Two-variable first order logic with modular predicates over words

LIPICS - Leibniz International Proceedings in Informatics, Feb 2013

We consider first order formulae over the signature consisting of the symbols of the alphabet, the symbol < (interpreted as a linear order) and the set MOD of modular numerical predicates. We study the expressive power of FO^2[<,MOD], the two-variable first order logic over this signature, interpreted over finite words. We give an algebraic characterization of the corresponding regular languages in terms of their syntactic morphisms and we also give simple unambiguous regular expressions for them. It follows that one can decide whether a given regular language is captured by FO^2[<,MOD]. Our proofs rely on a combination of arguments from semigroup theory (stamps), model theory (Ehrenfeucht-Fra�ss� games) and combinatorics.

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Two-variable first order logic with modular predicates over words

S TA C S ' Two-variable rst order logic with modular predicates over words? Luc Dartois 0 Charles Paperman 0 0 LIAFA, Universite Paris-Diderot and CNRS , Case 7014, 75205 Paris Cedex 13 , France We consider rst order formulae over the signature consisting of the symbols of the alphabet, the symbol < (interpreted as a linear order) and the set MOD of modular numerical predicates. We study the expressive power of FO2[<, MOD], the two-variable rst order logic over this signature, interpreted over nite words. We give an algebraic characterization of the corresponding regular languages in terms of their syntactic morphisms and we also give simple unambiguous regular expressions for them. It follows that one can decide whether a given regular language is captured by FO2[<, MOD]. Our proofs rely on a combination of arguments from semigroup theory (stamps), model theory (Ehrenfeucht-Frasse games) and combinatorics. and phrases First order logic; automata theory; semigroup; modular predicates - [<, MOD] ?1 Decidable [12, 21] Decidable [4] B?1 Decidable [17, 21] Decidable [4] FO2 Decidable [20] Decidable New result FO Decidable [11, 15] Decidable [18, 2] We also give an algebraic characterization of FO2[<, MOD] (Theorem 6), a description of the corresponding languages as unambiguous regular expressions (Proposition 31) and an equivalent de nition in terms of a suitable variant of temporal logic (Proposition 30). Our algebraic characterization QDA = FO2[<, MOD] can be viewed as an extension of two known results (a) QA = FO[<, MOD] proved in [2, 18] and (b) DA = FO2[<] proved in [5, 20]. However, it is not easy to extend the proofs of these equalities to our case. For instance, the proof of (a) makes use of the successor relation, which is not expressible in FO2[<]. Therefore our proof is closer to the proof of (b) but some technical di culties still have to be worked out (See Section 5). 1 1.1 Preliminaries Words and logic Let A be a nite alphabet. We denote by A? the set of all nite words over A and 1 the empty word. Given a word u = a0 ? ? ? an?1 of length n, we denote by ?(u) the set of letters of A occurring in u. We associate to u the relational structure Mu = {[0, n ? 1], ?}, where [i, j] is the set of integers between i and j and ? is the truth table of the predicates over u. Basic examples of predicates are the binary predicate <, which is the usual order on integers, and (a)a?A that are disjoint monadic predicates marking the positions of the letters over the structure. For instance, if u = aabbab, then a = {0, 1, 4} and b = {2, 3, 5}. We also consider the modular predicate M ODid, which holds at all positions equal to i modulo d, and the 0-ary predicate Did which is true if the word has length equals to i modulo d. For u = aabbab, we have M OD02 = {0, 2, 4}, and D13 is false whereas D03 is true. We denote by MOD the set of these modular predicates. First order formulae are interpreted on words in the usual way (see [18]). For instance the formula ?x ?y ?z a(x) ? b(y) ? a(z) ? x < y ? y < z de nes the language A?aA?bA?aA?. In this article, we focus on the rst order formulae containing only two di erent variables. The subsequent logic is denoted by FO2[<]. For instance the two-variable formula ?x ?y a(x) ? b(y) ? x < y ? (?x ? a(x) ? y < x) also de nes the language A?aA?bA?aA? of the previous example. The rst order logic with the order predicate can be enriched with modular predicates. We denote by FO[<, MOD] (resp. FO2[<, MOD]) the logic built with the same atomic propositions that FO[<] (resp. FO2[<]) except that we allow the modular predicates. For instance the formula ?x ?y ?z a(x) ? M OD02(x) ? b(y) ? a(z) ? x < y ? y < z de nes the language (A2)?aA?bA?aA?. Note that if required by context, we will specify the alphabet, denoting it between parentheses. For instance FO[<](B?) denotes the set of the languages of B? de nable by a formula of FO[<]. 1.2 Algebraic notions We recall in this section the algebraic notions used in this paper. 1.2.1 Semigroups and recognizable languages We refer to [13] for the standard de nitions of semigroup theory. A semigroup is a set equipped with a binary associative operation, which we will denote multiplicatively. A monoid is a semigroup with a neutral element 1. Given a semigroup S, we denote by S1 either S if S is already a monoid or the monoid obtained by adding a neutral element 1 to S otherwise. Recall that a monoid M divides another monoid N if M is a quotient of a submonoid of N . This de nes a partial order on nite monoids. A stamp is a surjective monoid morphism from A? onto a nite monoid. A language L is recognized by a nite monoid M if there exists a stamp ? : A? ? M and a subset P of M such that L = ??1(P ). A language is recognizable if it is recognized by a nite monoid. Kleene's theorem states that the set of recognizable languages is exactly the set of rational (or regular) languages. The syntactic congruence of a regular language L of A? is the equivalence relation ?L de ned as follow: u ?L v if and only if for all w, w0 ? A?, wuw0 ? L ? wvw0 ? L. The monoid A?/?L is the syntactic monoid of L and the morphism ? : A? ? A?/?L is the syntactic stamp. 1.2.2 Stability index, stable semigroup, stable automaton For a stamp ? : A? ? M , the set ?(A) is an element of the powerset monoid of M . As such it has an idempotent power. The stability index of a stamp is the least positive integer s such that ?(As) = ?(A2s). This set is therefore a semigroup called the stable semigroup of ?. Stable semigroups are strongly related to stable automata, de ned as follows. Let A = (Q, A, ?) be a deterministic automaton and let k be a positive integer. The kautomaton of A is the deterministic automaton Ak = (Q, Ak, ?k) where q ?k (a1a2 ? ? ? ak) = (? ? ? (q ? a1) ? a2) ? ? ? ) ? ak). Note that if M is the transition monoid of A, and Mk the transition monoid of Ak, then Mk is the submonoid of M generated by the image elements of words of length k in M . I De nition 1. Let A = (Q, B, ?) be a deterministic automaton. We say that A is stable if for any two-letter word, there exists a letter that has the same action over the set Q, and conversely for any letter of B, there exists a word of B2 that has the same action over Q. As shown in the next proposition, this de nition is a compatible translation of the stable semigroup for an automaton. I Proposition 2. Let A be a deterministic automaton. Then, there is an integer k such that the associated k-automaton is stable. The least k which satis es this proposition is called the stability index of the automaton. It is equal to the stability index of the associated stamp. 1.2.3 Stamps and varieties A (pseudo) variety of ( nite) monoids is a class of monoids closed under division and nite products. According to Eilenberg [ 6 ], a variety of languages V is a class of languages closed under nite union, intersection and complementation, and closed under inverse of monoid morphism. This means that, for any monoid morphism ? : A? ? B?, X ? V(B?) implies ??1(X) ? V(A?). Furthermore Eilenberg [ 6 ] proved that there is a one-to-one correspondence between varieties of monoids and varieties of languages. The class of languages FO2[<, MOD] is not closed under inverse morphisms, and the Eilenberg's varieties theory does not apply. Still, this class is closed under inverse of lengthmultiplying morphisms (shortened as lm-morphisms), and an algebraic characterization can be obtained by considering a more general framework : the theory of C-varieties independently introduced by Esik and Ito [7] and Straubing [19] and developed by Pin and Straubing [14]. Let us now recall the notion of variety of stamps. A morphism ? : A? ? B? is lengthmultiplying if there exists an integer n such that for any letter a of A, ?(a) is a word of Bn. Given two stamps ? : A? ? M and ? : A? ? N , the product stamp is the stamp ? : A? ? M ? N de ned by ?(a) = (?(a), ?(a)). A stamp ? : A? ? M lm-divides another stamp ? : B? ? N if and only if there exists a pair (?, ?) such that ? is a lm-morphism from A? to B?, ? : N ? M is a partial onto monoid morphism and ? = ? ? ? ? ?. The couple (?, ?) is called an lm-division. Then a lm-variety of stamps is a class of stamps containing the trivial stamp and closed under lm-division and nite product. Note that if V is a variety of monoids, then the class of all stamps whose image is a monoid in V forms a lm-variety of stamps, also denoted V. Moreover, given a lm-variety of stamps V, the class V of all languages recognized by a stamp in V is a lm-variety of languages. The correspondence V ? V is one-to-one and onto [19]. These notions are very useful to decide membership problems for regular languages. Let us recall a few examples. I Example 3. A monoid M is aperiodic if there exists an integer n such that for any x ? M , xn = xn+1. It has been proved by Schutzenberger [15] and McNaughton and Papert [11] that the class of aperiodic monoids forms a variety called A and the corresponding variety of languages is exactly the rst-order de nable languages, with the order and letter predicates. I Example 4. Let DA be the variety of monoids satisfying the equation (xy)? = (xy)?x(xy)? where ? is the idempotent power of the monoid. Alternatively DA is the variety of monoids whose regular D-classes are aperiodic semigroups. The corresponding variety of languages DA is the class of FO2[<]-de nable languages [20] or equivalently the unambiguous star-free languages [16]. I Example 5. Given a variety V, the set of all stamps whose stable semigroup is in V forms a lm-variety of stamps denoted by QV. A language L has its syntactic stamp in QV if and only if there is an automaton A recognizing L and a positive integer k such that the k-automaton of A has its transition monoid in V. Straubing proved in [18] that a language is de nable in FO[<, MOD] if and only if its syntactic stamp belongs to the lm-variety of stamps QA. We always denote by QV the lm-variety of languages associated to QV. 2 Main result Our main result extends the algebraic characterization of FO2[<]-de nable languages by Therien and Wilke [20] to FO2[<, MOD]-de nable languages. The next theorem states that the languages de nable in FO2[<, MOD] are exactly the languages whose syntactic stamp is in QDA. I Theorem 6. FO2[<, MOD] = QDA Given a regular language (given by a regular expression or by some nite automaton), one can e ectively compute the stable semigroup of its syntactic stamp. Since membership in DA is decidable we get the following corollary. I Corollary 7. Given a regular language L, one can decide whether L is FO2[<, MOD] de nable. In Section 3 we will give intuition of the power of the modular predicates. The rst inclusion FO2[<, MOD] ? QDA will be proved in Section 4, using general arguments on automata and logic. The second inclusion is proved in Section 5, using Ehrenfeucht-Frasse games and algebraic tools. We will extend our main result to several other characterizations in Section 6. 3 FO2[<] over an enriched alphabet Given an integer d > 1, let us denote by FO2[<, MODd] the fragment of FO2[<, MOD] restricted to congruences modulo d. For a given language, this restriction does not lead to any loss of generality. I Lemma 8. Let L be a language of FO2[<, MOD]. Then there exists an integer d such that L is in FO2[<, MODd]. We now x a positive integer d. I De nition 9 (Enriched alphabet). Let A be an alphabet. We call the set Ad = A ? (Z/dZ) the enriched alphabet of A, and we denote by ? : A?d ? A? the projection de ned by ?(a, i) = a for each (a, i) ? Ad. For example, the word (a, 2)(b, 1)(b, 2)(a, 0) is an enriched word of abba for d = 3. We say that abba is the underlying word of (a, 2)(b, 1)(b, 2)(a, 0). I De nition 10 (Well-formed words). A word (a0, i0)(a1, i1) ? ? ? (an, in) of Ad is well-formed if for 0 6 j 6 n, ij ? j mod d. We denote by K the set of all well-formed words of A?. d I De nition 11. For a word u = a0a1 ? ? ? an ? A?, the word u = (a0, 0)(a1, 1) ? ? ? (ai, i mod d) ? ? ? (an, n mod d) is called the well-formed word attached to u. I Remark. On well-formed structures, the projection ? is a one-to-one application. The enriched word (a, 0)(b, 1)(b, 2)(a, 0) is a well-formed word for d = 3. Thanks to the previous remark, it is the unique well-formed word having the word abba as underlying word. I Remark. The operation u ? u is not a morphism. Indeed, if |u| 6? 0 mod d then uv 6= u v. Thus we de ne the k-shift operation, denoted by uk, which maps the word u = u0 ? ? ? un to the enriched word (u0, k mod d)(u1, k + 1 mod d) ? ? ? (un, n + k mod d). Note that, if |u| ? k mod d, then uv = u vk. I Proposition 12. Let d be a positive integer. Then FO2[<, MODd](A?) = ?(FO2[<](A?d) ? K). The proof relies on a syntactic transformation of the formulae. We replace M ODid by a conjunction of enriched letters predicates. This can be done in the opposite direction as well, as we consider only well-formed words. We recall (see [10]) that two words are separated by a formula of FO2[<] with quanti er depth n if and only if Spoiler wins the n rounds Ehrenfeucht-Frasse game with two coloured pebbles. Thus one can state, in light of Proposition 12, the following assertion: I Proposition 13. Let u, v be words of A?. Then there exists a formula of FO2[<, MODd] of quanti er depth n that separates them if, and only if, Spoiler wins the n rounds Ehrenfeucht Frasse game for FO2[<] over the well-formed pair (u, v). 4 The inclusion FO2[<, MOD] ? QDA In this section, we prove one direction of the main theorem, using the enriched alphabet and the well-formed words. Let us rst study the language K of well-formed words. (a,i) i 6= 3 (a,3) 1 3 Consider the semigroup Bd = (Z/dZ ? Z/dZ) ? {?} where ? is a zero of Bd and for all (i, j) and (k, `) in Z/dZ ? Z/dZ, (i, j)(k, `) = ((i, `) ? if j = k otherwise. The monoid Bd1 is the transition monoid of the minimal automaton of K for d > 2. Let us denote by J1 the variety of idempotent and commutative monoids. I Proposition 14. The set of all well-formed words is recognized by a stamp in QJ1. I Lemma 15. Let L be a language of DA(A?d). Then the language L ? K is in QDA(A?d). Proof. This comes from the fact that L ? DA(A?d) ? QDA(A?d), and K ? QJ 1(A?d) ? QDA(A?d). J Now, we can use the previous result on well-formed words over modular predicates and prove the inclusion FO2[<, MOD] ? QDA. I Theorem 16. The syntactic stamp of a FO2[<, MOD]-de nable language belongs to QDA. As suggested by one the referees, this result can be proved by using Ehrenfeucht-Frasse games. The proof given below relies on nite automata and could easily be modi ed to recover the inclusion FO[<, MOD] ? QA [18] and similar results for other fragments of logic. Proof. Let L be a regular language de nable in FO2[<, MOD](A?). Then by Lemma 8, there exists an integer d such that L is de ned in FO2[<, MODd](A?). By Proposition 12, there exists a formula ? in FO2[<](A?d) such that, L = ?(L0) with L0 = L(?) ? K. Since FO2[<] = DA (see [20]), and thanks to Lemma 15, the language L0 is in QDA(A?d). Let A0 = (Q, Ad, ?, i, F ) be its minimal trim deterministic automaton. Since ? is one-to-one, the automaton ?(A0), obtained by dropping the integer component on the transitions of A0, recognizes L. As A0 is trim and recognizes only well-formed words, the labels of all the outgoing edges from a given state have the same second component. For 0 6 i < d, let Qi = {q ? Q | there exists a ? A such that q ? (a, i) is de ned } and let QE be the set of all states of fanout 0. Then Q is a disjoint union of the sets Qi (0 6 i < d) and QE. Observing that a word of length k can only send a state of Qi to a state of Qi+k mod d ? QE, the transition function of the d-automaton A0d is a subset of S Qi ? Add ? (Qi ? QE) . Then each set Qi induces a monoid Mi, which is a submonoid 06i<d of the transition monoid of A0d. Now, going back to the projected d-automaton ?(A0)d, one can see that the action of a word u ? Ad on the set Qi is the action of the word (u0, i) ? ? ? (ud, i ? 1) on Qi in the automaton A0d, described in Mi. A0 A0d Q0 . . . Qi . . . Qd?1 (a, 0)... Q...1 .?.. ?... ?... ?... (a, i)... ?... ?... Qi...+1 .?.. ?... (a, d ? 1) ? ? ? ? Q0 d-Automaton Q0 . . . Qi . . . Qd?1 u... Q...0 .?.. ?... ?... ?... ui... ?... ?... Q...i .?.. ?... ud?1 ? ? ? ? Qd?1 The inclusion QDA ? FO2[<, MOD] We now come to the second part of the proof of Theorem 6. We rst enrich the congruences de ned in [20] to take the modular predicates into account. 5.1 Congruence and syntactic operations over FO2[<, MOD] I De nition 17. Let u ? A? be a word, and let a ? A be a letter of u. We call left adecomposition of u the unique triple (u0, a, u1) such that u = u0au1 and u0 does not contain any a. We de ne the right decomposition in a symmetrical way. We recall the de nition of the congruence ?n on A? from [20]. I De nition 18. [20] Let u, v ? A? be words. Then we have u ?0 v. Moreover, u ?n v if and only if the following conditions hold: 1. ?(u) = ?(v), the two words have the same alphabet, 2. For each a occurring in u, if (u0, a, u1) is the left a-decomposition of u and (v0, a, v1) that of v, then u0 ?n v0 and u1 ?n?1 v1, 3. For each a occurring in u, if (u0, a, u1) is the right a-decomposition of u and (v0, a, v1) that of v, then u0 ?n?1 v0 and u1 ?n v1. The termination of these inductive de nitions has to be veri ed. Let suppose that u ?n v for some words u and v and some positive integer n. Then, thanks to the rst condition, the parameter n + |?(u)| is equal to n + |?(v)j. For any left or right decomposition we decompose the words in two parts for which the parameter strictly decreases. I Proposition 19. [20] The relation ?nis a congruence. This de nition can be extended to the enriched alphabet and well-formed words as follows. We say that u ?dn v if and only if u ?n v. I Lemma 20. Let n, d be two positive integers, and u and v two words such that u ?dn v. Then the following statements hold: 1. if u is the empty word, then so is v, 2. |u| ? |v| mod d, 3. if u = u0au1, v = v0bv1 with |u0a| ? |v0b| mod d and |u1| < d, |v1| < d, then a = b, u1 = v1 and u0 ?dn?1 v0, 4. if u = u0au1, v = v0bv1 with |u0| < d, |v0| < d and |au1| ? |bv1| mod d, then a = b, u0 = v0 and u1 ?dn?1 v1, 5. for any word w, uw ?dn vw and wu ?dn wv. I Corollary 21. The relation ?dn is a congruence on A?. We will now connect our congruence to the logic FO2[<, MODd] through the Ehrenfeucht-Frasse games for FO2[<](A?d) (cf. Proposition 13). I Theorem 22. Let u, v ? A? be words. If u 6?dn v then there is a formula of FO2[<, MODd] of quanti er depth at most n + |?(u)| that separates u from v. The proof makes use of Ehrenfeucht-Frasse games following the arguments of [20]. 5.2 Congruence and algebraic operations over QDA We now de ne a slightly modi ed version of the Green's preorders adapted to the stable semigroup. Let h : A? ? M be a stamp and let S be its stable semigroup. For any elements x and y in M let us write: x 6Rst y if and only if xM ? S ? yM ? S x 6Lst y if and only if M x ? S ? M y ? S x 6Hst y if and only if x 6Rst y and x 6Lst y. We also extend our de nitions to modi ed versions of the Green's relations. x Rst y if and only if x 6Rst y and y 6Rst x x Lst y if and only if x 6Lst y and y 6Lst x x Hst y if and only if x 6Hst y and y 6Hst x We say that the stamp h is length faithful if h?1(S1) = (Ad)?. This notion is shown to be necessary in the next lemma and does not involve a loss of generality, as shown in the proof of Corollary 29. I Lemma 23. Let h : A? ? M be a stamp and let S be its stable semigroup. If h is length faithful, then the restriction of 6Rst (resp. 6Lst ) to S is the usual Green relation 6R (resp. 6L) over S. Proof. Let x be an element of S, and y an element of M such that xy is in S. Then, since h is length faithful, h?1(xy) is contained in (Ad)?. Moreover, as x belongs to S, we also have h?1(x) ? (Ad)?. Thus for any word u such that h(u) = x, and any word v such that h(v) = y, we have |u| ? |uv| ? 0 mod d, so |v| ? 0 mod d. Therefore y is an element of S. This proves that for any x in S, xM ? S = xS, and consequently for any x, y in S, x 6Rst y if and only if x 6R y in the Green relation over S. The result for the 6Lst relation is obtained with a symmetric proof. J I Corollary 24. Let h : A? ? M be a length faithful stamp of QDA. Then, the restriction of the Hst-classes to S are trivial. We also de ne the Rst-decomposition : I De nition 25. Let u be a word and let h : A? ? M be a stamp. We call the Rstdecomposition of u the tuple (u0, a1, u1, . . . , as, us) such that u = u0a1u1 ? ? ? asus and: 1. |u0a1u1 ? ? ? aiui| ? 0 mod d for all 0 6 i < s 2. h(u0a1u1 ? ? ? ui?1ai) >Rst h(u0 ? ? ? uiai+1) 3. For every pre x v of ui of length multiple of d, h(u0 ? ? ? ui?1ai) Rst h(u0 ? ? ? aiv) 4. For every pre x v and v0 of u0 of length multiple of d, h(v) Rst h(v0) The positions occurring in the Rst-decomposition are the rst positions multiple of d after falling in the 6Rst -order. The two next lemmas will link our congruence ?dn to the Rst-decomposition of the lm-morphisms of QDA. I Lemma 26. Let h : A? ? M be a length faithful stamp in QDA, let S be its stable semigroup. Let u ? S and a, x ? M . If ax ? S, then uax Rst u implies uaxa Rst u. Proof. The elements u and uax are Rst-equivalent and h is length faithful. So thanks to Lemma 23 there is an element t of S such that u = uaxt. By iteration, we obtain u = u(axt)?. But S belongs to DA, hence it satis es the equation (xy)?x(xy)? = (xy)?. Thus, (axt)?ax(axt)? = (axt)?, then u = u(axt)?ax(axt)?. Shall we rewrite this last equation, we nally get u = uaxa(xt(axt)??1). And nally u ? uaxaM ? S, proving that u Rst uaxa. J I Corollary 27. Let h : A? ? M be a length faithful stamp in QDA and let u be a word. Then if (u0, a1, u1, . . . , as, us) is the Rst-decomposition of u then (ai+1, 0) 6? ?(aiui) for i < s. Proof. Let (u0, a1, u1, . . . , as, us) be the Rst-decomposition of u. Suppose now that there exists i such that (ai+1, 0) ? ?(aiui) for i < s. Then, thanks to the preceding Lemma, h(aiuiai+1) Rst h(aiui) which is in contradiction with the de nition of the Rst-decomposition of u. J We now have all the tools to prove the following theorem. I Theorem 28. Let h : A? ? M be a length faithful stamp of QDA and let d be its stability index. Then there exists an integer n such that for every words u and v, u ?dn v implies h(u) = h(v). Proof. Thanks to Lemma 20, if two words are equivalent for the congruence ?dn+1, then their su xes of length smaller than d are equal and the associated pre xes are equivalent for d the congruence ?n. Therefore it is su cient to prove the result for words of length multiple of d. Let u and v be two words of length multiple of d, and an integer n > |?(u)||S| such that u ?dn v. Let us prove by induction on |?(u)| that h(u) = h(v). If |?(u)| = 0, then u = v = 1. Consider the result to be true up to the rank k ? 1 and let u be such that |?(u)| = k. We write (u0, a1, u1, . . . , a`, u`) the Rst-decomposition of u. One can remark that ` 6 |S|, as each ai makes the word go down in the Rst-classes, whose number is bounded by the size of S. Using the preceding corollary, (ui, ai+1, ui+1 ? ? ? u`) is a left decomposition of xi = ui ? ? ? u` for i < `. As u ?dn v, there also exists a decomposition (v0, a1, . . . , a`, v`) of d v such that aiui ?n?i aivi where (ai+1, 0) 6? ?(aiui) and hence |?(aiui)| 6 |?(u)| ? 1. As i < `, we have n ? i > (k ? 1)|S| > |?(aiui)||S|. Using the induction hypothesis, for i < `, h(aiui) = h(aivi). And hence h(u) Rst h(u1 ? ? ? a`) = h(v1 ? ? ? a`) >Rst h(v). Symmetrically, we obtain that h(v) >Rst h(u) and thus h(u) Rst h(v). Using the left/right symmetry, we also get that h(v) Lst h(u) and hence h(v) Hst h(u). By Corollary 24, the Hst-classes are trivial in QDA over words of length multiple of d and hence h(u) = h(v). J I Corollary 29. QDA ? FO2[<, MOD] Proof. Let ? : A? ? M be the syntactic stamp of L and S be the stable semigroup of ?. Assume that ? is in QDA. We claim that the morphism h : A? ? M ? Z/dZ de ned, for all words u, by h(u) = (?(u), |u| mod d) is length faithful. Indeed, the stable semigroup of h is equal to S ? {0} and h?1(S ? {0}) = (Ad)?. By Theorem 28, there exists an integer n such that the congruence ?dn is thinner than the congruence induced by h which is itself thinner than the syntactic congruence of L. Therefore L is a nite union of ?dn- classes, each of them being, according to Theorem 22, de nable by a formula of FO2[<, MODd] of quanti er-depth at most n + |A|d. J 6 Other characterizations Several other characterizations of DA are known (see [5] for a survey). For example, consider the fragment TL[Xa, Ya] of the linear temporal logic de ned inductively as follow: ? ? > | ? ? ? | ? ? ? | ?? | Xa? | Ya?. The unary operator Xa stands for neXt a, and Ya stands for Yesterday a. For a word u and one of its positions x, we have (u, x) |= Xa? if ? is true at the next a after x. We say that the word u satis es Xa? if (u, ?1) |= Xa?. Symmetrically, we say that u satis es Ya? if (u, |u|) |= Ya?. It is a well known fact that the fragment TL[Xa, Ya] has the same expressiveness power as the variety DA. Therefore, it is natural to look at TL[Xar mod d, Yar mod d], where each predicate Xar mod d is de ned as follows. For a word u and one of its position x, we have (u, x) |= Xar mod d? if ? is true at the next a whose position is equal to r modulo d. As in Proposition 12 we can transfer a modular information from the predicates to the letters by changing the size of the alphabet. I Proposition 30. Let d be a non-zero integer. Then, TL[Xar mod d, Yar mod d](A?) = ?(TL[X(a,r mod d), Y(a,r mod d)](A?d) ? K). In [16], Schutzenberger de ned the monomials as the set of languages of the form B0?a1B1? ? ? ? anBn?, with ai ? A and Bi ? A. A monomial L is said to be unambiguous if for every word u in L, there exists only one decomposition u = u0a1u1 ? ? ? anun with ?(ui) ? Bi. Finally, Schutzenberger proved in [16] that a language is in DA if and only if it is a disjoint union of unambiguous monomials. We now give a similar de nition adapted to the modular predicates. We de ne the modular monomials as the languages of the form Proof. We know by Theorem 6 and Proposition 12 that a language L is in QDA(A?) if and only if there exists an integer d such that L is the projection of a set of well-formed words of a language L0 in DA(A?d). Then L0 is a disjoint union of unambiguous monomials. As the projection over well-formed words preserves disjoint union, it su ces to show that each unambiguous monomial projects into a disjoint union of modular monomials. Let B0?b1B1? ? ? ? bnBn? be an enriched unambiguous monomial with bi = (ai, ri). Then the projection of its well-formed words is the rational expression (A00 ? ? ? Ad?1)?A00 ? ? ? Ar01 a1(Ai1+1 ? ? ? Ai1)?Ai1+1 ? ? ? Ar12 a2 ? ? ? 0 with Aij = {a | (a, j) ? Bi}, which can be rewritten as a disjoint union of unambiguous modular monomials. J 7 Conclusion Our main results can now be summarized in a single statement, a consequence of Propositions 12, 30, 31 and Theorem 6. I Theorem 32. Let L be a regular language. Then, the following assertions are equivalent: L has its syntactic stamp in QDA, L is de nable in FO2[<, MOD], L is de nable in TL[Xar mod d, Yar mod d], L is a disjoint union of unambiguous modular monomials. Our results are an instance of a more general problem: given a fragment F of FO, what is the expressive power of F[<, MOD]. In particular, if F[<] has an algebraic characterization, is there also a natural algebraic description of F[<, MOD]? Further if F[<] is decidable, does it imply that F[<, MOD] is also decidable? These questions are related to non-trivial questions of semigroup theory [1]. There is some hope that, for some su ciently well-behaved fragment, F[<] corresponds to some variety of monoids V and that F[<, MOD] corresponds to the semidirect product V ? MOD where MOD denotes the variety of all stamps onto a cyclic group. This is the case for instance for the fragment ?1 and B?1, as shown in [4]. The decidability of V ? 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Luc Dartois, Charles Paperman. Two-variable first order logic with modular predicates over words, LIPICS - Leibniz International Proceedings in Informatics, 2013, 329-340, DOI: 10.4230/LIPIcs.STACS.2013.329