1. B 0 = def f;; A + g, 2. B n+1=2 = def POL(B n ) for n 0, and 3. B n+1 = def BC(B n+1=2 ) for n 0. For a language L A + and a minimal n with L 2 B n

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1 Languages of Dot{Depth 3=2 Christian Glaer? and Heinz Schmitz?? Theoretische Informatik, Universitat Wurzburg, Wurzburg, Germany Abstract. We prove an eective characterization of languages having dot{depth 3=2. Let B 3=2 denote this class, i.e., languages that can be written as nite unions of languages of the form u 0L 1u 1L 2u 2 L nu n, where u i 2 A and L i are languages of dot{depth one. Let F be a deterministic nite automaton accepting some language L. Resulting from a detailed study of the structure of B 3=2, we identify a pattern P (cf. Fig. 2) such that L belongs to B 3=2 if and only if F does not have pattern P in its transition graph. This yields an NL{algorithm for the membership problem for B 3=2. Due to known relations between the dot{depth hierarchy and symbolic logic, the decidability of the class of languages denable by 2{formulas of the logic FO[<; min; max; S; P ] follows. We give an algebraic interpretation of our result. 1 Introduction We contribute to the theory of nite automata and regular languages, with consequences in logic as well as in algebra. Particularly, we deal with starfree regular languages. These are languages constructed from alphabet letters using only Boolean operations together with concatenation. Alternating these two kinds of operations in order to distinguish combinatorial and sequential aspects, leads to the denition of concatenation hierarchies that exhaust the class of starfree languages. Prominent examples are the dot{depth hierarchy, rst studied in [CB71], and the Straubing{Therien hierarchy [Str81,The81,Str85]. Both are known to be strict [BK78] and closely related to each other [Str85]. Most naturally arising questions concerning these hierarchies are of major interest in dierent research areas since close connections have been exposed, e.g., to nite model theory, theory of nite semigroups, complexity theory and others. Here we deal with the dot{depth hierarchy. Let A be some nite alphabet with jaj 2. For a class C of languages over A + let POL(C) be its polynomial closure, i.e., the class of languages L A + that can be written as a nite union of languages u 0 L 1 u 1 L 2 u 2 L n u n, where u i 2 A, L i 2 C and n 0. Denote by BC(C) its Boolean closure, i.e., the closure of C under nite union, nite intersection and complementation. Then the dot{depth hierarchy can be dened as the following family of classes B n=2 (notations are adopted from [PW97]).? Supported by the Studienstiftung des Deutschen Volkes.?? Supported by the Deutsche Forschungsgemeinschaft (DFG), grant Wa 847/4-1.

2 1. B 0 = def f;; A + g, 2. B n+1=2 = def POL(B n ) for n 0, and 3. B n+1 = def BC(B n+1=2 ) for n 0. For a language L A + and a minimal n with L 2 B n=2 we say that L has dot{depth n=2. One obtains the same hierarchy classes when setting B n+1=2 = def POL(coB n?1=2 ) with cob n?1=2 = def L L 2 Bn?1=2 [Gla98]. By denition, all B n+1=2 are closed under union and it is known, that these classes are also closed under intersection [Arf91]. The question whether there exists an algorithm deciding `L(F ) 2 B n=2 ' for n 0 and deterministic nite automata F (dfa F, for short) is known as the dot{depth problem. Although many researchers believe the answer should be yes, some suspect the contrary. To our knowledge, only the classes B 0, B 1=2 and B 1 were known to be decidable [Kna83,PW97]. Especially, the case of dot{depth 3=2 was mentioned open in [Pin96,PW97]. This can be seen in contrast to the Straubing{Therien hierarchy, for which beside levels 0, 1=2 and 1 also level 3=2 is known to be decidable [Arf91,PW97]. Some partial results are known for level 2 of this hierarchy, e.g., its decidability in case of a 2-letter alphabet [Str88]. In this paper we prove an eective characterization of languages having dot{ depth 3=2. With an automata{theoretic approach we study the class B 3=2 in detail. Fix some k 0. We look at a word w as a word over A k+1 by taking together each k + 1 consecutive letters and call this the S k-decomposition of w. In this way we obtain classes B 3=2;k for which B 3=2 = B k0 3=2;k. The k- decomposition approach was used before in several contexts, e.g., when relating the dot{depth hierarchy and the Straubing{Therien hierarchy [Str85], and for a levelwise analysis of dot{depth one languages [Ste85]. We rst look at the family of classes B 3=2;k. With the help of a series of technical lemmas we prove a useful normalform representation for languages in B 3=2;k. Then we provide a combinatorial lemma which keeps control of the lengths of factors x; u; y of some input w to a dfa such that w = xuy and the state reached after input x 0 u has a loop with label u for all x 0. Iterated applications of this lemma are a basic tool in subsequent proofs. Let F be a dfa accepting some language L. We show that L belongs to B 3=2;k if and only if F does not have pattern Pk in its transition graph (cf. Fig. 1). This yields an NL{algorithm for the membership problem for B 3=2;k which looks for the non{existence of Pk in F. Since we encounter for k = 0 level 3=2 of the Straubing{Therien hierarchy, we provide as a by-product a self-contained reproof of the normalform and the decidability result for this class ([Arf91] and [PW97] use deep results from [Has83] and [Sim90], respectively). Our generalization to arbitrary k enables us to identify a pattern P such that L belongs to B 3=2 if and only if F does not have pattern P in its transition graph (cf. Fig. 2). So we can armatively answer the decidability question for B 3=2 even in an ecient way: looking for the non{existence of P in F yields an NL{algorithm for the membership problem of this class. Moreover, the proof is such that an algorithm can be derived to determine the exact level k of a given language inside B 3=2.

3 We draw some consequences which are due to various relations of the classes of the dot{depth hierarchy to other elds of research. The connection to rst{ order logic goes back to [MP71]. The dot{depth hierarchy is related to the rst{ order logic FO[<; min; max; S; P ] having unary relations for the alphabet symbols from A, the binary relation <, the successor (predecessor) function S (P, resp.), and constants min and max. Let n be the subclass of this logic which is dened by at most n?1 quantier alternations, starting with an existential quantier. It has been proved in [Tho82] (see also [PP86,PW97]) that n {formulas describe just the B n?1=2 languages and that the Boolean combinations of n { formulas describe just the B n languages. Due to this characterization we can conclude the decidability of the class of languages denable by 2 {formulas of the logic FO[<; min; max; S; P ]. We also give an algebraic interpretation of our result and characterize the languages of dot{depth 3=2 by a condition on their ordered syntactic semigroups. Spoken in algebraic terms, this yields an eective characterization of the variety of nite ordered semigroups corresponding to the positive variety of languages, as which B 3=2 can be understood. If one looks at our result in combination with the known forbidden pattern characterization of B 1=2 from [PW97] it is easy to discover regularities between both patterns. Continuing them leads to the denition of a pattern Rn for n 1, which denes decidable subclasses C n of starfree languages in a forbidden pattern manner. We conclude this paper with some informal arguments supporting the possibility that C n = B n?1=2 holds also for all n 3. A comprehensive treatment of the issues presented in this extended abstract is given in a self{contained way in [GS99], see 2 The Classes B 3=2;k Throughout the paper we consider languages as subsets of A +. Let k 0. We denote by A k the set of words from A + of length less or equal to k (similarly, A <k, A k, : : : ). It will be useful for us to look at w 2 A + as a word over A k+1 by taking together each k + 1 consecutive letters. We denote elements from A k+1 as ; ; ; : : : and subsets of A k+1 as ;?; : : :. Let w = a 1 a 2 a k+l 2 A + for some l 1. We call bw = def ( 1 ; 2 ; : : : ; l ) the k-decomposition of w if i = a i a i+k for 1 i l. Intuitively, k indicates by how many letters from A consecutive i overlap. We set (bw) = def f 1 ; 2 ; : : : ; l g. Next we dene languages that admit the same k-decomposition with respect to given elements and subsets of A k+1. Denition 1. Let k; n 0 and 1 ; : : : ; n 2 A k+1, 0 ; : : : ; n A k+1. For every w 2 A + we say w 2 ( 0 ; 1 ; 1 ; : : : ; n ; n ) k if and only if jwj k + 1, bw = ( 1 ; : : : ; l ) and there exist 0 = j 0 < j 1 < j 2 < : : : < j n < j n+1 = l + 1 such that (a) ji = i for 1 i n and (b) j 2 i for 0 i n and j i < j < j i+1 :

4 If we write the expression ( 0 ; 1 ; 1 ; : : : ; n ; n ) k we understand this as a syntactical object describing some language. We do not distinguish between this object and the language it stands for. So the language ( 0 ; 1 ; 1 ; : : : ; n ; n ) k consists of those words w 2 A k+1, whose k-decomposition starts with a number (possibly zero) of elements from 0, then 1, followed by a number (possibly zero) of elements from 1, then 2 and so on. Note that in case k = 0 we deal with the usual concatenation, e.g., (A 0 ) 0 = A + 0 and (A 0; a 1 ; A 1 ; a 2 ; A 2 ) 0 = A 0 a 1A 1 a 2A 2. For convenient notations we write (wj 0 ; 1 ; 1 ; : : : ; n ; n jv) k instead of? wa \ A v \ ( 0 ; 1 ; 1 ; : : : ; n ; n ) k. Denition 2. Let k 0 and m 1. Then B 1;(m;k) is the class of languages L A + that are in the Boolean algebra generated by languages L i such that L i = D with D A <k+m or L i = (wja k+1 ; 1 ; A k+1 ; 2 ; A k+1 ; : : : ; m ; A k+1 jv) k where j 2 A k+1 and w; v 2 A k. The denition of these classes is motivated by a characterization of B 1 in terms of the congruence m;k with k; m 0 introduced in [Sim72]. With the next theorem we recall in our notations the fact that the classes B 1;(m;k) rene B 1. Theorem 1 ([Sim72]). Let L A +. Then L 2 B 1 if and only if there exist k 0 and m 1 such that L 2 B 1;(m;k). For an overview on hierarchies which result from xing one or the other parameter see [Brz76]. Among others, a hierarchy in B 1 obtained by xing k has been studied [Sim72,Ste85]. Denition 3 ([Sim72]). Let k 0. Then B 1;k = def Sm1 B 1;(m;k). Proposition 1. B 1 = S k0 B 1;k and B 3=2 = S k0 POL(B 1;k). The rst equation follows from Theorem 1, while the latter is due to B 3=2 = POL(B 1 ) = POL( S k0 B 1;k) = S k0 POL(B 1;k), which gives rise to the starting point of our investigations. Denition 4. Let k 0. Then B 3=2;k = def POL(B 1;k ). By denition, languages in B 3=2;k are nite unions of concatenations of words with languages from B 1;k, which are in turn Boolean combinations of the languages L i from Denition 2, a somewhat unstructured representation. We give the following normalform. Theorem 2. Let k 0 and L A +. Then L 2 B 3=2;k if and only if L can be written as a nite union of languages L i such that L i = D for D A k or L i = ( 0 ; 1 ; 1 ; : : : ; n ; n ) k where n 1, j 2 A k+1 and j A k+1.

5 3 Finding Automata Loops in Words A useful tool in our proofs is the fact that we can nd factors in a word which lead to loops in a given dfa. It is important here to analyse the length needed to nd such a factor, depending on the size of the dfa in question. For this end, we dene a bounding function K(n) as K(n) = def (n + 1) (n+1)(n+1) and prove the following rather technical lemma. Let be the transition function of the given dfa F. Then every w 2 A induces a total mapping w : S! S on the set of states S with w (s) = def (s; w) for all s 2 S. Lemma 1. For every dfa F and for all v 0 ; : : : ; v n 2 A + there exist an m 0 and indices 0 = i 0 < i 1 < < i 2m+1 = n + 1 such that 1. i j+1? i j K(jF j) for 0 j 2m and 2. uu = u for all u = v ij v ij +1 v ij+1?1 with 1 j < 2m and j 1(2). To give some intuition, this means for factors of length one, that for every K(jF j) w = v 0 v 1 v n with v i 2 A there exist words w 0 ; : : : ; w m ; u 1 ; : : : ; u m 2 A such that w = w 0 u 1 w 1 u m w m and uiui = ui for 1 i m. We use Lemma 1 for arbitrary factors v i in the proof of Lemma 2 below. 4 Forbidden Pattern Characterization of B 3=2;k Let F be a dfa accepting a language L. We want to show that L is in B 3=2;k if and only if F does not have pattern Pk (cf. Fig. 1) in its transition graph. As usual v v s 0 u s 1 w s 2 z z s + s? Fig. 1. Pattern Pk with u; w; z 2 A ; v 2 A k+1, initial state s 0, accepting state s +, rejecting state s? and (dvwv) (cvv) for k-decompositions of words. in forbidden pattern proofs, there is on one hand an easier to prove implication (Theorem 3) and on the other hand a more dicult one (Theorem 4). Theorem 3. Let k 0. If a dfa F has pattern Pk, then L(F ) =2 B 3=2;k.

6 Proof (Sketch). Suppose F has pattern Pk and L(F ) 2 B 3=2;k. By Theorem 2 we have that L(F ) is a nite union of languages L i such that L i = D for D A k or L i = ( 0 ; 1 ; 1 ; : : : ; n ; n ) k. Since we can pump up uz 2 L(F ) to arbitrary uv j z 2 L(F ), we can determine a suciently large j, such that uv j z 62 A k and there is some l such that (cvv) 2 l. Because (dvwv) (cvv) by pattern Pk we can insert w without leaving L(F ), a contradiction to pattern Pk. ut The more complicated implication will be a consequence of Lemma 2 given below. There we derive from every x 2 L a subset of L which contains x and which can be described by expressions of bounded size. In particular, we consider expressions E of the form w 0 (v 1 j 1 jv 0 1) k w 1 (v n j n jv 0 n) k w n where w i 2 A +, v j ; vj 0 2 A k and j A k+1. We dene the size of E as jw 0 w 1 w n j and identify E with the language described by E. For a xed k 0 let us denote the set of all such expressions by E k. To analyse the size of expressions in Lemma 2 we make the following denition, where variables a; f; n will be associated with the size of the alphabet A, the size of the dfa F and the cardinality of ( bw) for a given word w, respectively. L(k; a; f; n) = def 8 >< >: k : if n = 0 2f f + k + 1 : if n = 1 3K(f) (5f f a k + 1) L(k; a; f; n? 1) : otherwise Lemma 2. Let k 0 and let F be a dfa which does not have pattern Pk. For every x 2 A + there exists an expression E x 2 E k of size L(k; jaj; jf j; j(bx)j) with x 2 E x and for all x 0 ; x 00 2 A we have x 0 xx 00 2 L(F ) =) x 0 E x x 00 L(F ): In the following, the term `short' (`long') means that the size of an expression can (can not, resp.) be bounded by a function in k; jaj; jf j and j(bx)j (we use `small' and `large' for cardinalities). Proof (Sketch). We prove the lemma by induction on j(bx)j. If j(bx)j = 0 then x is short and we are done. If j(bx)j = 1, it follows that x = a jxj for some letter a 2 A. Now it is easy to see that either x is short or a i? a k j a k+1 ja k k aj provides the desired expression for suitable choices of small i and j. In the induction step we consider x 2 A + with j(bx)j = n First of all we decompose x into factors s i (so-called `sectors') with j(bs i )j n as follows. Determine the longest prex s 1 of x such that j(bs 1 )j n. Now we start over with the remaining part of x and determine its longest prex s 2 such that j( bs 2 )j n and so on. Observe that we obtain a factorization of x into sectors s i such that ( \s i s i+1 ) = (bx). Furthermore, the induction hypothesis provides us with short expressions for sectors constructed in this way.

7 Note that neither the length of sectors, nor their number must be small. The main task of the induction step is to replace the large number of sectors by a small number of terms (vjjv 0 ) k in such a way that (i) we do not leave L(F ) if we started with x 2 L(F ) and (ii) we obtain an expression where only a small number of sectors is left. If the number of sectors is already small, we can replace each sector s with the expression E s provided by the induction hypothesis and we are done. If the number of sectors is large, we combine them to pairs p i = def s 2i?1 s 2i in order to have (bp i ) = (bx). Now we apply Lemma 1 to these pairs and get a partitioning of the sequence of the p i with x = w 0 u 1 w 1 u m w m such that (i) each partition u i (w i, resp.) consists of a small number of pairs and (ii) every partition u i leads to an u i -loop (i.e., uiui = ui ) with ( bu i ) = (bx). Now we assign to each u i a tag representing the mapping w0u1w1ui and the k-sux of u i. In a next step we want to nd maximal non{intersecting factors between some u i and u j having the same tags (so{called `regions'). Consider the simple greedy algorithm which chooses repeatedly a largest factor between some u i ; u j having the same tags, such that the region between u i and u j does not intersect an existing region (it stops if this is not possible). Since the number of dierent tags is small, it can be shown that this algorithm returns a small number of regions, such that the number of u i and w i in some gap between those regions is also small. It follows that the number of sectors in some gap between regions is bounded. Note that regions may contain a large number of sectors. We treat all regions of x from right to left. Consider a particular region between u i and u j. Then we replace this region with the term T = u i (pj(bx)js) k u j where p is the k-prex and s is the k-sux of the word w i u i+1 w i+1 u j?1 w j?1. Now we have reached a situation where we make use of the fact that F does not have pattern Pk. Using Lemma 1 and the tags, it can be observed that u i and any word from T lead to an u j -loop. It follows that if we leave L(F ) with some word from T, then we would nd pattern Pk in F, a contradiction. We can continue this substitution from right to left, region by region, without leaving L(F ), since the tags left to the substitution position remain valid. In this way we obtain an expression Ex 0 2 E k with x 2 Ex. 0 Since the whole argument is independent of prexes x 0 and suxes x 00, we can show x 0 xx 00 2 L(F ) =) x 0 Ex 0 x00 L(F ). Furthermore, the number of terms (vjjv 0 ) k and number of remaining sectors in Ex 0 is small. If we apply the induction hypothesis to these sectors, we obtain the desired expression E x. This completes the induction. ut Theorem 4. Let k 0. If a dfa F does not have pattern Pk, then L(F ) 2 B 3=2;k. Proof (Sketch). Let F be a dfa which does not have pattern Pk. By Lemma 2 we nd for every x 2 L(F ) a short expression E x L(F ) with x 2 E x. Since there is only a nite number of dierent expressions of E k having the same size, we can write L(F ) as a nite union of expressions of E k. With the help of Theorem 2 it can be shown that languages of E k are in B 3=2;k. ut Taking together Theorems 3 and 4 we obtain the main result of this section.

8 Theorem 5. Let k 0 and let F be a dfa. Then L(F ) 2 B 3=2;k if and only if F does not have pattern Pk. To see that this characterization is eective we provide an ecient algorithm to check the non{existence of pattern Pk in the transition graph of a given dfa. In particular, we show that the occurrence of pattern Pk can be decided in nondeterministic, logarithmic space (NL) for a xed k 0, which is a class closed under complementation. For this end we guess the states s 1 ; s 2 ; s + ; s? and check the existence of the words u; v; w; z applying the same technique that solves the graph accessibility problem. While guessing v and w we additionally store the last k letters which then enables us to determine all elements of (dvwv) and (cvv). Since k and the size of the alphabet can be considered as constants, all this can be done in NL. Furthermore, Theorem 5 allows a concise proof of the strictness of the hierarchy of classes B 3=2;k. We obtain B 3=2;k ( B 3=2;k+1 for k 0 with help of the witnessing languages L k = def? a k+1 b; a i ba k+1?i : 0 i k + 1 ; a k+1 b k+1. 5 Forbidden Pattern Characterization of B 3=2 We identify the pattern P given in Fig. 2 which characterizes B 3=2. u 2 u 2 u 2 u 2 u 1 w 2 v 2 u 1 w 2 v 2 u 1 v 1 um u 1 v 1 um w 1 wm+1 w 1 wm+1 u 0 v 0 vm+1 um+1 u 0 v 0 vm+1 um+1 s 0 u s 1 w 1 w 2 wm+1 s 2 z u 0 ; um+1 u 1 u 2 um z u 0 ; um+1 s + s? Fig. 2. Pattern P with initial state s 0, accepting state s +, rejecting state s?, m 0, u i; w i 2 A+ and u; z; v i 2 A. Theorem 6. Let F be a dfa accepting some language L. Then L 2 B 3=2 if and only if F does not have pattern P.

9 Proof (Sketch). We rst show that the existence of pattern P implies the existence of pattern Pk for every k 0. As witnessing words take u, z and v = def u k 0 v 0u k 0 w 1u k 1 v 1u k 1 w m+1 u k m+1 v m+1u k m+1 and w = def u k 0 w 1u k 1 w m+1 u k m+1. This denition ensures that each element of the k-decomposition of vwv overlaps at most two of the u k i. It follows that (dvwv) (cvv). Now suppose that F has pattern Pk for every k 0. In particular, we nd a pattern for k = def 3K(jF j) with jwj k + 1 (take vwv instead of w if necessary). K(jF j) By Lemma 1 we can write w as w = w 0 u 1 w 1 u l w l with words w i ; u i 2 A such that ui = uiui. Since (dvwv) (cvv) and ju i w i u i+1 j k we can nd each factor u i w i u i+1 in vv. This argument leads to pattern P. ut Since this proof establishes a bound on k in the size of the automaton, we can also nd an algorithm which determines the minimal k such that L(F ) 2 B 3=2;k for a given dfa F. As before in the case of the classes B 3=2;k, we exploit now Theorem 6 and construct an ecient algorithm which solves the membership problem for B 3=2. Looking for pattern P can also be done in NL since we may continuously check piecewise the occurrences of the respective subgraphs for w i u i v i u i and w i u i. Note that we do not need to bound m since no bound is required for the length of paths in an NL{computation. Theorem 7. The membership problem for B 3=2 is in NL. 6 Further Consequences Due to the various characterizations of the classes of the dot{depth hierarchy, Theorem 7 has immediate consequences in other elds of research. The correspondence of the class of languages denable by n {formulas of the logic FO[<; min; max; S; P ] and B n?1=2 from [Tho82] has already been mentioned in the introduction. Due to this characterization we have the following corollary. Corollary 1. Given a regular language L, it is decidable whether L is denable by a 2 {formula of the logic FO[<; min; max; S; P ]. An algebraic interpretation of our Theorem 6 can also be given. For an introduction to the algebraic theory of nite automata we refer to [Pin96]. Let L be a regular language of A + and let F L = (A; S; ; s 0 ; S 0 ) be its unique minimal dfa. We dene the syntactic semigroup of L via the transition semigroup of F L, i.e., as S L = def f w : w 2 A + g where the composition is dened as u v = def uv. By id S we denote the identity mapping on S. The syntactic semigroup S L can be considered as an ordered syntactic semigroup S L with order relation by setting if and only if (s 0 ) 2 S 0 implies (s 0 ) 2 S 0 for ; 2 S L [fid S g. If is an element of a nite semigroup, the minimal idempotent power of is denoted as!.

10 Theorem 8. Let L A + be a regular language and S L be its ordered syntactic semigroup. Then L 2 B 3=2 if and only if S L satises all inequalities fe m : m 0g for any choice of i ; i and i from S L where E m = def!!! with = def! 0 1! 1 2! 2 m+1! m+1 and = def! 0 0! 0 1! 1 1! 1 2! 2 2! 2 m+1! m+1 m+1! m+1 : Due to [Arf91] the class B 3=2 can be understood as a positive +{variety of languages when varying the alphabet A (for the denition of the notion of positive varieties we refer to [PW97]). An Eilenberg{like theorem was given for the case of positive varieties in [Pin95], which states that positive +{varieties of languages and varieties of nite ordered semigroups are in one{one correspondence via the operation of taking ordered syntactic semigroups. So the inequalities fe m : m 0g from Theorem 8 characterize the variety of nite ordered semigroups corresponding to B 3=2. It is known that this variety is equal to the Mal'cev product of the variety of nite semigroups corresponding to dot{depth one languages with a certain other variety of nite ordered semigroups, as stated in Theorem 5.8 in [PW97]. The benet of our characterization is that it is eective as follows from Theorem 7. 7 Conclusions It was conjectured in [PW97] that the decidability questions for the Straubing{ Therien hierarchy and the dot{depth hierarchy are related not only on levels n for integers n [Str85], but on all levels n=2. We conrm the latter with our work now also for n = 3, while the general case remains open. We see the contribution of our paper not only as a stand{alone result providing the decidability of B 3=2, but in what we can carry over to the general case. First we note that the nature of our proof is such that it bounds in a computable way (in terms of the automaton size and the alphabet size) the descriptional complexity of a language L, i.e., it bounds the length of an expression that witnesses that L is of dot{depth 3=2. Let us continue on an informal level. If we compare pattern P 0 from [PW97] characterizing B 1=2 and our pattern P characterizing B 3=2, we observe that subgraphs of type P 0 appear as u i v i u i in pattern P. Now one can repeat inductively this formation procedure using pattern P in a more complicated pattern in the same way as P 0 appears in P. This leads for n 1 to the denition of patterns Rn for which R 1 =P 0 and R 2 =P holds (This can also be done for the Straubing{Therien hierarchy with the same formation procedure, but starting with the pattern that characterizes level 1=2 there). Moreover, looking for the existence of Rn can be eectively carried out with a recursive application of our algorithm for testing pattern P. Denote by C n the class of languages that can be accepted exactly by those dfa's which do not have Rn in their transition graph. We believe that it is a

11 reasonable conjecture that B n?1=2 = C n holds for all n 1. As we know now, this is true for n = 1 and n = 2. We can further support this by some partial results, left without proofs here. First we note that for all n 1 it holds that C n is a subclass of starfree languages. Moreover, one can show with a generalization of Theorem 3 that C n contains B n?1=2. Finally, we see that C n ( C n+1 using the languages L n from [BK78] witnessing the strictness of the dot{depth hierarchy. Acknowledgements. For discussions related to our subject we thank Bernd Borchert, Klaus W. Wagner and Thomas Wilke. References [Arf91] M. Ar. Operations polynomiales et hierarchies de concatenation. Theoretical Computer Science, 91:71{84, [BK78] J. A. Brzozowski and R. Knast. The dot-depth hierarchy of star-free languages is innite. Journal of Computer and System Sciences, 16:37{55, [Brz76] J. A. Brzozowski. Hierarchies of aperiodic languages. RAIRO Inform. Theor., 10:33{49, [CB71] R. S. Cohen and J. A. Brzozowski. Dot-depth of star-free events. Journal of Computer and System Sciences, 5:1{16, [CK96] C. Chorut and J. Karhumaki. Combinatorics of words. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 329{438. Springer, [Gla98] C. Glaer. A normalform for classes of concatenation hierarchies. Technical Report 216, Inst. fur Informatik, Univ. Wurzburg, [GS99] C. Glaer and H. Schmitz. Languages of dot-depth 3/2. Technical Report 243, Inst. fur Informatik, Univ. Wurzburg, [Has83] K. Hashiguchi. Representation theorems on regular languages. Journal of Computer and System Sciences, 27:101{115, [Hig52] G. Higman. Ordering by divisibility in abstract algebras. In Proc. London Math. Soc., volume 3, pages 326{336, [Kna83] R. Knast. A semigroup characterization of dot-depth one languages. RAIRO Inform. Theor., 17:321{330, [MP71] R. McNaughton and S. Papert. Counterfree Automata. MIT Press, Cambridge, [Pin95] J. E. Pin. A variety theorem without complementation. Russian Math., 39:74{ 83, [Pin96] J. E. Pin. Syntactic semigroups. In G.Rozenberg and A.Salomaa, editors, Handbook of formal languages, volume I, pages 679{746. Springer, [PP86] D. Perrin and J. E. Pin. First-order logic and star-free sets. Journal of Computer and System Sciences, 32:393{406, [PW97] J. E. Pin and P. Weil. Polynomial closure and unambiguous product. Theory of computing systems, 30:383{422, [Sim72] I. Simon. Hierarchies of events with dot-depth one. PhD thesis, University of Waterloo, [Sim90] I. Simon. Factorization forests of nite height. Theoretical Computer Science, 72:65{94, 1990.

12 [SS83] J. Sakarovitch and I. Simon. Subwords. In M. Lothaire, editor, Combinatorics on Words, Encyclopedia of mathematics and its applications, pages 105{142. Addison-Wesley, [Ste85] J. Stern. Characterizations of some classes of regular events. Theoretical Computer Science, 35:17{42, [Str81] H. Straubing. A generalization of the Schutzenberger product of nite monoids. Theoretical Computer Science, 13:137{150, [Str85] H. Straubing. Finite semigroups varieties of the form V * D. J.Pure Appl.Algebra, 36:53{94, [Str88] H. Straubing. Semigroups and languages of dot-depth two. Theoretical Computer Science, 58:361{378, [The81] D. Therien. Classication of nite monoids: the language approach. Theoretical Computer Science, 14:195{208, [Tho82] W. Thomas. Classifying regular events in symbolic logic. Journal of Computer and System Sciences, 25:360{376, 1982.