### Arithmetic, first-order logic, and counting quantifiers

Nicole Schweikardt
2005 ACM Transactions on Computational Logic
This paper gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers. Precisely, this means that for every first-order formula ϕ(y, z) over the signature {<, +} there is a first-order formula ψ(x, z) which expresses over the structure N, <, + (respectively, over initial segments of this structure) that the variable x is interpreted
more » ... ctly by the number of possible interpretations of the variable y for which the formula ϕ(y, z) is satisfied. Applying this theorem, we obtain an easy proof of Ruhl's result that reachability (and similarly, connectivity) in finite graphs is not expressible in first-order logic with unary counting quantifiers and addition. Furthermore, the above result on Presburger arithmetic helps to show the failure of a particular version of the Crane Beach conjecture. · 3 its initial segments first-order logic is indeed as expressive as FOunC. As applications of this result we obtain the failure of a particular version of the so-called Crane Beach conjecture, and we obtain an easy proof of Ruhl's result [Ruhl 1999 ] that reachability in finite graphs is not expressible in FOunC(+) and, similarly, that connectivity of finite graphs is not definable in FOunC(+). Via communication with Leonid Libkin the author learned that the result on Presburger arithmetic was independently discovered, but not yet published, by H. J. Keisler. Let us mention some more papers that deal with unary counting quantifiers and with FO(+), respectively: Benedikt and Keisler [Benedikt and Keisler 1997] investigated several different kinds of unary counting quantifiers. Implicitly, they show that, under certain presumptions, such unary counting quantifiers can be eliminated (cf., Lemma 19 in the appendix of [Benedikt and Keisler 1997] ). However, their result does not deal with Presburger arithmetic and its initial segments, and their proofs are non-elementary, using hyperfinite structures. Pugh [Pugh 1994 ] deals with Presburger arithmetic Z, <, + and counting quantifiers from a different point of view. He presents a way of how a symbolic math package such as Maple or Mathematica may compute symbolic sums of the form {p( y, z) : y ∈ Z and Z, <, + |= ϕ( y, z)}, where p is a polynomial in the variables y, z and ϕ is a FO(<, +)-formula. The FOk-aryC-formulas considered in the present paper correspond to the simplest such sums in which the polynomial p is the constant 1. For other related work that deals with first-order logic, counting, and/or arithmetic from different points of view see, e.g., the articles [Lee 2003; Llima 1998; Krynicki and Zdanowski 2003; Mostowski 2001] and the references therein.