Short Presburger arithmetic is hard

Forum of Mathematics, Sigma

Pak

2018

Self Cite

We give complexity analysis for the class of short generating functions. Assuming #P ⊆ FP/poly, we show that this class is not closed under taking many intersections, unions or projections of generating functions, in the sense that these operations can increase the bit length of coefficients of generating functions by a super-polynomial factor. We also prove that truncated theta functions are hard for this class.

Section: We Define Higher Classes σ Gmentioning

confidence: 94%

Complexity of Short Generating functions

Forum of Mathematics, Sigma

Pak

2018

Self Cite

“…Our strategy is to represent the set T as a union of arithmetic progressions (APs). In [NP17b], we already gave a method to define any union of APs by a short Presburger formula of polynomial size. For each 1 ≤ i ≤ d, let J i = {j : 0 ≤ j < 2 d , i ∈ S j }.…”

Section: Proofsmentioning

confidence: 99%

“…which is a union of d APs. Using the construction from [NP17b], we can define T ′ by a short Presburger formula:…”

Section: Proofsmentioning

confidence: 99%

“…, a k ] encode the starting and ending points of our d APs. We refer to Section 4 in [NP17b] for the details. Each term a k in that construction is at most the product of the largest terms in the d APs we want to encode.…”

Section: Proofsmentioning

confidence: 99%

“…The construction in Theorem 2 uses a number-theoretic technique that employs continued fractions to encode a union of many arithmetic progressions. This technique was explored earlier in [NP17b] to show that various decision problems with short Presburger sentences are intractable. In this construction we need to pick a prime roughly larger than 4 d , which can be done in probabilistic polynomial time in d. This can be modified to a deterministic algorithm with run-time polynomial in d, at the cost of increasing φ(F ):…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

VC-Dimensions of Short Presburger Formulas

Pak²

2019

Combinatorica

Self Cite

We study VC-dimension of short formulas in Presburger Arithmetic, defined to have a bounded number of variables, quantifiers and atoms. We give both lower and upper bounds, which are tight up to a polynomial factor in the bit length of the formula.1.2. NIP theories and bounds on VC-dimension/density. It is of interest to distinguish the first-order theories in which VC-dimension and VC-density behave nicely. Let L be a first-order language and M be an L-structure. Consider a partitioned L-formula

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Bogart

Goodrick

Mathematical Logic Qtrly

et al. 2019

Self Cite

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.