Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

Abstract

Parametric Presburger arithmetic concerns families of sets S t ⊆ Z d , for t ∈ N, that are defined using addition, inequalities, constants in Z, Boolean operations, multiplication by t, and quantifiers on variables ranging over Z. That is, such families are defined using quantifiers and Boolean combinations of formulas of the formRecent results of Chen, Li, Sam; Calegari, Walker; Roune, Woods; and Shen concern specific families in parametric Presburger arithmetic that exhibit quasi-polynomial behavior. For example, |S t | might be a quasi-polynomial function of t or an element x(t) ∈ S t might be specifiable as a function with quasi-polynomial coordinates, for sufficiently large t. Woods conjectured that all parametric Presburger sets exhibit this quasi-polynomial behavior. Here, we prove this conjecture, using various tools from logic and combinatorics.

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

“…In , the parameter t takes values in instead of . However, one can see that the same proofs and conclusions also hold when t ranges over .…”

“…Example 1.5 is a family where |S t | is an EQP. Theorem 1.6 (Bogart,Goodrick,& Woods;[3]) Let {S t : t ∈ Z} be a 1-parametric PA family. There exists an EQP g : Z → N such that, if S t has finite cardinality, then g(t) = |S t |.…”

“…There are several other forms of quasi-polynomial behavior that 1-parametric PA families exhibit (such as possessing EQP Skolem functions; cf. [3]). Here we focus on the cardinality, |S t |.…”

“…We are seeing many types of “nice” functions in these examples, and the question is now how to generalize. In fact, Example generalizes to any family in 1‐parametic Presburger arithmetic , as described in the next section.…”

“…where ψ(x, t) is a Boolean combination of equations s 1 (x, t) = s 2 (x, t) and divisibility relations s3 (t)|s 1 (x, t), where s 1 (x, t),s 2 (x, t), and s 3 (t) are L 2 -terms, i.e., expressions built up using only the operations in L 2 and the displayed parameters and variables. P r o o f o f T h e o r e m 4.1 Say ϕ t (x) defines a k-parametric unordered PA family in Z d .…”