σ +

14 700 Szabolcs Iván and Judit Nay-Györy with (p, q)x = (p, q). This is urther equivalent to ask whether there is a (p, q) with p, q bein in the same sink such that (p, q) is in a nontrivial component o B. Computin B and its components takes O(n 2 ) time, and (since we still have the component raph o A) checkin this condition takes constant time or each state (p, q) o B, the alorithm consumes a total o O(n 2 ) time. Hence we have a low-deree polynomial-time upper bound: Theorem 5. Problem GenDe can be solved in O(n 2 ) deterministic time in the RAM model o computation. 5 Conclusion, urther directions The orbidden pattern characterization o eneralized deinite lanuaes we ave is not surprisin, based on the identities o the pseudovariety o (syntactic) semiroups correspondin to this variety o lanuaes. Still, usin this characterization one can derive eicient alorithms or checkin whether a iven automaton reconizes such a lanuae. Thouh we could not compute an exact unction or the syntactic complexity, we still manaed to show that these lanuaes are not more complex than deinite lanuaes under this metric. Also, we ave a new upper bound or that. The exact syntactic complexity o deinite lanuaes is still open, as well as or other lanuae classes hiher in the dot-depth hierarchy e.. the locally (threshold) testable and the star-ree lanuaes. Reerences [1] R. S. Cohen, J. Brzozowski. Dot-Depth o Star-Free Events. Journal o Computer and System Sciences 5(1), 1971, [2] J. Brzozowski, D. Liu. Syntactic Complexity o Finite/Coinite, Deinite, and Reverse Deinite Lanuaes. [3] S. Cho, D. T. Huynh. The parallel complexity o inite-state automata problems. Inorm. Comput. 97, 122, [4] M. Čirič, B. Imreh, M. Steinby. Subdirectly irreducible deinite, reverse deinite and eneralized deinite automata. Publ. Electrotechn. Fak. Ser. Mat., 10, 1999, [5] F. Gécse, B. Imreh. On isomorphic representations o eneralized deinite automata. Acta Cybernetica 15, 2001, [6] A. Ginzbur. About some properties o deinite, reverse-deinite and related automata. IEEE Trans. Electronic Computers EC-15, 1966, [7] M. Holzer, B. Köni. On deterministic inite automata and syntactic monoid size. Theoretical Computer Science 327(3), , 2004.

15 Nonpermutational transormation semiroups and syntactic complexity 701 [8] Neil D. Jones, Y. Edmund Lien and William T. Laaser: New problems complete or nondeterministic lo space. THEORY OF COMPUTING SYSTEMS Volume 10, Number 1 (1976), [9] O. Klíma, L. Polák. Alternative Automata Characterization o Piecewise Testable Lanuaes. Accepted to DLT [10] B. Krawetz, J. Lawrence, J. Shallit. State Complexity and the Monoid o Transormations o a Finite Set. Proc. o Implementation and Application o Automata, LNCS 3317, 2005, [11] B. Li. Syntactic Complexities o Nine Subclasses o Reular Lanuaes. Master s Thesis. [12] D. Perrin. Sur certains semiroupes syntactiques. Séminaires de l IRIA, Loiques et Automates, Paris, 1971, [13] T. Petkovič, M. Čirič, S. Bodanovič. Decomposition o automata and transition semiroups. Acta Cybernetica 13, 1998, [14] J-É. Pin. Syntactic semiroups. Chapter 10 in Handbook o Formal Lanuaes, Vol. I, G. Rozenber et A. Salomaa (eds.), Spriner Verla, 1997, [15] M. Steinby. On deinite automata and related systems. Ann. Acad. Sci. Fenn., Ser. A I 444, [16] J. Stern. Complexity o some problems rom the theory o automata. Inormation and Control 66, 1985, [17] A. N. Trahtman. Piecewise and local threshold testability o DFA. Proc. o FCT 2001, LNCS 2038 (2001), Received 19th July 2014