From Ultrafilters on Words to the Expressive Power of 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 profinite equations characterizing B ∩Reg via ultrafilter equations satisfied by B. 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 [5] 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 [6] 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 summarized by saying that Boolean algebras of languages can be defined by ultrafilter equations and by profinite equations in the regular case. Restricted instances of Result 1 have proved to be very successful long before the result was stated 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. 3 In [5], these were denoted by u ↔ v. 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 generalize 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 4.7) 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 summarized as follows. First we find a set of ultrafilter equations satisfied by FO[N0,N u 1 ] (Theorem 3.2). These equations do not necessarily suffice to characterize FO[N0,N u 1 ] , but projecting ultrafilters onto profinite words, we convert our ultrafilter equations to profinite equations for FO[N0,N u 1 ] ∩ Reg (Theorems 3.3 and 3.4). The last step consists in verifying that the set of profinite equations thus obtained suffices to characterize FO[N0,N u 1 ] ∩ Reg (Theorem 4.7). Now, a closer look at our proof shows that we are far from making use of the potential power of ultrafilters. For instance, difficult combinatorial results like Szemeredi’s theorem on arithmetic progressions can be formulated in terms of ultrafilters. Thus it is quite possible that more sophisticated arguments are required to extend our results to larger fragments of logic, including FO[N ]. 4 We recently proved that these equations actually do suffice to characterize FO[N0,N u 1 ], but this will be the topic of another paper.