Ultrafilters on words for a fragment of logic

We give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let B be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving ultrafilter equations characterising B and then projecting these to obtain profinite equations characterising B∩Reg. This suffices to establish the decidability of the membership problem for B ∩Reg. In two earlier papers, Gehrke, Grigorieff, and Pin proved the following results: Result 1 [4] Any Boolean algebra of regular languages can be defined by a set of equations of the form u ↔ v, where u and v are profinite words. Result 2 [5] Any Boolean algebra of languages can be defined by a set of equations of the form u ↔ v, where u and v are ultrafilters on the set of words. These two results can be summarised by saying that Boolean algebras of languages can be defined by ultrafilter equations and by profinite equations in the regular case. When a Boolean algebra is closed under quotients, we use the notation u = v instead of u ↔ v, for a reason that will be fully explained in Section 1.2. Restricted instances of Result 1 have been obtained and applied very successfully long before the result was stated and proved in full generality. It is in particular a powerful tool for characterizing classes of regular languages or for determining the expressive power of various fragments of logic, see the book of Almeida [2] or the survey [9] for more information. Result 2 however is still awaiting convincing applications and even an idea of how to apply it in a concrete situation. The main problem in putting it into practice is to cope with ultrafilters, a difficulty nicely illustrated by Jan van Mill, who cooked up the nickname three headed monster for the set of ultrafilters on N. Facing this obstacle, the authors thought of using Results 1 and 2 simultaneously to obtain a new proof of the equality FO[N ] ∩ Reg = J (xy) = (xy) for x, y words of the same length K (1) ⋆ Work supported by the project ANR 2010 BLAN 0202 02 FREC. This formula gives the profinite equations characterizing the regular languages captured by FO[N ], the first order logic using arbitrary numerical predicates and the usual letter predicates. This result follows from the work of Barrington, Straubing and Thérien [3] and Straubing [10] and is strongly related to circuit complexity. Indeed its proof makes use of the equality between FO[N ] and AC, the class of languages accepted by unbounded fan-in, polynomial size, constantdepth Boolean circuits [11, Theorem IX.2.1, p. 161]. See also [7] for similar results and problems. However, before attacking this problem in earnest we have to tackle the following questions: how does one get hold of an ultrafilter equation given the non-constructibility of each one of them (save the trivial ones given by pairs of words)? In particular, how does one generalise the powerful use in the regular setting of x? And how does one project such ultrafilter equations to the regular fragment? In answering these questions and facing these challenges, we have chosen to consider a smaller and simpler logic fragment first. Our choice was dictated by two parameters: we wanted to be able to handle the corresponding ultrafilters and we wished to obtain a reasonably understandable list of profinite equations. Finally, we opted for FO[N0,N u 1 ], the restriction of FO[N ] to constant numerical predicates and to uniform unary numerical predicates. Here we obtain the following result (Theorem 5.16) FO[N0,N u 1 ] ∩ Reg = J(x s)(xt) = (xt)(xs), (xs) = (xs) for x, s, t words of the same length K (2) which shows in particular that membership in FO[N0,N u 1 ] is decidable for regular languages. Although this result is of interest in itself, we claim that our proof method is more important than the result. Indeed, this case study demonstrates for the first time the workability of the ultrafilter approach. This method can be summarised as follows. First we find a set of ultrafilter equations characterising FO[N0,N u 1 ] (Theorems 3.2, 3.3, and 4.7). Projecting these ultrafilter equations onto profinite words, we obtain profinite equations characterising FO[N0,N u 1 ] ∩ Reg (Theorem 5.2). Finally we show that the simpler class (2) generates the full family of projections of our ultrafilter equations to obtain Theorem 5.16. In the proceedings version of this paper [6], we had only proved the validity in B of the equations given in Section 3. Here we also prove their completeness in Section 4. As a consequence, we get a new completeness result for B ∩ Reg obtained by projection in Section 5.1. This leads to a new proof of decidability of membership in B ∩ Reg in Section 5.2. The completeness result expressed by equation (2) above is then obtained from the completeness result in Section 5.1 by rewriting in Section 5.3 . In [6], the completeness part of (2) was proved by traditional automata theoretic means.