Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1,…,tk. A formula in this language defines a parametric set Sboldt⊆double-struckZd as boldt varies in Zk, and we examine the counting function false|Sboldtfalse| as a function of t. For a single parameter, it is known that false|Stfalse| can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming P≠NP) we construct a parametric set St1,t2 such that false|St1,t2false| is not even polynomial‐time computable on input (t1,t2). In contrast, for parametric sets Sboldt⊆double-struckZd with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that false|Sboldtfalse| is always polynomial‐time computable in the size of t, and in fact can be represented using the gcd and similar functions.

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“…(v) On the other hand, if we allow no multiplication , even by parameters (cf. Example ), then

false| S_{boldt} false|

will be computable in polynomial time; in fact, it has a nice form as a piecewise‐defined quasi‐polynomial .…”

Section: Summary Of Complexity Resultsmentioning

confidence: 99%

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Bogart

Goodrick

Nguyen

et al. 2019

Mathematical Logic Qtrly

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“…is a Presburger counting function, which is a piecewise quasi-polynomial from [22,Theorem 1.10]. Hence #S n agrees with a piecewise quasi-polynomial for all sufficiently large n. Remark 6.1.…”

Section: Asymptotic Behaviour Of Symbolic Multi-rees Algebras Of Mono...mentioning

confidence: 95%

“…For the next result we need to introduce some notations which we borrow from [22]. A map σ : N r → Q is said to be periodic if there exists an α ∈ N such that…”

Section: Asymptotic Behaviour Of Symbolic Multi-rees Algebras Of Mono...mentioning

confidence: 99%

Asymptotic growth of symbolic multi-Rees algebras

Das¹,

Roy²

2021

Preprint

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We study the question of finite generation of symbolic multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the symbolic multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated length function is eventually a polynomial. We further prove that the symbolic multi-Rees algebra of monomial ideals is finitely generated with no assumptions on the analytic spread. In this case, the corresponding length function is shown to exhibit piecewise quasi-polynomial behaviour.Proposition 4.1. [13, Lemma 3.4] There exists a vector h ∈ N r such that Ann R M n = Ann R M h for all n ≥ h. In particular, dim M n = dim M h for all n ≥ h.

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“…Versions of these properties always hold for sets defined in Presburger arithmetic. For example, Woods [19] gave several properties of Presburger formulas that hold even for sets defined with multivariate parameters, t ∈ N n : Theorem 1.25 (from Theorems 1 and 2 of Woods [19]). Suppose F (s, t) is a Presburger formula, with s and t collections of free variables.…”

Section: We Can Compute the Generating Functionmentioning

confidence: 99%

The Unreasonable Ubiquitousness of Quasi-Polynomials

Woods

2014

Electron. J. Combin.

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A function g, with domain the natural numbers, is a quasi-polynomial if there exists a period m and polynomials p 0 , p 1 , . . . , p m−1 such that g(t) = p i (t) for t ≡ i mod m. Quasi-polynomials classically -and "reasonably" -appear in Ehrhart theory and in other contexts where one examines a family of polyhedra, parametrized by a variable t, and defined by linear inequalities of the form a 1 x 1 +· · ·+a d x d b(t).Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a quasi-polynomial structure in several problems where the a i are also allowed to vary with t. We discuss these "unreasonable" results and conjecture a general class of sets that exhibit various (eventual) quasi-polynomial behaviors: sets S t ⊆ N d that are defined with quantifiers (∀, ∃), boolean operations (and, or, not), and statements of the form a 1 (t)x 1 + · · · + a d (t)x d b(t), where a i (t) and b(t) are polynomials in t. These sets are a generalization of sets defined in the Presburger arithmetic. We prove several relationships between our conjectures, and we prove several special cases of the conjectures. The title is a play on Eugene Wigner's "The unreasonable effectiveness of mathematics in the natural sciences".

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Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

Cited by 14 publications

References 42 publications

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Asymptotic growth of symbolic multi-Rees algebras

The Unreasonable Ubiquitousness of Quasi-Polynomials

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