On sets of relations definable by addition

Abstract

For every κ ∈ ω, there is an infinite set Aκ ⊆ ω and a d(κ) ∈ ω such that for all Q0, Q1, ⊆ Aκ where ∣Q0∣ = ∣ Q1∣, or d(κ) < ∣Q0∣, Q1∣ < ℵ0, the structures ‹ω, +, Q0› and ‹ω, +, Q1› are indistinguishable by first-order sentences of quantifier depth κ whose atomic formulas are of the form u = v, u + v = w, and Q(u), where u, v, and w are variables.

Addition-Invariant FO and Regularity

“…The following is a generalization of a result by Lynch [12] which was proved in [18] and which establishes a winning strategy for the duplicator in an EF-game on particular structures which have a built-in addition relation.…”

“…In order to do so, we use the following generalization of a result by Lynch [12], which was proved in [18] and which allows us to reduce the existence of a winning strategy in a game with addition to the existence of a winning strategy in another game, where addition is not present. Proposition 3.5 (Immediate from [18]).…”

Randomisation and Derandomisation in Descriptive Complexity Theory

“…Theorem 4.7 (Lynch [26]). For every k ∈ AE there is an infinite set A k ⊆ AE and a d k ∈ AE such that for all finite…”

Counting and addition cannot express deterministic transitive closure

“…Most of them are related to techniques used by Presburger for the quantifier elimination of T h(Z) [23]; a very readable exposition of such a strategy can be found in section 3.3 of [12]. Lynch [21] also gave a similar strategy when proving inexpressibility results for first-order logic on structures with an addition relation.…”

“…First we consider the easier point move case. As mentioned in the introduction, strategies for this case can also be found in [12,21]. Then we will state the set move strategy which will be a non-trivial generalization of the point move.…”